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Monographs in Modern Logic



Review : This is a clear and straightforward summary of the history of formal logic from the time of Aristotle to that of Nidditch discusses the four main trends at the root of modern logic : Aristotle's theory of the syllogism; the idea of a universal language ; the idea of the parts of mathematics forming deductive systems ; and the discoveries in mathematics in the early nineteenth century. He goes on to outline the chief ideas and theories of the main writers on mathematical logic, including Jevons, Peirce, Boole, Russell, and Whitehead. The text is easy to read and gives the beginning student a valuable perspective on mathematical logic.

Back Cover : Monographs in Modern Logic. Edited by Geoffrey Keene.
Modern Logic, like mathematics, is a highly-specialised subject with an extensive technical literature. but unlike mathematics it is a relative newcomer to the academic scene. While it s true that the initial stages of its develoment took place several hundred years ago, it made its bebut only as recetnly as the beginning of the present century. Since then it has progressed so rapidly that few textbooks on the subject can hope to be comprehensive. A compromise between breadth of coverage and depth of treatment seems inevitable. The present series is designed to meet this problem. Each monograph devtes separate attetion to a particular branch of mdern logic, the level of treatment being intermediate between the elementary and the advanced. In this way a wider coverage of htis subject will, it is hoped, be made more acessible to a larger number of readers.
Routledge & Kegan Paul - - - Dover Publications


1.1. Purpose of the book.   The purpose of the present book is to give such an account of Mathematical Logic as will make clear in the framework of its history some of the chief directions of its ideas and teachings. It is these directions of the mass of detail forming the theory and its history which are important to the rest of philosophy and are important, in addition, from the point of view of a general education. In the limits of space we are able to give attention only to what has been, or seems as if it may be fertile or of special value in some other way. More than this, Mathematical Logic having no small number of important developments, a selection of material is necessary, the selection being guided by the rule to take up the simpler questions, other things being roughly equal. Facts have to be looked at in the light of ones' purpose. Though they may hall have the same value simply as facts, they are no at all equal as judged b the profit and the pleasure that thought, and not least the thought of the learner, is able to get from them.

1.2. Language of book.   The writing of all this discussion of Mathematical Logic is in Ogden's Basic English. One is forced when keeping to the apparatus of this form of the English language (which is a body of only 850 root words -- not taking into account 51 international words, and names of numbers, weights and measures, of sciences and of other branches of learning, and up to 50 words for any field of science or learning, all of which words may be used in addition to the 850 -- and, in comparison with those of normal English, a very small number of rules limiting greatly the uses of the given words and the structures of statements that may be formed from them), to take more care than one commonly does to make the dark and complex thoughts that are at the back of one's mind as clear and simple as possible. We will be attempting in what is to come to get across to the reader the substance of the story of Mathematical Logic. In this attempt Basic English will certainly be of some help to some, possibly even to almost all, readers, and will certainly not make things harder for any.
    All the special words of logic which are not among the 850 words of Basic English are put in sloping print when they are first made use of here and their senses are made clear by Basic English.
    However, though we have done our best, by using Basic English, to keep the account as clear and simple as possible, it will not be surprising if there are bits of it which are hard fully to get a grip of in a first reading ; this is specially true when the newer developments are being quickly talked about (9.1 to the end of the book). The reason for this is that the tendency of the growth of all sciences is towards the more and more complex and towards an ever-increasing number of high-level ideas in their organization. Certainly the growth of Mathematical Logic has been like this. But our hope is to have said nothing that will not be completely clear after some thought -- thought will sometimes be needed . No earlier knowledge of logic will in any degree be necssary and only such a knowledge of high school mathematics as the reader stil has in the mists of his memory.
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2.1 . Mathematical Logic produced by four lines of thought.   Mathematical Logic is the outcome of the joining together of four different lines of thought. These are the old logic, the invention of Aristotle ; the idea of a complete and automatic language for reasoning ; the new developments in algebra and geometry which took place after 1825 ; and the idea of the parts of mathematics as being systems of deductions, that is of chains of reasoning in agree ment with rules of logic, these rules giving one the power to go from a statement s1 to another statement s2 when s2 is necessarily true if s1 is taken to be true. We will say something about these four in turn.

2.2 . Aristotle's work on logic.   Aristotle (384 - 322 B.C.) first, then. There are at least five sides to Aristotle's writings on logic. In these writings are : discussions of common language, chiefly in relation to the different sorts of words and their connection with the possible orders of existence (substance, quality, place, time and so on) ; a body of suggestions on the art of argument -- the art of causing the destruction of the arguments of those who are not in agreement with oneself and of stopping one's arguments from being open to attacks ; a group of teacings on the way of science, on how an increase of knowledge of physical laws may come about by the work of the natural sciences ; a number of views on the right organization os system in the science of mathematics ; and a theory of that form of certain reasoning whch was named by Aristotle syllogisitic reasoning. It is this last theory and its later offspring that one normally has in mind when talking of "the old logic", ' the common logic' or ' the logic of Aristotle', and it is this side of his writings on logic that was important for the start and early growth of Mathematical Logic ; so in what we now say about Aristotle we will be limiting ourselves to his Syllogistic.

2.3 . Reasoning, implication and validity.   Complex statements such as ' if all animals are in need of food and all men are animals, then all men are in need of food' and ' if the sides joining any three points in a plane make a right-angle at one of the points, then the measure of the square on the side opposite the right-angle is equal to the measure got by the addition of the measures of the squares on the other two sides' are the sort of statements put forward is examples of reasoning that is certain : reasoning which is to the effect that a given statement is certainly true if a number of other given statements offered as conditions for it are true. There is no suggestion that all reasoning put forward as ceretain is in fact certain ; it s common knowledge that errors of reasoning are frequently made.
    Now, more generally, let s1, s2, . . ., sn be any n statements. In the logic of certain, as against for example probable reasoning, a statement of the form ' if s1 and s2 and . . . and sn-1, then necessarily sn ' is said to be an implication. ('Implication' by itself, without ' an' or ' the, is used as a name for the ' if-then necessarly' relation.) s1 to sn-1 are the conditions and sn is the outcome of the implication. An implication is said to have validity if only and only if it is necessary for the outcome to be true when all the conditions are true. The reader is not to let the idea of an implication be mixed iwth the idea of an argument. An argument is a complex statement of the form ' s1 is true and s2 is true and . . . and sn-1 is true, so necessrily sn is true ' which has validity if and only if the parallel implication has validity. An argument is different from an implication in so far as the conditions on which the outcome is dependent are judged to be true and because of this the outcome is put forward as being true ; in an implication, on the other hand, the outcome is put forward as being true if the conditions are true, but these are not judged to be in fact true, or false. ('Argument' is used, in addition, for any chain of reasoning ; this might be an implication or it might be an argument in the narrower sense.)

2.4 . General statements.   In Aristotle's logic only four sorts of statement may be used as conditions or outcomes of implications. They are general statements, in a special sense of ' general ', which have the structure ' all S is P ', ' no S is P ', ' some S is P ' or ' some S is not P ' ; later, those of the first sort were said to be A, those of the second sort E, those of the third sort I and those of the fourth sort O statements, while for all of them S was named the subject and P the predicate. The way of Aristotle himself in talking about these four forms of statement ws somewhat different from that which was made common by the Schoolmen. For him ' P is truly said of all S ' or ' P is a part of whatever is S ' took the place of ' all S is P '. ' P is truly said of no S ' or ' P is a part of nothing that is S ' took the place of ' no S is P ' and so on.
. . . In Aristotle's opinion the only name which make sense in a general statement are gerneral names, for example ' man'' and ' flower' and ' green', not names of persons or plaes or of other things that might be viewed as if they were units of being. In harmony with this opionion Aristotle did not let statements wuhas Socrates is going to be dead some day; and ' Callias has a turned-up nose' -- examples he frequently makes us of in is wwritings on philosophy -- be conditions or out-comes of implications.
   It is only right to say in addition that Aristoltle gave modal general statements and syllogistic reasonings made of these an important place in his chief work on Syllogistic, the Prior Analystics ; such modal statements and ones about the necessary or the possible, for example ' allS is necessarily P ', ' no S is necessaryily P ' and ' some S is possibly P '. Aristotle's modal logic had little effect on later thought and no effect, it seems, on the birth of Mathematical Logic. We will give no more attention to it.

2.5 . Aristotle's Syllogistic.   Aristotle's Syllogistic is a theory of syllogistic implications. A syllogistic implication is an implication with two and only two conditions and one outcome, the conditions and the outcome being general statements. What Aristotle made the attempt to do in his Syllogistic was to give a complete account of the different possible detailed forms of syllogistic implicatons and a complete body of rules as tests of the validity of any give syllogistic implication. Being a first attempt, it is not surprising that Aristotle's theory is not free from error and is not the best possible one. But, without any doubt, he made a solid start. It is sad that for hundreds of years after his ideas on Syllogistic had got a wide distribuion among those
with an interest in the theory of reasoning, almost no one was able to let himself say outright in what ways Aristotle's answers were right or wrong and, much more important, say that the questions they were designed as answers to were only some of the questions needing to be answered. It is true that some specially among the later Schoolmen did make important discoveries in logic and that their work here was not limited to Syllogistic. However, their logic was one of rules whose statement was in everyday language (Latin) ; no special signs for the operations of reasoning were used and the Schoolmen seem to have had no idea that it was possible for logic to be turned into a sort of mathematics. So they took no step forward in the direction of turning logic into Matematical Logic. It would certainly have been hard for them to take seriously the idea of logic as a sort of mathematics because their view of logic and mathematics kept these quite separate. Logic (ars logica) was one of the three Arts of Language forming the the trivium (' the three-branched way') which was the field of learning that one had to go through first when at the university ; the other two branches of the trivium were the art of using language rightly (ars grammatics and the art of using language well (ars rhetorica). The teaching of the triviu was done on the Arts side of the universities. On the other hand, the business of mathematics was not seen as being with language at all and its teaching was given on the Science side of the universities.
    Reasons for the over-great respect for Aristotle's logic were at first his system's being the only one of which one had records making possible detailed knowledge, together with the authority of the Church of Rome which had taken up, with some adjustments, much of Aristotle's teaching in philosophy ; and later the crushing effect of higher eduction whlch was controlled by those who had given their years to the learning of the languages, history and writings of the Greeks and Romans and so naturally put the highest value on such knowledge, a value much higher than they were ready to give to new sciences and to new, natural knowledge.

    An example of a syllogistic implication having validity is ' if all animal is going to die and all man is animal, then necessarily all man is going to die'. This implication is of the form ' if P is truly said of all M and M is truly said of all S, then necessarily P is truly said of all S'. (The reason for saying 'man' and 'animal' and not 'men' and 'animals' will be made clear later, in 6.2.)

2.6 . The use of variables by Aristotle.   It was in such forms of syllogistic implications hat Aristotle was interested and rightly for Syllogistic because the business of this is with the laws of syllogistic implications, and if these laws are to be general, it is necessary for them to be put forward in relations to the structures and not in relation to material examples ot the implications. One has to give Aristotle great credit for being fully conscious of this and for seeing that the way to general laws is by the use of variables, that is letters which are signs for every and any thing whatever in a certain range of things : a range of qualities, substances, relations, numbers or of any other sort or form of existence . The S, P and M higher up are examples of the variables used in Syllogistic. In Aristotle's theory the range of such variables is, at the level of language, the range of all possible general names and, parallel to this, he variables are, at the natural level, the representatives of any qualities and of any sort of substance, like star and metal and backboned animal.
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    If one keeps in mind that the Greeks were very uncertain about 'and very far from letting variables take the place of numbers or number words in algebra, which was why they made little headway in that branch of mathematics though they were good at geometry and so why the invention of a geometry based on algebra was not made till number variables had become common and were no longer strange, in the time of Fermat and Descartes, then there will be less danger of Aristotle's invention of variables for use in Syllogistic being overlooked or undervalued. Because of this idea of his, logic was sent off from the very start on the right lines. But not before Boole, 2200 years later, who saw in the old Syllogistic the seeds of an algebra of logic, was the important step taken of changing Syllogistic into a mathematics of certain reasoning. Only when logic was married to mathematics did it become fertile. On its becoming the mother of a good number of ideas and teachings of profit, logic's position as the poor relation of philosophy came to an end, the change going so far as to make the new logic widely looked at and respected as the head of the family. In fact, it has been the desire of some persons to see the rest of the family dead. This is true of the supporters of the Vienna Circle that was flowering in the 1920's and 1930's : it was their belief that logic, in the wider sense that takes into account, in addition to Mathematical Logic, the logic of chance and the logic of the natural sciences -- logics which have again, like Mathematical Logic, deep connections with mathematics -- is the one part of philosophy that is of serious interest and value, because it is the only one in which true knowledge is possible ; in the other parts man's thought is wasted on systems of fictions caused by the tricks of language and on talk which, though put forward as offering a key to the secrets of all that there is, is without good sense, by reason of its being framed in such a way that its powers of opening the locks of existence are unable to he publicly tested by experience.
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2.7. Aristotle's errors.   What are the points of Aristotle's Syllogistic were he has gone wrong or where one might do better ? These points may be put into two groups, one group having those which come inside Syllogistic itself and the other having those points which are outside Syllogistic.
    Some points in the first group are these. In the old logic only two general statements as conditions and, in reasonings having validity, not more than three general names may be used in implications. But there is no good reason for so limiting implications while there is a good one for ot so doing, this being that the business of logic is with every sort of implication having validity and there are implications having validity which have more than two general statements as conditions and make use of more than three general names ; an example is ' if all animal with wings is backboned, no self-moving thing is rooted in the earth and all that is backboned is self-moving, then necessarily no animal with wings is rooted in the earth'. Again, Aristotle's logic is not a well worked out system like the systems of mathematics ; in these the different statements are or are made to be dependent on one another : one takes a selection of some of them as starting-points from which the rest have to be got by deductions. Though, it is true, Aristotle's Syllogistic is different only in degree from for example Euclid's system of geometry, still the degree between them is enough for it not to be unkind to say that Aristotle's theory is nearer to a simple grouping than to an ordering and organization of his material. In addition, its use of signs other than words
not being taken far, the old logic was not an algebra and so it was not much help in the art of quickly getting the right answers to questions about reasoning and of making fully clear that the answers were right because of the processes of deduction by which they were produced, processes based on a body of rules fixed at the start.
    There are two chief points in the second group about which we will say something, again without going into details. Aristotle and his school of logic made no attempt to give an account of the theory of implications having structures whose validity is dependent on the force of words like ' not' , ' and', ' or' and ' only if ' which are used for forming more complex statements from given ones ; an example of such a structure is ' if s1 is false but s1 is true or s2 is true, then necessarily s2 is true', and another example is ' if s1 only if s2 but s2 is false, then necessarily s1 is false'. This branch of logic may be named the logic of statement connections. It was first given attention by the Megarians and Stoics, working quite separately from Aristotle ; see 2.8. Secondly, the old logic did not take into account implications such as ' if A is greater than B and F is greater than C, then necessarily A is greater than C', ' if A is B, then necessarily A of X is B of X'. If song is music, then necessarily the song of a bird is the music of a bird') and if A is B, then necessarily X of A is X of B' (if Christ is the Highest Being, then necessarily the mother of Christ is the mother of the Highest Being', an example given by Leibniz), which are dependent on words or ideas for relations between things. The expansion of the old logic so that at least some par of the logic of relations might be covered was first undertaken by De Morgan in the 1840's ; in his view the logic of relations was more important because wider than Syllogistic, this being nothing but a small part of that logic.
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2.8 The Megarian and Stoic logic of statement connections.   We will here say something about the logic of the Megarians and Stoics ; that logic was first formed at about the same time as Aristotle's, and ir will be best if this present history of the development of logic keeps as far as possible to the order in time in which the different schools came into existence.
    The Megarian work on logic was done between, roughly, 400 and 275 and the Stoic between, roughly, 300 and 200 before Christ. There were two chief ways in which the Megarian-Stoic theory was different from Aristotle's Firstly, it was interested in forms of reasoning having the structure of an argument and not the structure of an implication. Secondly and more importantly, it as a logic of statement and connections and not one of general and suchlike statements.
    The sorts of reasoning to which the Megarians and Stoics gave attention were these; ' if s1 then s2, and s1 ; so s2', ' if s1 then s2, and not s2 ; so not s1' ; 'not the two of s1 and s2, and s1 ; so not s2' ; ' s1 or s2, and s1 ; so not ' s2' ; and ' s1 or s2, and not s1 ; so s2'.
    Some of these and other forms have an ' if-then' statement as one of the bases of the argument. Because one may rightly put forward as true what comes at the end of an argument only when all the statements on which it is based are true, it is necessary for an answer to be given to the question : when is an ' if-then' statement true ? This question was much talked about in the Meganian-Stoic school, so much so that someone (Callimachus) said : 'Even the birds on the rooftops are having discussions on the question.' A number of different answers to it were offered ; we will give three of them.
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Philo said that all ' if-then' statements are true but for those which have a true ' if ' part and a false ' then' part, these being false, so ' if 2+2=5, then it is day' is true at all times, and when it is daylight ' if it is night, then it is day' is true while ' if it is day, then it is night' is false. Philo's sort of ' if-then' relation has been that normally used in Mathematical Logic ; it has been named by Russell 'material implication'. Again, Diodorus said that an ' if-then' statement is true if and only if it is one which at no time has a true ' if ' part and a false ' then' part ; so ' if it is night, then it is day' is not a true statement even if made in daylight because their are times -- at night -- when ' it is night' is true and ' it is day' is false. Because, in Diodorus's opinion, an ' if-then' statement is true if and only if it is at all times a true material implication, his sort of ' if-then' relation may be named ' all-time material implication'. And thirdly, there were some who had the belief that it is right to put forward an ' if-then' statement as true only when there is some connection between what its no parts are about; they said that ' if s1 then s2' may be judged to be true when and only when s2, is necessarily true if s1 is true, or, putting it another way, when it is not possible that s1 is true and s2 is false. This sort of ' if-then' relation is what was named simply 'implication' in 2.3.
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3.1 . Lull's 'Ars Magna'.   The first step in the direction of a complete and automatic language for reasoning was taken by Ramon Lull (1235 - 1315) in about 1270, in his book Ars Magna ('The Great Art'). Lull's belief was that all knowledge in the science is a joinng together of a umber of root ideas : Knowledge is a complex of simples. . There were only 54 of these root ideas, about a third of them having tg do with the field of religion or of theories about right and wrong behaviour. The joining together of groups of these ideas is 't he great art ' by which scientia generalis (' the substance of science') is to be effected. Lull did not get much further than working out the number of ways in which complexes of these ideas might be formed. He gave no rules, or only foolish ones, for judging the value as knowledge of the different possible complexes. It seems to have been his opinion that no knowledge in the sciences has any need of sense-experience as a guide and support, as if the discovery and the testing of the discovery of what is under the sea might be made without stepping from the land : the fishing-boats of knowledge may be kept safely and with profit in harbour and do not have to be sent sailing with their nets over the deep waters of possible experience. In having this opinion, Lull was representative of much of the thought of his time. - 14 -
. . .

The following has not been edited !!!
Sections 5.1 to 6.2 may be okay. 6.3 and 6.4 are still bad.
Sections 6.5 and 6.6 are good.

3.2 . Dalgarno's and Wilkin's Languages   After Lull suggestions for a general language for a general science were not uncommon in philosophy. But the development of such a language was not made till the 1660's when Ars Signorum (' The Art of Signs') of George Dalgarno (1626-1687) and the Essay towards a Real Character and a Philosophical Langauge of John Wilkins (1614-1672) and were printed. The ideas at the ase of Wilkins's Essay were, to some degree, copied from Dalgarno. However. it is not by chance or without good reason that Wilkins's book has been better kept in memory than Dalgarno's. The working out in detail of the ideas for a new consciously made language is more valued than an outline of them and the Essay, the last and separate part of which (by William Lloyd) gives a full list of the English words of the day and their senses in Wilkins's language, is on a much greater scale than Dalgarno's. Things were not so hard for Wilkins, who at one time or another was a Head of Colleges at Oxford and Cambridge -- the only man ever to have been so at the two places -- and who had a high position in the Church of England ; he had the backing of the newly formed Royal Society, of which he was one of the Secretaries, and his book was printed by the order and the necessary money came out of the pocket of the Society. Dalgarno, on the other hand, was a private school teacher.
    Wilkins's purpose was, he says, to give a regular list and account of all those things and ideas on which marks or names might be put in agreement with their natural properties, is being the end and design of the different branches of science that all things and ideas be placed in such a frame as may make clear their natureal order, their ways of being dependent on one another and their relations. (It is easy to see here the - 15 - effect of Bacon's teaching on the roads and limits to knowledge in the sciences : science is the discovery and ordering of natural groups of facts, not the building of delicate theories of them.) Wilkins goes on to say that an art of natural rules for forming statements and questions is needed from which one will get ' such helps and instrument s' as are necessary for the framing of the simpler ideas into smooth and unbroken talk or writing. The smaller and simpler this body of natural rules is, the better. Wilkins had three points in mind : the rules of his language were to be 'natural" ; the 57817 :un > A n .-..P' s . 4_ .!9.l?9. o n 41 I I l | I 4 : 8 .. gSSgfvwvulpg n 4- .-,--q'1'1 Wi ~ ... | l l ..t .:. ._ of which they I 's * *rr* 5 QI these . *"* n L 1. . H . , so that the names would in some way be in agreement with the things they were signs of. In addition, Wilkins, like Dalgarno before him, had the opinion that a system of shorthand writing, in which the shorthand outlines are dependent only on the sounds of the words, was needed for a specially designed language such as his and no small part of his book is given to the development of this side of his language,
    The great trouble Wilkins took to make his language a living instrument of science (and international trade and religion) was completely wasted-but for its helping to keep before men's minds the thought that it is possible for a language better than the everyday ones to he formed by conscious art. The offspring of Wilkins's brain was dead at birth. One reason for this was that - 16 - because of discoveries in science and the changes in common knowledge caused by these it came to be seen that his accounts of things in relation to 40 chief headings and under these headings an attempted grouping of all other things by divisions of higher and lower levels, in this way giving an account of everything by what it has in common with certain other things and how it is different from them, as in taxonomy-were false or without support, and, equally serious, ir was seen that the new ideas and sorts of facts that were the outcome of the new knowledge were not covered by Wilkins at all, while they put in the shade much of what was covered, for example about the Highest Being and other such ideas that were a necessary part of the beliefs of a man having authority in the Church.. . .+ ; A a =i,,* why W|.lkin.'s,sg11||1:|;nn,,,l:nd. no "'; "T "" .U 11. .m. . . ' 1.ii ..| \ . 1 1 natural blow . dge was very much o 'ii d by him. On condition that a language is ab to be used for making as sharp and as detailed recordings of observations as is desired and is able to undergo expansion, letting new ranges of fact and of thought be talked about without unnecessary trouble being caused by the language apparatus, then it is not generally important for the natural sciences if the Language is not as good as possible in other ways, for example in having forms that are not completely regular. On the other hand, languages, such as those of mathematics, having short and simple signs and for which there are rules for building statements which put forward two groups of numbers or of other things as being equal or being unequal and for going from some given statements of this sort to other ones, by processes of deduction, do have great powers of increasing and ordering knowledge. But this knowledge is of possible - 17- relations between things, not of facts themselves. Wilkins did not see that if a designed language was to be of use in the way he was picturing to himself, it would have to be a sort of arithmetic or algebra. However, he was not expert enough in mathematics to be able to make the necessary developments for this new purpose .

3.3 . Descartes' Idea of a General Language   It seems that Descartes (1596-1650) was the first person to have the idea of a general language -- ;une langue universelle' are his words rot it -- as a sort of arithmetic. In s a letter to Mersenne of 20 November 1629 he said that the invention of a language is possible in which ; an order is formed between all the thoughts which ..Ig . .1.. ;... . .. . 5 . . .. . m; ... . . . . .. 1. ... . . . w . ". H . .. . ` sltessrhfsa ur - . .. Il .. . * 8 "o I sl .--1 e . . eng, are na . ur 0 4 . . . .. jggptmone tlzon one daY...||.. dl these . 4 ;.;g v . . . o . r . . .y . . . . . "i1q,hdgg . . . m i . . . . 1 fog take about all the other things coming in our minds. . . .The invention of this language is dependent on the true philosophy, but for this it is not possible for all the thoughts of men to be listed and put in order or even to be made separate from one another so that they become clear and simple, which is in my opinion the great secret of getting well-based knowledge. And if someone was to give a good account of which are the simple ideas that are in men's minds of which all their thoughts are made up . . . there would he almost no chance of going wrong' No doubt wisely Descartes made no attempt to give a list of all our simple thoughts and to put them in order so that an arithmetic of reasoning might be formed that would let one get complete and certain knowledge of whatever is true. - 18 -

3.4 . Leibniz's Idea of a Mathematics of Thought   A little later Leibniz (1646-1716) had designs for a new and general language that were not unlike Descartes', but he took them some stages further. From as early as his De Arte Combinatqria ('On the Art of Complex Forms') , printed in 1666, Leibniz made suggestions fot. a m .L1 I atics of ideas . . . . J ; . . . -. a. ~= . ~- | ...4."}*.. ....~.......g;=1&. :4 1 1 ran F . .2.,*31 5, T, I . " . . .... I . u.. . 8 . i . . 4. . 1 . | . . diiniaunsg . . .. l . . : . r0|4ibJ1=@w~iatb .. . . e a . . . . |g|4| he .in that . . J . \.. x|3,.3.5=-5;-c7) all . . T 1d4mh4H8hIIid1HuIG. .. . . 0 p . . . . . . . . . . . .axe ybc nned byp1|3.:t.ii:gtogetl1er aomeof thesilmghn!p. . . u . . . . .-. e simple .nletedhy numbers in " A7 .. . . 3898s . . . u 4; 4439 'S=,`?',iI,13,...Whi1tlieuncip&::r . . .. . ..- . . . 1 1.1 *1 . . . =~ . . in 41' ml# 6 . . . 5 I 4. . . J 1 - EM. is r3x5), 21 (_-32 * u . . I . . . Ti . ~4+.r . H. . . . . lc: idea being iependsrnnx on te *..L.. I 1 . . . hi - " 5; . i| . | . . .. *...: . . . . I . .I ... | . si an ABB.a.|lun:Iu4a . 8 . . . . . 4 r . I v 9 . g v .. . . .. J . . 1. _mwu\gn=hihmlawm& l 2 . Q l r.. . ,aminatmment for the discovery land * 24G...
    Leibniz was ruled by a belief in mathematics as the *.scanned.toe~..v... . . . . #I 5 l nv: l 9; are able 9 undergo 1 . . f* "* . I .&=:t.d49no~|m.s11 s@stn~=sls=a= . . . . . .4 . 4 .=| . . ntrd "m ao far as such decisions are by - 19 - reasoning from the given facts. Because though some *i expenences are ever needed as a base for reasoning, when these experiences have been given we would get from them everything that any person would ever be able to get from them, and we would even make the discovery of what more experience is necessary if our minds are to become free from the rest of our doubts. . . . If we had a body of signs that were right for the purpose of our talking about all our ideas as clearly and in as true and as detailed a way as numbers are talked about in Arithmetic or lines are talked about in the Geometry of Analysis, we would be able to do for every question, in so far as it is under the control of reasoning, that one is able to do in Arithmetic and . pendant ' and 'I of signs and by a sort of Algebra; an effect me 0 this is that the discovery of facts of great interest and attraction would become quite straightforward. It would not be necessary for our heads to be broken in bard work as much as they are now and we would certainly be able to get all the knowledge possible from the material given. In addition, we would have everyone in agreement about whatever would have been worked out , because it would be simple to have the working gone into by doing it again or by attempting tests like that of "putting out nines" in Arithmetic. And if anyone had doubts of one of my statements I would say to him : ict us do the question by using numbers, and in this way, taking pen and ink, we would quickly co me to an answer.' - 20 -

3.5 . Leibniz's Errors   The idea of what we have been naming a complete and automatic language for reasoning is made very clear by these words of Leibniz. But though he frequently gave attention to the idea of such - 20 - a language he did little with it. He was at the same distance from having a new language system at the end as he had been in 1666. His design got no further than being designs -- and rough ones. Leibniz went wrong in two ways. Firstly, he was unwise not to have msfr much .%Fef tI~fe be desired inention. There is no hope of covering from the very start every part of what might be reasoned about ; the building of any such apparatus as was pictured by Leibniz, as of any theory in science, comes only with a slow process of growth, not jumping suddenly into a condition of full development at birth like Athena out of the head of Zeus. It is necessary to go one step at a time, starting with what is simplest; this is the best way of getting knowledge that is of value in itself and of value as a base for making wider the range of facts or thoughts taken into account. Leibniz would have done much better by limiting himself to an attempted forming of a language for no more than one or two branches of the mathematics of his day. Desiring to have a system that was a guide to everything, he was unable to put together a system that was a guide to anything. At the back of this desire was the belief, by which Descartes earlier had been gripped, that to a person like himself who had made such new and important discoveries in mathematics the invention of an instrument of logic having the power to give the answers, without error and with- out waste, to all that is a question for man's reason, was a work of little trouble and of little time. This belief was one side of the thought then current: in living memory a great increase of knowledge had taken place and the mind had become much more free and open, had become forward-looking and full of hope, sharply conscious of its powers. - 21 -
    Secondly, Leibniz went wrong by desireing to have 4 J I. I an . I '.~~ i .., \ . 1: v =1r\lln41e a : 1 :Wt ~ m<>1d.,=iru1s4r:.:sese the .. ~. " ~~a 4 ~tem in wbish #li '=1\!f=~U1l11* be . xit:?Wdu1d 'Be or ei-EH-a" system of . "~M"i>1=@==~w1>hs=. The reward would have been greater if he had made the decision to keep to the logic, which is so much more straightforward and which would have given him the chance of turning the metal of the old logic into the gold of a new and better one, for example one whose laws, unlike those of Syllogistic, are enough and have the right properties so that the validity of arguments in at least some of the theories of mathematics may be tested and judged. Or, if Leibniz had the feeling that the material in addition to the form of reasoning had to be a part of his language system, he might have done well -- much better than he did -- by letting himself be interested in the bricks of only mathematics, having as his purpose an organization of mathematics based on its simplest ideas and starting- points. A road to the full development of such an organization was not seen till about 200 years later : by Frege, Peano, Whitehead and Russell, in the years between 1875 and 1910.
    It would not be right to get the idea that Leibniz's work was of no value at all because his controlling thought of a new mathematics-like language came to nothing. Having that thought, it was natural that his mind was transported in the direction of logic itself and there he made a number of important discoveries some of them in Syllogistic but most of them in fields of 1logic that were new or had been given little attention befpre. However, these discoveries were not taken seriously till much later because the separate bits were
- 22 -
not united in a general theory and so, not being complete enough, they were not put in print or offered to the public in Leibniz's time or even till long after; by then they were common knowledge, not as his teachings but as the teachings of theories worked out in the years after 1840, theories whose start and development were in no way dependent on Leibniz's work on logic.


4.1 . Peacock's Algebra   Between 1825 and 1900 algebra and geometry underwent great changes, changes producing by 1900 a completely different general outlook in the philosophy of mathematics from that which had been common before. The changed forms and purposes of algebra and geometry had a strong effect on all the parts and at every stage in the growth of Mathematical Logic.
    An equation is a statement that two things, normally two groups of number. or of signs representative of numbers, are equal. Till 1825 or a little after that, .=.=.=.~.:w... 4s I o I ll >... . ;. .....,..;4 *.#&'4l81%;*9l.9x.7<:. -q ,re Q , . . ..4- ... .7."= "*~ . " .. ...|n 1: .. . -A | x ~i;"""l *%~8'iiW .,. Fda learning about in high school, for example 3x-2y = 11 - 23 - and ax'+bx+c =0. The business of the theory was to get a knowledge of how such equations may be worked out to give number values which make them true and M get a knowledge of the conditions controlling the existence of and relations between the number values. The four operations of addition and so on were dOne, as they are still done by schoolboys, more or less un- consciously by making whatever moves seemed right and natural, the rules supporting these moves resting in the dark. There was no thought that a statement of the rules was necessary or might be a help in the development of algebra. It is strange that though geometry had early in its history been turned into a structure of fixed starting-points and of reasonings from these, the position of algebra, like arithmetic, was dilferent. The need of forming a table of and of being consciously guided by the laws of algebra which say what are the properties of the operations in algebra was seen only by Peacock (1791-1858). In his book A Treatise on Algebra (l830; second writing, in two parts, 1842-1845) the idea was put forward that algebra is, rightly viewed, a science of deductions like geometry. (At about the same time -1837- the first account of Newton's mechanics as a science of deductions, copying Euclid's Elements, was given by Whewell, a friend of Peacock, in his Mechanical Euclid.) Peacock had two chief points which are to be noted here. Firstly, all the processes of algebra have to l.;e...based _on acomolete stat nt t h e b o d of 13y-whiQ138e_about the I operations used in those processes, no pro erty of an operation_b__ejg,g,_3;_eg.l.j t e     Examples of the laws .g1 9 i * . _of 419 4 is | af . t l. . .. - . 44 . 14 j ' . . Qi L ..; I ti i ? n outcomis || . " I i . . . 48 ` . ... gg.8 7 " . . .. . u 4 U 11 r + vanabfes is changed: a+b . . .1" . . `u8.4 1"* ; . ! F . . t " *d9 ii . . . . rmreiagg ., 1 n| ! .: jat _in outcome is l g . 4 . tile g. ;.;. H. .. 4 . 4. . 4 3; IF e . w .* 4$ > * se . . !r.;. . . * *..wa . v . y *|i4--*--- w5. rQ"1 * ii . a . . . . .. . . . . t r H* . . . . .r 4: . .3r I1* .. . . . 1. """"3* l r sae I 110 I 1. n I av ` r 5=ax c. P -25-

4.2 . Equations of the Fifth Degree   Another event causing the movement in the direction of Abstract Algebra and away from the limits of the old theory of equations was the surprising discovery br Abel and others n numlbtt . ..w''8llll%istA ..9..H.! ."=:is. ..t..s8?!"e3s...4: . 1 . . . .. . : I 0 .91 : . m. . ..9r?" we * _8 1 . .. 5..$ 8* . . . F; |1 . .\. |" | be y can Nzetie- H prooenews - 25 - specially the taking of square and higher roots-which make possible the automatic getting of number values for equations of degree 1, 2, 3 or 4. The belief had become fixed that answers to questions about the number values for equations of higher degree than the fourth would be (and anyhow that it would not be possible to make certain that they may not be) got in the same simple way as for those equations of no higher than the fourth degree. Abel's discovery was seriously damaging to the belief, having a wide distribution among workers in science and philosophy, that at least the heart of future knowledge will be in agreement with the expert knowledge of their time. This belief had to take blow after blow, not only in mathematics but in the natural sciences, through the years after this discovery in algebra, some of the blows coming from Mathematical Logic.
    Galois, in attempting t ; |;.|. 1 ' ' ` to Nwmwagntai _ I I F =;&4; -.. : nm5 a u *";2.Zi;3v ~;. I ... &. > < 5 4 o I ri"*!~ . . . m. >. .. su : . u -~ . .. 4: * 8"1'l1|* / si. . . + . ..p=x e " 1 . ku quickly into being as the first new branch of Abstract Algebra. In this way the changes in algebra taking place in England came together in harmony with those taking place in other countries of Europe.

4.3 . Vector and Matrix Algebra   To those who were pained by the discoveries of Abel and Galois, worse was to come. They were given more, violent shocks by the systems of Vector Algebra and Matrix Algebra worked out not long after the death of Galois in 1812. These systems were company for Group Theory as new branches of Abstract Algebra. - 26 - ' I U i 4 . ... . * *lux MIL jr . 41.s . ur . n 1 . . . . .. ... ..\. - " . " " ...n4 ;i ..3.. 5W *&iv .. ... ,.; 4;:5 4|lt=\r r . . * 11. .. . . . .. . - I . . r'==. 1 ....z....e..-..r.*w:~,~4 4 : . .4 q 9. I . - 1 ( .an ri n . mu .4 r . . ; ... .. . . . " " uw. . . . . 9 "...%...... ..._.).;... ..m I 6 . . i IL l n . . .. . .x 4*1'W'*t...|4|-Hin0. .4 I gr - P in a p 1 e . i .. .*.. 1 . . W ER. 8.. -. L *vw IM. tt: 3 . w1.-= *W* Lyw.8, H=m1=m4 .. 1 . *I l 1 + i 2 I | . . * the J* . . . . . . . ..; . M gg.. . . 4 *is s.: li3I .4I4h.l|9.| I tally .+ tu . . ... .. '*= " . .... . . . . . 89%... . 8 .. .. :rams- . J* r ways like l Ol .._ . .5 . . .=,@.. .: . 1 . a. . n . . 4 | 1. . * 5 I 8 %. Ple i um . . . M8 E- 061 1 . ,1ggh|4um 18 that ax c)-=(axh}+(axc). (Signs for direction-properties and for operations on them are normally put in blacker print.)
    The 1850's saw I . * . 1 ..i . t 85 dna,; 4 il . k-. .. . Q; &1.*2 = 1 | | a. $...i . I. 0 . . . 088.831 .'>. . 9 .9 . . . ;" 4.3 . 4 | . " . . 0. . . . . . . . . 1 1 . . as, a . . .. ... . _ *.. .i . . . .. . I . r .r . . . . . . . s . 4. i pa - .. '. . . ". I .E . an .. . Algebra M lhwi wr 8"|3BxA is not generally true. To take a simple 3 example: let A=(l 2) and B=( ) ;then by the sense 4 given to :=< in connection with number tables AxB - (1 =(1.3+2.4]=ll1), while on the other hand (3) .. 6 BxA (1 2)=(3 I his |,"h;ttl1ue. _ (8) 4 8) U .. . . . 2 . . . s . . . . . . -a. ... ... . \. .. .S|{..1. ...M |e-:1=1e..|: . *4 l. * .... .8. T 4 . JE'.j .4.. .. .." .. . a e .. ss .v . . ... 2. . _ " v . . . ;L. .: ; .l I rifihw hnmdmummnmMxnMHmumm4d9 - 27 - . 1 . F g,. ...- 4 .4 . ni t. . . . r . . 8 9. * . :. . .. 1 " ' *gag " 1 ..` 4 = . ". H. - " |4 . i =M=ti11s pount 8 ,rr V74 Yu I hr | any * > . .. .* .r~ . r.. * " " . . .I . 1: L ..* . . 5 +e!? | . .. *984. . vi ". . 9 : . .. . " . x 4 . : T r - an-pnI=@in1||\un:d33i * .- 4 f . .. . . .. . 4 .. . In ; . J 9. . 4. . . . . . . . 81lM!\lih2n1cQnsd0W H P ra - 4 a. *rcllsar what mm 3h*-3ilF;lgg * : . . . . . . ..4.= ... . a.. used in the - ..c .. ;1i&rnnt. Belds . . 4 . . 8 . a.. . 1 M||gt9|=Q8 f . . "aa . . 41 .. . 1Im. 2 .. ;2 \ . . 2 . e mane i. M*r i | | Qn, . . This 1138586 .in algebra, whose full . > ==a~r. . . . "Q": * .. . ... . . . . ). vwna Qigxeat. wish*. . 4 . v in the development of almost every other part of mathematics. To give only one example of the way in w c Mathematical Logic was touched by it: because of the new current of ideas Schroeder and Whitehead in the 1890's were caused to give an account of the algebra of logic in the form of a science of deductions using to no small degree the signs and processes of algebra and of its latest directions in Abstract Algebra. These workers, in their turn, were copied by others, and it became the normal thing in Mathematical Logic to take note of the latest theories in algebra, by which it is offered new food for thought that keeps it a living branch of philosophy and science.
- 28 -

4.4 . Geometries of Euclid and Lobachevsky   1 as Geometries of Euclid and Lobachevsky. Fo6 . . * . . .. I ,s\=rss=1h4~*e** . . . . U r... . . .A I I ng; oin8~ Thi Ent in! . . . . ik := . 4 time .. . . . . .* . * .. . . ua be true without the support of reamonmg. y 's doing this, geometry was changed and kept from being nothing but a mass of separate statements and
- 28 - was turned into a united body of knowledge, all the parts of which were dependent and in connection with one another. Because of Euclid's work geometry was_ looked at down the years as the most complete branch of mathematics and its deduction form was looked at as offering the best way to true knowledge and, outside religon, the highest knowledge.
    --~ -~'~t was thc invention by.||...~.,:...... it | | e ml. J 4 33 \ |=|.la*q ~"s...~gp..n..34 a~...~ ' .. 4.. 'L' I 1 . glggqg8 4. ...4.*...~w.,_._.~~;..~1.~ 1 =~>=~1~ 1 4-~ ` e -. s More than one l I | ' 1 parallel to the given line. Other ways in which they are different are outcomes of this; for example the law of Pythagoras is true and may be got by deduction in Euclid's but is false and may not be got by deduction in Lobachevsky's geometry. The fact that one was able to have a system other than Euclid's came as a deep shock, producing in a quite short time a view of what mathematics is about, what its purpose is and what its relations to physical space and things in that space are which was completely unlike the normal, old view in the philosophy of mathematics. When it was seen that one may have two geometries where what is true in one is false in the other, the old opinions such as Plato's, the Schoolmen's, Spinoza's, Leibniz's and Kant's by which mathematics is a science giving completely certain knowledge of existence -- natural or higher-than- natural, and if not of things-in-themselves, then of things as they seem to be -- were not able to be put forward with reason on their side.
    The effect of the changed outlook on mathematics and specially geometry was to make theories in mathematics viewed not as in themselves natural knowledge
- 29 - sw ~. .r2:w:ij. iriir .,.4;s' -be -ia.|--4-4--4 a hj hlneauU|unineh I . aiiml iinteflaqugahelpinilnmmmilacieneu. 1 . llldnemnmes I due nnennpm to get lllolutniunnbersandspa1eesandsucl:|like\bings,tonuake inmueatixuug dileovelies about their properties and rehtium, and to put the supported by these dis- ouveries into systems of reasoning. Mathematics is the hee invention of the mind, limited only by possible uses in natural science that may be desired and by rules of logic that one is willing or is forced, as a person of good sense, to be guided by.
    The third turning-point in the history of geometry is best talked about in connection with what comes under the fourth and last of the headings given in 2.1, so we will now go on to that.
- 30 -
Sections 5.1 to 6.2 may be okay. 6.3 and 6.4 are bad. 6.5 and 6.6 are good.


5.1 . Mathematics and Systems of Deductions : Consistency   The idea of the parts of mathematics as systems of deductions goes back to Euclid's work in geometry, geometry being the first part of mathematics that was made into a system of this sort. What a system Of deductions is has been outlined clearly enough to the reader for us not to have to go into more detail
- 30 - about it at this stage, though more will be said later. What we will have to say something about here is another side or level of a system of deductions, the idea of which has been very importance for Mathematical Logic.
    Some attention in relation to geometry had been given before the 1890's to the question : Is it possible to get the end of one chain of deduction in a system a statement S, which becomes by this fact a theorem -- a true statement -- of the system and to get at the end of another such chain the theorem not-s which says that s is after all, false ? A system of deductions is said to have the property of consistency if and only if it does not have two opposite theorems s and not-s. The amount of attention that had been given to the question of consistency had been small because the question had not been taken very seriously, the belief being general in mathematics that any expert was able quickly to see the answer to the question of a system's consistency without his having or needing a supporting argument. But in the 1890's the question was viewed as interesting and important by that Italian school of mathematics that was headed by Peano. From the point of view of logic there is no doubt the question is an important one because any statement whatever abut the field under discussion becomes a theorem in a system that does not have the property of consistency, the statements being true ones or being false ones about the field, so htat what is true would not be able to be kept separate from what is false, while the very purpose of a system is to have true nad only true statements about the field as theorems. Because the question of consistency is important, how the answer is made and supported is important. Peano's opinion was that reasoning and clear public tests are the right instruments for making decisions about consistency. (See end of 9.4.) - 31 -
    He had the same opinion about another question that might be put about a system of deductions. Giving the name axiom to a statement that is taken to be true, without the support of reasoning, at the base of such a system, this question is : Are an of the axioms of the system dependent on the others in the sense that it is possible to get an axiom as a theorem at the end of a deduction in which the axiom it self is not used?
    Some thought had been given to this question and the question of consistency from quite early times in connection with attempts to get Euclid's axiom of parallels as a step in a deduction not resting on that axiom, these attempts being made to let one see that that axiom is dependent on the other axioms : which it is not. However, only in the 1890's were these questions regularly put, and put about all systems of deductions in mathematics, and were ways designed for answering them by arguments. Peano and his school were chiefly. responsible for this, though from late in the 1890's the work, in the same direction, that was done by Hilbert in Germany had a greater effect on later developments.

5.2. Metamathematics. From this work came a new branch of Mathematical Logic : metamathematics. The business of metamathematics is with what may be truly said about some system ( or group of systems) of deductions, for example that it has the property of consistency or that no axiom is dependent on the others. If S is a system whose field under discussion is F, then the purpose of S is the building of a framework of supportable theorems about F, the theorems saying what true relations there are between the things in F. If S is an arithmetic, then a theorem in S such as ' 7 + 5= -12 ' is - 32 - about numbers. On the other hand, ' "'7 + 5 = 12 " is a theorem in S; ' is not about numbers ; it is a statement about a statement about numbers. ' 7 + 5 = 12 ' is a theorem in S ; ' "7 + 5 = 12 " is a theorem in S' is not a theorem in S but a theorem in the metamathematics S' of S. It is necessary for the reader to see what this delicate point is that makes S and S' different from one another. Every system S has its metamathematics S', sometimes named its ' logic'\ ', which is formed of the true statement, and the argueements for them, that may be made about about S, while S itself is formed of the true statements and the arguments for them be made about F ; for example, if SF is an arithmetic of the common so , then F is the body of all the numbers such as 34, 1/2, -6/79. S is normally a part of mathematics, though systems of deductions outside mathematics are possible ; S' is a part of logic: the logic of that part S of mathematics. The birth of Mathematical Logic was in the 1840's. Most of its more important and more interesting developments after its first 50 years of growth have been at the level of metamathematics. - 33 -


6.1 . Boole and the older Logics of Aristotle and Hamilton - De Morgan   Now that the four currents of thought named in 2.1 have been talked about, we may have a look at that stretch of Mathematical Logic which their motion and driving-force have so far produced ;
as one might say, the look will be given from the footway at the side of the river and not from the point of view of one having a swim underwater, because clearly we are not able to go deeply into Mathematical Logic here.
    What was the start of Mathematical Logic ? The shortest and simplest answer is George Boole's Mathematical Analysis of Logic (full name: The Mathematical Analysis of Logic, being an essay towards a calculus of of deductive reasoning) There is nothing completely new under the sun. Every birth is the outcome of earlier events. So Boole's little book of 82 pages was only marking a stage in an unbroken line of thought from the past. However, though ir certainly had connections with what some others had done, it was still different enough from the rest for ir rightly to be seen as starting a quite new theory -- the theory of Mathematical Logic -- and not as being another step in an old one .
    The earlier teachings in logic of which Boole had a knowledge and which had an effect on him were, on the one hand, those of Sir William Hamilton (1788-1856) and De Morgan (1806-1871) on the theory that was based on the changing of the four, A, E, I, and O forms into a greater number of forms in which the amount of the predicate is given ; for example, Hamilton has two A forms, one being ' all S is all P ' and the other being ' all S is some P '. For later developments two points were important in the theory of Hamilton and De Morgan (the two of whom had a bitter fight, Hamilton protesting that his ideas had been taken from him without his being given credit for them, and De Morgan answering that his account was not at all dependent on anything that Hamilton had ever said). One point - 34 - was the weight placed by it on the amounts of predicates in statements the old logic noting only the amounts of subjects. The value of this was that every statement of the subject-predicate form was able to be turned into an equation or into a statement that an equation was false. By placing this weight on the idea of an equation, logic was moved near to algebra. Secondly, what had been named by Aristotle the two ' ends ', S and P of a statement had generally been viewed as signs. of qualities. In the new theory of Hamilton and De Morgan S and P were changed into being signs of the things themselves that have the qualities. An example of an A statement in the old logic is ' all leaf is green ', this having the sense that the quality of green is a 'part' of the quality of leaf. From the point of view of the new theory this statement had the sense of saying that whatever thing is a leaf has the property of being green: 'all leaves are green things'; or, putting it another way, 'the group of all the things that are leaves is a "part" of all the things that are green'. Putting the change shortly, it was a change from 'all S is P' to 'all S's are P's'. Though the reading of 'all S is P' as 'all S's are P's' had sometimes taken place before Hamilton and De Morgan (for example in Whately's Logak), the reasons for so doing, when there were any conscious reasons, were not reasons of logic. And the greater minds, such as Leibniz's, had kept to the S and P form, not making use of the S's and P's form. It was because they kept to the S and P form that they were stopped from building a better logic. The better logic they had hopes of building was to be a logic of mathematics, even if for more than of mathematics, and it was to be one based on mathematics. However, reasoning in mathematics is generally not about qualities as such but about the things that have the qualities. - 35 - For example, geometry is not a theory of the qualities ' being a point ', ' being a line ' and ' being a plane ', it is a theory of the things that are points, lines and planes .
    The name used in logic and mathematics for a group of all the things that have a certain simple or complex property is class and the things that have the propertyr are said to be the elements of the class. The ideas of class and class elements are root ideas in all present day mathematics. The outcome of the Hamilton-De Morgan theory was to make possible a view of logic as being, at least in one of its branches, an algebra of classes. Boole was the first man to have this view clearly.

6.2 . Boole the Man   Boole (1815-1864) was, till he went across the sea to Ireland as the first Professor of Mathematics at Queen's College, Cork, when this was started in 1849, an English private schoolteacher, whose schooling had come to an end when he was only a boy and who had at no time any sort of higher education. His father was in trade on a very small scale, as a shoe-maker, and is family had little money. The older Boole was a man of regular behaviour, living with care, and having a love of learning and a deep interest in the sciences. These qualities were handed down to his son, who got from his father, in addition, a knowledge of the simpler parts of mathematics. George Boole beaume a teacher for his living when he was only 16 years old. By teaching himself from books in his free time hom teaching others, he quickly became a person of wide learning, specially expert in mathematics. When he was 24 he was writing papers on new mathematics for the Cambridge Mathematical Journal and five years later the Royal Society gave him a special mark of reward for the very first paper he sent it. He is a good example of what might be done by self-help.

6.3 . Boole's 'Mathematical Analysis of Logic'   Boole's first of four books was The Mathematical Analysis of Logic, whose writing took only some weeks in the spring of 1847, when he was 31. This is a work offering a logic bsed on the mathematics, chiefly algebra ; - w* . i2:1 . .. L;; .;,.== ~:g4.;~ macs 9 ; 4 1 . ** ,gi ; . 4 ..aa 4 1 @"H@@ 4. a" - r . 4 . . . ... *B " l . . : . ..4.$ M .. . j *Q I ... .. .. . .. n* s . f. **'** i=~ .. l ... I * alof; . dhNheuscni " - . . l 4 . . . .. . p . . . 4 ;===. . wnmnf . I . . . ..." an . T 1 M -. * . ? 4 a u he. P111 in thn imn eg . .. . . w .. 4 d . 'Har " . fi==u@pk_**_. . . 4. h.. 4 by GT! - m." 1. " . . .. . . :~ . . m . . "9 .q .. " . . g, : . w . ,max lie gof . .. .. &amr1 'no S's ar P's'; how s 1. . .AV " . ` lnnrpluwlilnnrl . :> . .. may be tested by . . * | . . . D l . . 5. . 2 '9=====&. x. 18481184889 ,.1|,g,,.," ~,. @9emnf 8. .. } .. . . .. . . H equations and seeing if't&c" .. Qmes\Inl|3g-:9,ua&i1n . . .. ..... ..:. W ... . pt 1311 . . Ilf|9Q3l . . 1 .9. . . . ...... . .. . . . s !k .I . ~*. . iwd *hw . nnenunganm I . .. l .I 4. 9 1 4 butnanuma *i . . ll,..i& 4 a.;:: 9344 44 outcome of them from their equations -- if they have any necessary outcome.
    But, and this was a great step forward, Boole gave in addition an account of the logic of statement connections (2.7) as an algebra ; unlike Leibniz, Hamilton and De Morgan, he saw that this part of logic was important, aland where others had been completely at a loss he was able to give a good theory of it. It is interesting that his theory was the very same in form as that of the algebra of classes. Boole was the first man to give a united theory of logic.
    We have said that Boole 's algebra of statement connections is the very same in form as his algebra of classes. This is true of the body of the book. But after this had been printed an addition of some pages to make clear a number of points in it was put in, and on the very last page Boole made a note of the fact that there is one way in which the two parts of logic are different. The algebra of statement connections has all the laws of algebra of classes and it has one more law in addition.
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6.4 . Class and statement readings of Boole's Algebra   At the start of his opening discussion in The Mathematical Analysis of Logic Boole says that those who are in touch with the present condition of ' theory of Symbolical Algebra ' are conscious of the fact .1 . .*. . . n ... . . . . .. :.. U .:.= . .. .. 1 .n 1 ..I. e l . . .. . + 4 9.. kL v lg .Le =* "'.; w i :.. . " . . . ; . 8 * cI ... 1. I b W4-~=== an *vr* vs. . ... . 3. . .." .. i : Hr " W ?1:.s=s . : ;1. .. I .. I I ~ i l g . br t . . 1"- J $i * iv . .. I I . 1 . J"I 4 11118 . - . .@'4c*ls=8s4 1 . \ . . . u . I ... . . In as ang; other 4 . v 4 * ,become one statements 1 . . 1. il*" . 7"! . 114 . with . . v;!1i1= * . t . 1 " . . . . . : a weis? . Fw emu1s.P$9._4H> -- ... a .. tr . 4 .. ." . w I I System ~. . 448148 . ,her . a..4a 1... A. . I 10 a qucstilm. on . \ . .... .. pro .. . .." I ~.** .e - - so ro 2* . a . gym tib .. .. . . and under a third to that of a question of dynamics or optics. .
    Boole's algebra of logic is one theory; there are (at least) two systems of readings of it, one in connection with material about classes, the other in connection 4 with material about statements. For example, in the l * v 4 the 2 * 11; i . " -. L *. ... . wisest . 'I . i . .1ii.1{ i M . 4 . . .......Q_IW 1 .. .. 55 i sa Ur .La f 1 . - i.tun d aliout,th8 .:. 8 . * $ : ... - ... \ . . . T |"* . | . 1 . .l ..4 . . . Of . .- froln MI; . . . . being the class having nothing which is an element of - 38 - as ag clenmllla5.-l-y il 'i . : . .. one e]|emenl8 aie . n. ; . . * " - | i$ 4W A Q ;I 4 l e. e su":. " saw-is .A s. .... .. .. . gg ' \ . 4 . .. . i..4 . * .. . m . . u N 4 ts . . 1 . . . .. ? " I 41 1 "l""*"* \. g * i s y'8 . I * . . . "?* utnat c 1 * 8 " *u " . ; l* 9 . . . . . . . ; . . E . .I . wgoae e M . 4 93 4 :4 =h=4m1> PJ .. v J. ~ 4. I . g.. I 8? . * .;.}- a;||, . e t ul o *.. J, and .if 5 .. .. 34,4 1HMi!l | . . 4 ...*.y..".in.. .. .3 under 4, . . " . . . . . . nl .. . d' ... sign o is. The-Eil&4B4|Ehl=&|Bl|I%g& Q 4 . .". .. . 1 . ,wmnh . iS . H44iHm=t@if4!lFIW a . 1 4 onJr-48.91119., butnoQ~thwo of them . . . a I *$a?lMl'*&ui#x "' . 1 *
    For Boole a statement 'x' is separate form the statement. ' "x" is true ' in which 'x' is put forward as being true ; the second statement is separate from the first if only because the first might be put forward as being false and not as being true. In Boole's algebras of statement connections the statement ' "x" is true ' becomes the equation x = 1, while the statement ' "x" is false ' becomes the equation x = 0. Because every x is true or false, a law of this algebra is: x = 0 or x = 1. This law is specific to the algebra of statement connections ( see end of 6.4).
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6.5 . Boole's 'Laws of Thought'   Boole's ideas about the algebra of logic were only outlined in The Mathematical Analysis of Logic. His complete statement of them was given in 1854 in his 424-page book The Laws of Thought (full name : An Investigation of the Laws of Thought, on which are founded the mathematical theories of logic and probabilities). (The other two of Boole's four books were not on logic but on higher branches of mathematics.) The first half of The Laws of Thought is on the algebra of logic, While the second half is on the uses of this algebra in the theory of probable reasoning and in the mathematics of chance. Most of the teachings in the Brat half are roughly the same as, though often given more fully than, those in his earlier book ; together with them this time is a detailed account of the writer's opinions on their connections with and value for mathematics and philosophy. On a small number of quite important points, however, Boole's thought had Undergone a change, sometimes for the better, sometimes not. Two such changes may be noted here. One is about the operation of division in his algebra ; the other is about his views on the relations to time of a statements being true or being false.
    In The Laws of Thought Boole went very much further than he had done before in attempting to make use of the operations and processes of mathematics in building his theory of logic. It was his desire to let any idea of arithmetic or number algebra be used in his algebra of logic if it was a help to getting the right answers, even though some of the statements got from some of the processes of mathematics in the middle of the working out of a question in logic had no sense from the point of view of logic itself. Boole's belief was that his way was completely all right on condition that the last line of the working -- the line giving the answer -- did have sense for logic, An effect of this belief was that some of the rules of his algebra were guided by ideas in mathematics more than by the needs of a good theory of logic. No small number of steps in working out questions by his algebra are based on an operation of division where what is got by this has no sense for logic; and a specially strong protest is rightly to be made against his frequent use of signs, taken from what he givu the name of Arithmetic to (but which these days - 40 - would be named Mathematical Analysis), like 0/0, 0/1 and1/0, whose senses and uses even in Arithmetic were then not very clear or free from error.
    On the other point, about time, Boole says : let us make use of the letters X, Y and Z as marks of the simple statements on which we have the desire to put some value in relation to what is true and false, or among which we have the desire to put forward some relation in the form of a complex statement (such as ' X is true and Y is false ' or ' if X is true then Y is true ']. And let us make use of the small letters x, y and z in this way: let x be representative of an act of the mind by which one's looking at that stretch of time in which X is true is fixed; and let this be the sense to be given to the putting forward of x as the name of the time in which X is true. Let us further, Boole says, make use of the signs of connection +, -, = and × in this way: let x+y be the name of the group of those stretches of time in which X is true or Y is true, those times being completely separate from one another, let x - y be the name of the rest of the time got when one takes away from the stretch of time in which X is true the stretch of time, which is a part of that first stretch, in which Y is true; let x = y be the name of the statement that the time in which X is true is the very same as the time in which Y is true ; and, lastly, let x × y be the name of the stretch of time in which the statements X and 2" are true together. It is readily seen that such laws of number algebra as x + y =y + x , x × y = y × x , and x × (y + z)=(x × y) + (x × z) are true in this algebra of time : it is into an algebra of time that the algebra of statement connections has been turned by Boole, that is to say for the purpose of an account of the logic of statements he gave a time system of readings to his algebra of logic.
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6.6 . Boole's Algebra and Number Algebra   It is natural to put the questions : Are all the laws of Boole's general algebra of logic laws of the general algebra of numbers? and Are all the laws of the general algebra of numbers laws of Boole's general algebra of logic ? The answer to the first question is, No. There is a law of Boole's algebra which is not a law of number algebra. This law is x × x = x : in the algebra of classes it is true at the class of things common to x and to x is simply the class x itself and in the algebra of statement connections looked at from the point of view of times it is true that the stretch of time in which X is true and X is true is simply the time in which X is true, while in number algebra it is not true that 2 × 2 =2 and, more generally, if x is any number greater than 1, it is not true that x × x = x. And the answer to the second question is the same as to the first, No. The law of number algebra that if x × y = 0, then x = 0 or y = 0 (and so the law, which is true if and only if that law is true, that when z is not 0, then x=y if z × x=z × y) is not true in Boole's algebra of logic, for example, under the class system of readings, if All is the class of living persons, x is the class of mothers and y is the class of fathers, then x × y is equal to Nothing but x is not equal to Nothing and y is not equal to Nothing: though x × y = 0, x not= 0 and y not= 0.
    One of the chief points made by Boole in The Laws of Though is that though his algebra of logic is not the same as the general algebra of numbers, it is the same from end to end as a more limited algebra, this being the algebra of the two numbers 0 and 1. The algebra of 0 and 1 is formed of the laws of algebra which are true when the variables used are representatives of only 0 and l (so, in fact, all the laws of high school algebra are laws of this algebra). Any one of the things of which a variable is a representative is said to be a 'value' of the - 42 - variable; a variable in the algebra of 0 and 1 has only two possible values, 0 and 1. A necessary condition of Boole's algebra being the same in form as this algebra is that the somewhat strange law x × x=x which is true in the algebra of logic is true in the algebra of the numbers 0 and 1. The question is, then : Is the equation x × x=x true for the two possible values of x, that is, is it true that 0 × 0 = 0 and that 1 × 1 = 1? The answer is, Yes. But facing one necessary condition without a fall is not enough. To get full marks in a test the first question that is answered has to be answered rightly, however, being able to give the right answer to this part of the test paper is not all that is needed because, in addition, the rest of one's answers have to be right. Boole is in a like position, he has more troubles to overcome if his idea is going to get through our test without any loss. Let us now put another question: Is there a law of the algebra of 0 and I which is not a law of the algebra of logic? If there is such a law, Boole made an error in saying that his algebra was the same as that limited number algebra. There is such a law : when z not= 0, x = y if z × x = z × y. In the algebra of 0 and 1, when z not= 0, z = 1, and if 1 × x = 1 × y, then certainly x = y. In the algebra of logic, on the other hand, the implication ' if z not= 0 and z × x = z × y, then necessarily x = y ' does not have validity. The root reason why it does not have validity is that in the algebra of logic, when z is not 0, z is not necessarily 1. This is clear straight away from the class system of readings of Boole's algebra. Under this system of readings not every class is All or Nothing ; there is a place for classes which have some elements of All without having all the elements of All as their elements. However, though it is not true that the general algebra of logic is the same as the algebra of 0 and 1, it is true that the algebra of statement connections is the same as that number algebra. The reader has seen that in the algebra of statement connections, for every statement x, x = 0 or x = 1: as in the algebra of 0 and 1, every variable has only two possible values, 0 and 1.

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