The Development of Mathematical Logic

P. H. Nidditch

• Publisher: Routledge
• Year: 2019
• Print ISBN: 9781032808727
• eBook ISBN: 9781000737080
• Edition: 1
• Topic: Philosophy
• Subtopic: Business General

Originally published in 1962. A clear and simple account of the growth and structure of Mathematical Logic, no earlier knowledge of logic being required. After outlining the four lines of thought that have been its roots - the logic of Aristotle, the idea of all the parts of mathematics as systems to be designed on the same sort of plan as that used by Euclid and his Elements, and the discoveries in algebra and geometry in 1800-1860 - the book goes on to give some of the main ideas and theories of the chief writers on Mathematical Logic: De Morgan, Boole, Jevons, Pierce, Frege, Peano, Whitehead, Russell, Post, Hilbert and Goebel. Written to assist readers who require a general picture of current logic, it will also be a guide for those who will later be going more deeply into the expert details of this field.

1

PURPOSE AND LANGUAGE OF THE BOOK

1.1. Purpose of 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, not the mass of detail forming the theory and its history, which are important for the rest of philosophy and are important, in addition, from the point of view of a general education. In the limits of our 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 one’s purpose. Though they may all have the same value simply as facts, they are not at all equal as judged by 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 necessary and only such a knowledge of high school mathematics as the reader still has in the mists of his memory.

2

ARISTOTLE’S SYLLOGISTIC

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 agreement 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 s₁is 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 one- self and of stopping one’s arguments from being open to like attacks; a group of teachings 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 of a system in the science of mathematics; and a theory of that form of certain reasoning which was named by Aristotle syllogistic 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 as 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 certain is in fact certain; it is common knowledge that errors of reasoning are frequently made.

Now, more generally, let s₁ s₂…, s_n 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 s_n-1, then necessarily s_n’ is said to be an implication. (‘Implication’ by itself, without ‘an’ or ‘the’, is used as a name for the ‘if-then necessarily’ relation.) s₁ to s_n-1 are the conditions and s₁ to is the outcome of the implication. An implication is said to have validity if 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 with the idea of an argument. An argument is a complex statement of the form ‘s₁ is true and s₂ is true and … and s_n-1 is true, so necessarily s_n 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 was 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 names which make sense in a general statement are general names, for example ‘man’ and ‘flower’ and ‘green’, not names of persons or places or of other things that might be viewed as if they were units of being. In harmony with this opinion Aristotle did not let statements such as ‘Socrates is going to be dead some day’ and ‘Callias has a turned-up nose’—examples he frequently makes use of in his writings on philosophy—be conditions or outcomes of implications.

It is only right to say in addition that Aristotle gave modal general statements and syllogistic reasonings made of these an important place in his chief work on Syllogistic, the Prior Analytics; such modal statements are ones about the necessary or the possible, for example ‘all S is necessarily P’, ‘no S is necessarily 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 implications and a complete body of rules as tests of the validity of any given 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 distribution 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 Mathematical 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 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 grammatica) and the art of using language well (ars rhetorica). The teaching of the trivium 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 education which 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 trulv 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 that 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 relation to the structures and not in relation to material examples of 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, the variables are, at the natural level, the representatives of any qualities and of any sort of substances, like star and metal and backboned animal.

If one keeps in mind that the Greeks were very uncertain about and very far from letting variables ...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Original Title Page
• Original Copyright Page
• Contents
• 1. Purpose and Language of the Book
• 2. Aristotle’s Syllogistic
• 3. The Idea of a Complete, Automatic Language for Reasoning
• 4. Changes in Algebra and Geometry, 1825–1900
• 5. Consistency and Metamathematics
• 6. Boole’s Algebra of Logic
• 7. The Algebra of Logic after Boole: Jevons, Peirce and Schroeder
• 8. Frege’s Logic
• 9. Cantor’s Arithmetic of Classes
• 10. Peano’s Logic
• 11. Whitehead and Russell’s ‘Principia Mathematica’
• 12. Mathematical Logic after ‘Principia Mathematica’: Hilbert’s Metamathematics
• Further Reading
• Index