A Beginner's Guide to Mathematical Logic

Index

Part I

General Background

1

Genesis

Just what is Mathematical Logic? More generally, what is Logic, whether mathematical or not? According to Tweedledee in Lewis Carroll’s Through the Looking Glass, If it was so, it might be, and if it were so, it would be, but since it isn’t, it ain’t. That’s logic.

In The 13 Clocks by James Thurber, the author says, Since it is possible to touch a clock without stopping it, it follows that one can start a clock without touching it. That is logic, as I understand it.

A particularly delightful characterization of logic was given by Ambrose Bierce, in his book The Devil’s Dictionary. This is really a wonderful book that I highly recommend, which contains such delightful definitions as that of an egotist: An egotist is one who thinks more of himself than he does of me. His definition of logic is "Logic; n. The art of thinking and reasoning in strict accordance with the limitations and incapacities of the human misunderstanding. The basis of logic is the syllogism, consisting of a major and a minor premise and a conclusion thus:

The philosopher and logician Bertrand Russell defines mathematical logic as The subject in which nobody knows what one is talking about, nor whether what one is saying is true.

Many people have asked me what mathematical logic is, and what its purpose is. Unfortunately, no simple definition can give one the remotest idea of what the subject is all about. Only after going into the subject will its nature become apparent. As to purpose, there are many purposes, but again, one can understand them only after some study of the subject. However, there is one purpose that I can tell you right now, and that is to make precise the notion of a proof.

I like to illustrate the need for this as follows: Suppose a geometry student has handed in to his teacher a paper in which he was asked to give a proof of, say, the Pythagorean Theorem. The teachers hands back the paper with the comment, This is no proof ! If the student is sophisticated, he could well say to the teacher, "How do you know that this is not a proof? You have never defined just what is meant by a proof! Yes, with admirable precision, you have defined geometrical notions such as triangles, congruence, perpendicularity, but never in the course did you define just what is meant by a proof. How would you prove that what I have handed you is not a proof?"

The student’s point is well-taken! Just what is meant by the word proof? As I understand it, on the one hand it has a popular meaning, but on the other hand, it has a very precise meaning, but only relative to a so-called formal mathematical system, and thus the meaning of proof varies from one formal system to another. It seems to me that in the everyday popular sense, a proof is simply an argument that carries conviction. However, this notion is rather subjective, since different people are convinced by different arguments. I recall that someone once said to me, "I can prove that liberalism is an incorrect political philosophy! I replied, I’m sure you can prove this to your satisfaction, and to the satisfaction of those who share your values, but without even hearing your proof, I can assure you that your so-called proof would carry not the slightest conviction to those with a liberal philosophy! He then gave me his proof," and indeed it seemed perfectly valid to him, but obviously would not make the slightest dent on a liberal.

Speaking of logic, here is a little something for you to think about: I once saw a sign in a restaurant which read, Good food is not cheap. Cheap food is not good.

Problem 1. Do those two statements say different things, or the same thing?

Note that solutions to problems are given at the end of the chapters.

Mathematical Logic is sometimes also referred to as Symbolic Logic. Indeed, one of the most prominent journals on the subject is entitled The Journal of Symbolic Logic. How did the subject even start? Well, it was preceded by logic of a non-symbolic nature. The name Aristotle obviously comes to mind, for that famous ancient Greek philosopher was the person who introduced the notion of the syllogism. It is important to understand the difference between a syllogism being valid and a syllogism being sound. A valid syllogism is one in which the conclusion is a logical consequence of the premises, regardless of whether the premises are true or not. A sound syllogism is a syllogism which is not only valid, but in which, in addition, the premises are true. An example of a sound syllogism is the well-known:

All men are mortal.

Socrates was a man.

Therefore, Socrates was mortal.

The following is an example of a syllogism which, though obviously unsound, is nevertheless valid:

All bats can fly.

Socrates is a bat.

Therefore, Socrates can fly.

Clearly, the minor premise (the second one) is false, and also, of course, so is the conclusion. Nevertheless, the syllogism is valid – the conclusion really is a logical consequence of the two premises. If Socrates were a bat, then he really could fly.

I am amused by syllogisms that appear to be obviously invalid, but are actually valid! Here are two examples:

Everyone loves my baby.

My baby loves only me.

Therefore, I am my own baby.

Doesn’t that sound ridiculous? But it really is valid! Here is why:

Since everyone loves my baby, then my baby, being also a person, loves my baby. Thus my baby loves my baby. But also, my baby loves only me (minor premise). Since my baby loves only me, then there is only one person my baby loves (namely me), but since my baby loves my baby, that one person must be me. Thus I must be my own baby.

Here is another example of a valid syllogism which seems invalid. We define a lover as one who loves at least one person.

Everyone loves a lover.

Romeo loves Juliet.

Therefore: Iago loves Othello.

Here is why the syllogism is valid. Since Romeo loves Juliet (second premise), then Romeo is a lover. Since Romeo is a lover, then everyone loves Romeo (by the first premise). Since everyone loves Romeo, then everyone is a lover. Since each person is a lover, then everyone loves that person (by the first premise). Thus it follows that everyone loves everyone! In particular, Iago loves Othello.

Once, the eminent logician and philosopher Bertrand Russell was asked, What is really new in the conclusion of a syllogism? Russell replied that logically, the conclusion may contain nothing new, but the conclusion can nevertheless have psychological novelty, and he then told the following story to illustrate his point.

At a certain party, one man told a somewhat risqué story. Someone else told him, Please be careful. The abbot is here! The abbot then said, We men of the cloth are not as naïve as you might think! Why, the things I have heard in the confessional . . . my first penitent was a murderer! Shortly after, an aristocrat came late to the party, and the host wanted to introduce him to the abbot, and asked if he knew him. The aristocrat said, Of course I know him! I was his first penitent.

Aristotelian logic flourished through the ages. In the seventeenth century the philosopher Leibnitz envisioned the possibility of a symbolic calculating machine that would settle all questions – mathematical, philosophical and even sociological. Wars would then be unnecessary, since the opposing sides could instead say, Let us sit down and calculate.

There is the story told that Leibnitz was undecided whether or not to marry a certain lady, and so he made one list of advantages and one of disadvantages. The disadvantage list turned out to be longer, and so he decided not to marry her.

As we will see in the course of this volume, an amazing discovery of the logician Kurt Gödel [1931] showed that Leibnitz’ dream was unrealizable even in pure mathematics.

Symbolic Logic proper can be said to have its beginnings in the nineteenth century, through such thinkers as Pierce, Jevons, Schroeder, Venn, De Morgan and particularly George Boole, after whom Boolean Algebra is named. Boole was a completely self-educated school master, who wrote An Investigation of the Laws of Thought [republished, 2009], in which he described his purpose in writing the book in the following opening words:

The design of the following treatise is to investigate the fundamental laws of the mind by which reasoning is performed, to give expression to them in the symbolic language of a Calculus, and upon this foundation to establish the science of Logic and construct its method; to make that method itself the basis of a general method for the application of the mathematical doctrine of Probabilities, and finally, to collect from the various elements of truth brought to view in the course of these inquiries some probable intimations concerning the nature and constitution of the human mind.

The book is an interesting mixture of precise mathematical and symbolic reasoning with philosophical considerations. Boole attempts to put purely philosophical arguments into symbolic form – especially in his chapter on the philosophers Clarke and Spinoza. Towards the beginning of the chapter in which he does that, he says,

In the pursuit of these objects it will not devolve upon me to inquire, except accidentally, how far the metaphysical principles laid down in these celebrated productions are worthy of confidence, but only to ascertain what conclusions may justly be drawn from the given premises.

Thus Boole’s purpose in that chapter is not to decide whether the philosopher’s premises (and hence also the conclusions) are true, but only whether the conclusions are really logical consequences of the premises – in other words, not if the philosopher’s arguments are sound, but only whether they are valid.

Boole becomes quite philosophical in the last chapter of the book. To my great delight he writes the following beautiful words:

If the constitution of the material frame is mathematical, it is not merely so. If the mind, in its capacity of formal reasoning, obeys, whether consciously or unconsciously, mathematical laws, it claims through its other capacities of sentiment and action, through its perception of beauty and of moral fitness, through its deep springs of emotion and affection, to hold relation to a different order of things. Even the revelation of the material universe in its boundless magnitude, and pervading order, and constancy of law, is not necessarily the most fully apprehended by him who has traced with minutest accuracy the steps of the great demonstration. And if we embrace in our survey the interest and duties of life, how little does any process of mere ratiocination enable us to comprehend the weightier questions which they present! As truly, therefore, as the cultivation of the mathematical or deductive faculty is a part of intellectual discipline, it is only a part.

Set Theory

The beginnings of mathematical logic went pretty much hand in hand with the nineteenth century development of set theory – particularly the theory of infinite sets founded by the remarkable mathematician Georg Cantor. Before discussing infinite sets, we must first look at some of the basic theory of sets in general.

A set is any collection of objects whatsoever. The basic notion of set theory is that of membership. A set A is a bunch of things, and to say that an object x is a member of A, or an element of A, or that x belongs to A, or that A contains x, is to say that x is one of those things. For example, if S is the set of all positive integers from 1 to 10, then the number 7 is a member of A (so is 4), but 12 is not a member of A. The standard notation for membership is the symbol ∈ (epsilon), and the expression "x is a member of A is abbreviated x ∈ A".

A set A is said to be a subset of a set B if every member of A is also a member of B. Unfortunately, many beginners learning about sets confuse subset with membership. As an example of the difference, let H be the set of all human beings and let W be the set of all women. Obviously W is a subset of H, since every woman is also a human being. But W is hardly a member of H, as W is obviously not an individual human being. The symbol for subset is the so-called inclusion sign ⊆. Thus, for any pair of sets A and B, the phrase "A is a subset of B" is abbreviated A ⊆ B. If A is a subset of B, then B is called a superset of A. Thus a superset of A is a set that contains all elements of A, and possibly other elements as well. A subset A of B is called a proper subset of B if A is not the whole of B, in other words, if B contains some elements that are not in A.

A set A is the same as a set B if and only if they contain exactly the same elements, in other words, if and only if each is a subset of the other. The only way two sets can be different is if one of them contains at least one element that is not in the other. The only way a set A can fail to be a subset of a set B is when A contains at least one element that is not in B.

A set is called empty if it contains no elements at all, such as the set of all people in a theater after everyone has left. There can be only one empty set, because if A and B are empty sets, they contain exactly the same elements – namely no elements at all. Put another way, if A and B are both empty, then neither one contains any element not in the other, since neither one contains any elements at all. Thus if A and B are both empty sets, then A and B are the same set. Thus there is only one empty set, and it will be denoted in this work by the symbol Ø.

The empty set has one characteristic which seems quite strange to those who encounter it for the first time. As a preliminary illustration, consider a club whose president says that all Frenchmen in the club wear berets. But suppose it turns out that there are no Frenchmen in the club. Should the president’s statement then be regarded as true, false, or neither? More generally, given an arbitrary property P, should it be regarded as true, false, or neither, to say that all members of the empty set have property P? Here we have to make a choice, once and for all, and the choice universally agreed upon by mathematicians and logicians is that such a statement should be regarded as true! One reason for such a decision is this: Given any set S and any property P, the only way that P can fail to hold for all elements of S is that there be at least one element of S for which P doesn’t hold. The empty set is to be regarded as no exception to the statement just made, and so the only way that P can fail to hold for all elements of the empty set is that there is at least one element of the empty set that doesn’t have the property P, but that cannot be, since there is no element of the empty set at all! [As the late mathematician Paul Halmos would say, "If you don’t believe that P holds for all elements of the empty set, just try to find me an element of the empty set for which P doesn’t hold!"] Thus we shall henceforth regard it as true that for any property P, all elements of the empty set have property P. Here is another way of looking at it, which anticipates an important principle of Propositional Logic, which we study in Part II of this volume, namely the logical use of the word implies or if – then.

The phrase if – then, as it is used in classical logic, is a bit of a shock to those encountering it for the first time, and rightfully so, since it is highly questionable whether it really corresponds to the way the phrase is commonly used.

Suppose a man tells a girl, If I get a job next summer then I will marry you. If he gets a job next summer and marries her, he has clearly kept his word. If he gets a job and doesn’t marry her, he has obviously broken his word. Now, suppose he doesn’t get a job but marries her anyway. I doubt that anyone would say that he has broken his word! And so in this case too, we will say that he has kept his word. The crucial case is that he neither gets a job nor marries her. What would you say about this case – has he kept his word? Broken his word? Or neither? Suppose the girl complains, You said that if you got a job you would marry me, and you didn’t get any job, nor will you marry me! The man could rightfully say, "I haven’t broken my word! I never said that I will marry you – all I said is that if I get a job, then I will marry you. Since If didn’t get a job, then I have not broken my word."

Well, as I said, I believe that you would not be uncomfortable with saying that he has not broken his word in this case, but I imagine many of you would be uncomfortable with saying that he has kept his word.

Well, we want all statements of the form if – then to be either true or false, regardless of whether the if – part or the then-part is true or false. Under this rule, since we have decided that the man did not break his word, we have no option but to say that he has kept his word, strange as it may seem!

Thus in classical logic, for any pair of propositions p and q, the statement "if p, then q (also stated p implies q") is to be regarded as false only if p is true and q is false. In other words, "p implies q is synonymous with it is not the case that p is true and q is false, or equivalently, either p is false or p and q are both true, which is also equivalent to p is false or q is true."

This type of implication is more specifically called material implication, and it does have the strange property that a false proposition implies any proposition! For example, the statement If Paris is the capital of England, then 2 + 2 = 5 is to be regarded as true!

I must tell you an amusing incident: Someone once asked Bertrand Russell, You say that a false proposition implies any proposition. For example, from the statement 2 + 2 = 5, could you prove that you are the Pope? Russell replied, Yes, and gave the following proof. Suppose 2 + 2 = 5. We also know that 2 + 2 = 4, from which it follows that 5 = 4. Subtracting 3 from both sides of the equation, it follows that 2 = 1. Now, the Pope and I are two. Since two equals one, then the Pope and I are one! Therefore, I am the Pope.

Material implication, with all its oddity, really has its advantages, which I would like to illustrate as follows. Suppose I take a card from a deck and put it face down on the table and say, If the card is the Queen of Spades, then it is black. Do you agree? Surely you would agree. Then I turn the card over, and it is a red card – say the Jack of Diamonds. Would you then say that you were wrong regarding my statement as true? My case rests!

Now, how is all this about implication relevant to the statement that any property P holds for all elements of the empty set? Well, to say of a given set S and a property P that all elements of S have property P, is to say that for every element x, if x is in S, then x has property P. In particular, to say that all elements of the empty set Ø have property P is to say that for any element x, if x is in Ø, then x has the property P. Well, for any x, it is false that x is in Ø, and since a false proposition implies any proposition, it is true that if x is in Ø, then x has the property P. Thus for all x, if x ∈ Ø, then P(x), which means that P holds for all elements of Ø.

Problem 2. Is the empty set a subset of every set?

Finite sets are often displayed by enclosing the names of their elements in curly brackets – for example {2, 5, 16} is the set whose elements are the three numbers, 2, 5, and 16. Sometimes the empty set is denoted { }, and I will sometimes use this notation in contexts in which we are describing members of a set by listing them inside curly brackets.

Boolean Operations on Sets

Unions

For any pair of sets A and B, by the union of A and B, denoted A B, is meant the set of all things that belong either to A or to B, or to both. For example, if P is the set of all non-negative integers (i.e. the positive integers and zero) and N is the set of all negative integers and I is the set of all integers, then P N = I.

{2, 3, 7, 24} = {1, 2, 3, 7, 18, 24}.

Problem 3. Which, if any, of the following statements are true?

(1)   If A B = B, then B ⊆A.

(2)   If A B = B, then A ⊆B.

(3)   If A ⊆ B, then A B = B.

(4)   If A ⊆ B, then A B = A.

We can think of A B as the result of adding the elements of A to the set B, or what is the same thing, adding the elements of B to A. Thus A B = B A. It is also obvious that for any three sets A, B and C, A (B C) = (A BC – if we add the elements of A to the set B C, we get the same set as when we add the elements of A B to the set C. It is equally obvious that A A = A. Also obvious is the fact that A Ø = A (we recall that Ø is the empty set).

Intersections

For any pair of sets A and B, by their intersection – denoted A B – is meant the set of all elements that are in both A and B. For example, suppose A = {2, 5, 18, 20} and B = {2, 4, 18, 25}. Then A B = {2, 18}, since 2 and 18 are the only numbers common to both A and B. The following facts are obvious:

(a)   A A = A;

(b)   A B = B A;

(c)   A (B C) = (A BC;

(d)   A Ø = Ø.

Problem 4. Which of the following statements are true?

(1)   If A B = B, then B ⊆ A.

(2)   If A B = B, then A ⊆ B.

(3)   If A ⊆ B, then A B = B.

(4)   If A ⊆ B, then A B = A.

Problem 5. Suppose A and B are sets such that A B = A B. Does it necessarily follow that A and B must be the same set?

Complementation

We now consider as fixed for the discussion a set I, which we will call the universe of discourse. What the set I is will vary from one application to another. For example, in plane geometry, I could be the set of all points on a plane. For number theory, the set I could be the set of all whole numbers. In applications to sociology the set I could be the set of all people. In the discussion of Boole’s general theory of sets we are now embarking upon, the set I is a completely arbitrary set, and we will be considering all subsets of I.

For any subset A of I, by its complement (relative to I, which will be understood) is meant the set of all elements of I that are not in A. For example, if I is the set of all whole numbers, and E is the set of all even numbers, then the complement of E is the set of all odd numbers. The complement of a set A is denoted A′, or sometimes Ā or Ã.

It is obvious that A″ (the complement of the complement of A) is A itself.

Problem 6. Which, if either, of the following statements is true?

(1)   If A ⊆ B, then A′ ⊆ B′.

(2)   If A ⊆ B, then B′ ⊆ A′.

The operations of union, intersection and complementation are the fundamental Boolean operations on sets. Other operations are definable by iterating these fundamental operations. For example, the set denoted A –B [the so-called difference of A and B], which is the set of all elements of A that are not in B, can be defined in terms of the three fundamental operations, since A –B = A B′.

Venn Diagrams

Boolean operations on sets can be graphically illustrated by what are known as Venn diagrams, in which the universe of discourse I is represented by the set of all points in the interior of a square, subsets A, B, C etc. of I are represented by circles within the square, and Boolean operations are represented by shading appropriate portions of the circles. For example,

Boolean Equations

We will use the capital letters A, B, C, D, E, with or without subscripts, to stand for arbitrary sets (just as in algebra we use the lower-case letters x, y, z to stand for arbitrary numbers). We call these capital letters (with or without subscripts) set variables. By a term we mean any expression constructed according to the following rules:

(1)   Each set variable standing alone is a term.

(2)   For any terms t1 and t2, the expressions (tt2), (ttare again terms.

Examples of terms are A (B C′), and A(B A″), and (A B)′.

It is necessary to use parentheses to avoid ambiguity. For example, suppose we didn’t use parentheses in writing the expression A B C. It is not possible to tell whether this means the union of A B with C or the intersection of A with B C. If we mean the former, we should write expression (A BC, if the latter, we should write expression A (B C). Sometimes parentheses can be deleted if no ambiguity can result. For example, outer parentheses can be dropped, as we did above by writing (A BC instead of ((A BC).

By a Boolean equation we mean an expression of the form t1 = t2, where t1 and t2 are Boolean terms. As some examples, consider:

(1)   A B = A B

(2)   A′ = B

(3)   A B = B A

(4)   A B′ = AB′

(5)   (A B)′ = (A BC

(6)   A (B C) = (A B(C (A B))

A Boolean equation is called valid if it is true no matter what sets the set variables represent. For example, (3) is valid, since for any pair of sets A and B, it is true that A B = B A. None of the other five equations above are valid.

Testing Boolean Equations

Suppose we wish to test whether or not a given Boolean equation is valid. Is there a systematic way of going about it, or is ingenuity required? The answer is that it can be done systematically. One well-known way is by use of Venn diagrams, but I have found another way [Smullyan, 2007], the method of indexing, to which we now turn.

As a simple starter, let A and B be subsets of I.

I (the interior of the square):

In this diagram I is divided up into the four sets of A B, A B′, AB and AB′, which we have labeled (indexed) by the numbers 1, 2, 3 and 4 respectively. Every element x of I belongs to one of these four regions. Let us call these regions the basic regions.

Now, let us identify any union of the basic regions with its set of indices. So for example,

Actually, of course, we could write A 2 and B 3, but we are dropping the union sign to make the indices easier to focus on.

Then A B = (1, 2, 3); and A B = (1), since (1) is the only region common to A and B. Also, A′ = (3, 4), and since there is nothing in common between (1, 2) and (3, 4), then A A′ = Ø, i.e. A A′ is the empty set. Also, A A(3, 4) = (1, 2, 3, 4). Thus A A′ = I.

Now, suppose we wish to verify the De Morgan law, (A B)′ = AB′ by the method of indexing. The idea is to find first the set of indices of (A B)′ and then the set of indices of AB′ and see if the two sets are the same.

Thus, (4) is the set of indices of both (A B)′ and AB′; hence (A B)′ = AB′.

Let’s now try an equation with three sets A, B and C. These three sets divide I into eight basic regions, as the following figure shows:

I (the interior of the rectangle):

Thus:

Suppose we wish to show that A (B C) = (A B(A C). Again we reduce each side of the equality sign to its set of indices and see if the two sets are the same.

Thus both sides of the set equation reduce to (1, 2, 3, 4, 5) and we have won the case.

Now let’s try the equation A (B C) = (A B(A C).

Thus the equation reduces to (1, 2, 3) = (1, 2, 3) and we see the equation is valid.

What do we do if we have more than three unknowns, say A, B, C and D? Well, we can no longer draw circles, but still the four sets divide I into 16 basic regions, and we can number them in such a way that:

Once we number the basic regions so that we know what our original sets are in terms of unions of basic regions, we can operate as we have seen with the set of indices for these four sets.

For five unknowns, A, B, C, D and E we have 32 basic regions. In general, for any n equal or greater than 2, the sets A1, A2, . . ., An divide I into 2n basic regions, and we can assign an index to each of these regions, say by taking A1 to be the first half of the integers 1, 2, . . . 2n, then taking A2 to be every other quarter (starting with the first), A3 to be every other eighth, and so forth. For example, for n = 5, we take:

Exercise 1. By the method of indexing, prove the following Boolean equation to be valid:

Some other Boolean operations are:

A → B, which is AB, and

A ≡ B, which is (A B(AB′).

Exercise 2. By the method of indexing, prove that the following equations are valid.

(1)   A → B = (A –B)′

(2)   A ≡ B = ((A → B(B → A))

(3)   (A (A –B)′) = Ø

(4)   A ≡ (A → B)) = A B

The Boolean theory of sets is but the beginning of the field known as Set Theory. The deeper aspects of the subject concern infinite sets, which is the subject of the next chapter.

Solutions to the Problems of Chapter 1

1.   Logically they say the same thing, namely that no food is both good and cheap, but psychologically they convey quite different images. The statement good food is not cheap tends to create the image of good expensive food, whereas cheap food is not good makes on think of cheap rotten food.

2.   It is true that the empty set is a subset of every set S, because the property of being an element of S (like any other property) holds for all elements of the empty set, as we have seen. Thus every element of the empty set is an element of S, which means that the