Introduction to Symbolic Logic

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James D. Carney A rizona State University

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Prentice-Hall, Inc. Englewood Oliffs, New Jersey

PREFACE

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Library of Congress Catalog Card No.: 7l-97926 13-498709-8

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Printed in the United States of America

This book is an effort at providing an introductory text to symbolic logic on the university level. No prior knowledge of either philosophy or mathematics is assumed. It is hoped that sections have been so set up with their accompanying exercises that a diligent and intelligent reader will find the book to a large extent self-teaching. To aid in this end, partial answers to the exercises are supplied with most sets of exercises. The aim of this book is to provide a beginning student with a working knowledge of the foundations of modern logic-the lower predicate calculus with identity. Accordingly, Part I emphasizes the formal language, transformations, proof construction in a natural deduction system, and translation. I have tried to develop the natural deduction system for predicate logic as formally as is in keeping with the general aims of the book. The natural deduction system for the sentential calculus is an adaptation of that found in E. J. Lemmon's Beginning Logic, which is, in turn, an adaptation of Gentzen's 1934-35 paper. The quantification rules of the predicate calculus are the standard, non-independent four. To simplify these rules somewhat, I have stated them in terms of limited and non-limited individual constants. Neither system has independent rules, but sacrificing simplicity leads to easier development of the system. Of special interest is Beth's semantic tableaux method for constructing proofs in logic. This is introduced in three exercise sections. The first section (at the end of Chapter Four) deals with sentential logic, the second (at the end of Chapter Six) deals with predicate logic, and the last (at the end of Chapter Seven) deals with predicate logic with identity. The rules and procedures for constructing trees are taken from Richard C. Jeffrey's Formal Logic: Its Scope and Limits. Part II emphasizes logical axiom systems and metalogic, stopping short of the completeness proof for the predicate calculus. The completeness proof of the predicate calculus is regarded as the demarcation point between beginning logic and advanced logic. Three axiom systems are presented in Part II: one for syllogistics, taken from .}:Jukasiewicz; the sentential calculus, found in Principia; and an infinite axiom system

vi

Preface

for the predicate calculus, taken from Ohurch. Very few theorems are derived, though the exercises call on the reader to derive a legion of theorems. In the development of the sentential calculus little more is done than deriving the machinery to prove the replacement theorem. The development of the axiom system for the predicate calculus is aimed at helping the student become acquainted with axiom schemes and what it is like to construct scheme proofs. For some semester courses Part I is all that can be covered, especially if most of the exercises are done. But courses vary; many will find time to do some or all of Part II. Not only is it fruitless to work each and every exercise, but it is also advisable to omit some sets of exercises within sections. From time to time, usually at the end of an exercise section, exercises are introduced which go beyond the analysis in the section. These exercises can be omitted without the loss of continuity. The manner of presentation in the book owes a great deal to Margaris, Ohurch, Lemmon, Mates, and Hughes and Londey (see Further Readings). For ideas for notation and organization, I am especially indebted to Angelo Margaris' First Orde'r Mathematical Logic. I wish to express my appreciation to those who helped in the preparation of the manuscript: Alan Lesure, the understanding editor, Professor Phillip Von Bretzel and Lt. Walter Jones, who did some of the proofreading, Mrs. Ruth Bardrick, for typing assistance, and my wife, who has been able to raise a family, teach French, and do expert typing on a difficult manuscript, all at the same time.

TABLE OF CONTENTS

Part I: NATURAL DEDUCTION SYSTEMS Chapter 1: 1.1 1.2 1.3 1.4 1.5

Arguments Necessarily true statements Valid logical formulas Formal language The use/mention distinction

Chapter 2: 2.1 2.2 2.3 2.4

2.5

2.6

3.3 3.4 3.5

3.6 3.7

3 6 9

13 14

Sentential Language

Conjunction and denial Other statement connectives Determining the truth-value of compound statements Stat~ment connectives and the natural language Puttmg the natural language into symbols Formation rules for the sentential language

Chapter 3: 3.1 3.2

Validity

16 19

21 24

28 31

Decision Procedures

Truth-tables Valid, inconsistent, and contingent formulas Arguments and truth-tables A short-cut truth-table method Logical equivalence and transformations Other two-place operators Normal forms and testing for validity Supplement: Logical networks vii

36 38 40 45 47 54

57 61

viii

Table of Contents

Chapter 4: 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8

5.1 5.2 5.3

5.4 5.5

5.6 5.7 5.8

68

Part II: LOGICAL AXIOMATIC SYSTEMS

72

76 83

86 89 91 97

Predicate Language

Predicates Quantifiers Interpretation of quantifiers Valid predicate formulas Proving the invalidity of predicate formulas Proving the validity of predicate formulas Polyadic predicates Formation rules for the predicate language

Chapter 6:

102 106 110 113 116 118

120 128

Chapter 8: 8.1 8.2 8.3 8.4 8.5

Chapter 9: 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8

6.3 6.4 6.5 6.6 6.7

7.1 7.2 7.3 7.4

7.5

''1'.6

134 136 140 147

150 152 155

10.1 10.2 10.3 10.4 10.5 10.6

10.7

Preliminary discussion of PM System PM Development of PM Derived rules Oonsistency and soundness of PM Independence of the axioms of PM Completeness of PM Completeness of SC

Identity Some properties of two-place predicates Predicate logic with identity: System POI Proofs for arguments Symbolizing using the identity sign Definite descriptions

160 162 166 169

171 175

204 206 207 210

214 217 220 224

Axioms jor the Predicate Calculus with Identity

Preliminary discussion of LPC System LPO Development of LPC LPC with identity ConSistency of LPC with identity Soundness of LPC with identity Oompleteness of predicate logic theories

Further readings

Predicate Logic with Identity

186 187 190 194 199

Axioms jor the Sentential Calculus

Natural Deduction System PC

Individual constants and tautologies Universal elimination and existential introduction Existential elimination and universal introduction Strategies in using quantification rules System PO Theorems of PC Soundness, consistency, and completeness of PC

Chapter 7:

Formal Axiom Systems

The development of geometry Formal axiomatic theories System CS Metalogical properties of CS Axiom systems and logic

Chapter 10: 6.1 6.2

ix

Natural Deduction System SC

Rules of inference and proofs Four additional rules of inference Assumption discharging rules Useful strategies System SO Some theorems of SO Two derived rules Soundness, consistency, and completeness of SO

Chapter 5:

Table of Contents

227 230 232

236 238 239 242 245

Index 248

I NATURAL DEDUCTION SYSTEMS

,.

validity

1.1

All of us reason. That is, we draw conclusions from what we take to be true. Sometimes we reason correctly and sometimes we do not. That is, even when what we take as true is true, sometimes our conclusions follow and sometimes they do not. Gradually most of us become aware that there are certain norms for reasoning correctly, and if we follow them and reason from satisfactory data our conclusions can be relied on. The study of these norms or principles of reason is the aim of this book. And we will use symbolic or mathematical methods to systematically set out these norms. It would be futile at this time to attempt to define 'symbolic logic'. Rather we will try in this chapter to give a more or less intuitive account of the central idea and purpose of logic.

Arguments

If we assume that the statements Sophocles wrote Oedipus Rex. If Sophocles wrote Oedipus Rex, then he was a Greek. are true, then our logical intuition tells us that we should be able to conclude that Sophocles was a Greek. In this type of reasoning we take the two statements above as assumptions or premises and infer the conclusion, Sophocles was a Greek. The word argument will be used to designate a set of premises and an accompanying con/clusion. We commonly indicate the presence of an argument by using words such as 'therefore' and 'so' between the premises and the conclusion. The 3

4

Validity

argument corresponding to the above inference may thus be explicitly set down as follows:

.1

Arguments

Furthermore we may have a valid argument made up of one or more f aI se . premIses and a true conclusion. Thus in the argument: All U.S. Senators are charismatic. McCarthy is a U.s. Senator. Therefore McCarthy is charismatic.

Sophocles wrote Oedipus Rex. If Sophocles wrote Oedipus Rex, then he was a Greek. Therefore Sophocles was a Greek. Noone has difficulty seeing that the conclusion of the above argument follows from the premises. In turn, few have difficulty seeing that the conclusion of the next argument does not follow from the premises. All Communists are atheists. Bertrand Russell is an atheist. Therefore Bertrand Russell is a Communist. We may thus have arguments in which the conclusion follows from the premises and arguments in which the conclusion does not follow. We need not be reminded that there are arguments in which it is not so easy to see whether the conclusion follows. In fact it is such shortcomings in our logical intuition that, in part, motivate our interest in logic. In logic we are interested in the relation between the conclusion of an argument and its premises. We are concerned with whether a conclusion does or does not follow from the given premises. If the conclusion does follow from the premises the argument is said to be a valid or sound argument. If the conclusion does not follow from the premises, then the argument is said to be invalid or unsound. It is well to note at the outset that there can be valid arguments that have one or more false premises, and valid arguments with a false conclusion, and there can even be valid arguments with both false premises and a false conclusion. For example each premise and the conclusion of the next argument are false; nevertheless this argument is valid: All creatures that fly have wings. All wallabies fly. Therefore all wallabies have wings. In turn, the correct appraisal of an argument as invalid does not imply that any of its components are false. An invalid argument may be made up exclusively of true components, as the next example illustrates:

we .fin~ a true conclusion that validly follows from premises, the first of whICh IS false and the second true. What is ruled out if an. argument is valid is the comb'Inat'IOn 0 f true . prmll1ses and false conclusIOn. The only combinatl'on th t . " . a cannot OCCur if an argument IS valId IS that of true premises and a J.a £' I ' . . se coneI USlOn The validIty of an argument only assures us that I'f th . . e premIses are true, then the conclusion is also true In other word . ... . . .. .' s an argument IS vahd Iff. (If and only If) It IS ImpossIble for its premises to b e t rue an d ItS . conclusIOn false.

Exercises 1. Which of the following are valid arguments? (a) Every conservative believes in stability. Every fascist believes in stability. Therefore every conservative is a fascist. (b) For any number x, x + 0 = x. Therefore there is a number x such that 0 + x = 0 (c) With Wilson we had a Democrat and a war' with FD' R d T h dD ,an ruman we a. emocrats and wars. With Kennedy and LBJ we have had th thmg. Thus Democratic Presidents cause wars. e same (d) If I ~ad my way, national primaries would take the place of pa t ventlOns. r y con2. Give examples of valid arguments in which: (a) All the components are false. (b) One or m~re of the premises are false and the conclusion is true. (c) The premIses are true and the conclusion is true. 3. Test your logical intuition by seeing if you can tell which of the foll . statements are correct. Let P Q Rand S st d £ owmg £' ' " an or any statement (a ) If P lollows from Q, then not Q follows from not P . (b) If P follows from Q and R follows from P, then R follows from Q (c) If Q follows from P and also not Q follows from P, then R follows fr~m P (d) If Q follows from P and if not P obtains, then not Q. . (e) If Q follows from P, if R follows from not P and if not R £ II f not Q, then Q. , 0 ows rom (f)

ifromR £pollows from P, not S from Q, and Q from R, then not S follows .

Answers All birds have wings. All roadrunners have wings. Therefore all roadrunners are birds.

1. (a) is invalid, (b) is valid, and (c) is invalid; (d) is not an argument. 3. Only (d) is incorrect.

6

Validity

1.2

Necessarily True Statements

7 1.2

Necessarily true statements

Logic makes precise the conditions for valid argumentation. Before introducing the apparatus needed for this task we need to see the close connection between a valid argument and a necessarily true statement. What, first of all, is a statement? Sentences are usually classified as declarative, exclamatory, interrogative, or imperative. With respect to some types of declarative sentences it is meaningful to assign either the truth-value true or the truth-value false. By a statement will be meant a declarativ/e/ sentence that is either true or false. Some examples of statements are: ~

'--

-

(1) 2

-

+2=

4.

would be true. nonnecessary statements may b e sal'd t 0 b e con. ;All . tingent. Contmgent statements are thus those for which there can be circumstances in which they could be true or false ..iFor example: If children are to obey, force must be used on th The United States is a republic. em.

are both contingent. The first happens to be false and th dh b e secon appens to be true, . ut the world could be such that their truth-values would have been dIfferent. To make the connection between validity and necessal'l'1 y true statements, we must next introduce the conditional t . A ." sen ence. conihhonal sentence IS one that has the form

(2) All triangles are isosceles. (3) There is life on other planets. (1) is true, (2) is false, and (3) is a conjecture whose truth or falsity is unknown at this time. In our discussion of logic we shall confine ourselves to statements. The components of arguments, for example, are statements. We will call truth and falsity truth-values. The truth-value of a statement is thus its truth or falsity. If a statement is true, its truth-value is true and if it is false its truth-value is false. In some cases, as the third example above illustrates, we may not know the truth-value of a statement. But if anything is a statement it must be either true or false. A statement may be either necessarily true, necessarily false, or co-ntingent. Every statement, in fact, is either necessarily true, necessarily false, or contingent. A statement is neceSSMY if no state of affairs could possibly exist that could change its truth-value. As G. W. Leibniz, the seventeenth-century German mathematician, logician, and philosopher, put it, aJlecessary statement is one which is true (or false) , in all possible worlds. IThus, to illustrate,

r

,then---

If

It will be found useful to have a way of talking ab ou t w h a t·IS £ound In . the first blank and what is found in the second bl nk f t t . . -. a 0 a s a ement that has a condItIOnal form. What is found in the fir s t bl ank WI'11 b e called the antecedent of the conditional while what is fou d' th d . ' n m e secon blank wIll be called the consequent of the conditional. For any argu men t we ·can construct a corresponding condI't'IOnaI . . . h t sen ence b y conJommg t e premises using 'and's a d t t' th . n rea mg e result as the antecedent m the conditional form and b k' h . . ' y ma mg t e conclusIOn the consequent m .the conditional form . To 1'11us t rat e, b eIow are two arguments and then corresponding conditionals:

I

If Frances is wise and witty, then Frances is wise. If Frances is wise then someone is wise.

ARGUMENT:

Some men are mortal. Therefore some mortals are men. CORRESPONDING CONDITIONAL:

If some men are mortal, then some mortals are men. ARGUMENT:

6 7

< <

7 (reads: "6 is less than 7"). 10.

Therefore 6

<

10.

are necessarily true, whereas CORRESPONDING CONDITIONAL:

Some square objects are round. is

n~gessarily

false-we could imagine no set of circumstances in which it

If 6

< 7 and 7 < 10, then 6 < 10.

We can now make tl1e connec t'IOn t h at IS . Important . for thO d' . CUSSIOn the t' b . IS IS, connec IOn etween a valid argument and a necessarily true

8

Validity

statement. It is simply this: An argument is valid if and only if its corresponding conditional is necessarily true. The conditionals corresponding to the last two arguments are necessarily true; thus the arguments are valid.

1.3

Valid Logical Formulas

9

Answers 1. Only (b). 4. Only (al and (d).

Exercises 1.3

1. Which of the following are statements~ (a) Pass the tanis root. (b) Reading Kant isn't easy. (c) Why don't you think up a nice sentence~ (d) Where is the master key~ (e) Put the boxes in here. (f) Back to stacking books.

When a statement is necessarily true, it is usually the case that statements with the same form are likewise necessarily true. But here we must make it clear what we are talking about when we talk about a form that a statement has.

2. Try to classify the following as necessarily true, necessarily false, contingently true, contingently false, or none of these. (a) Every brother is a sibling. y = z. (b) For every number x and y there is a z such that x (c) Somebody loves everybody. (d) Everybody is loved by somebody. (e) There is an honest and intelligent U.S. politician. (f) The first man to land on the moon is alive today. (g) If G. Frege read this book, he would express his disapproval. (h) If x is next to y, then it is false that y is next to x. (i) There is intelligent life on a planet other than Earth. (j) There is no intelligent life on Earth.

+

3. Write the corresponding conditionals for the following arguments. {a) Only idiots would buy calls or puts. Alfred bought some calls. Therefore Alfred is an idiot. (b) If Alfred sells short, the market goes up. If Alfred sells long, the market goes down. Since Alfred is broke, he cannot sell short or long. So either the market won't go up or it won't go down. (c) For any number x there is a y such that x is less than y. Therefore there is a y such that y is greater than any number. . (d) Since someone is admired by everyone, it follows that everyone admIres someone or other. 4. Which of the conditionals corresponding to the above arguments are necessarily true statements~ 5. Why are the following arguments valid, given the account of 'valid arguments' found in Section 1, and why are their corresponding conditionals necessarily true statements1 (a) Hotel Utah is in Salt Lake City. Therefore 2 + 2 = 4. (b) Hotel Utah is in Salt Lake City and Hotel Utah is not in Salt Lake City. Therefore 2 2 = 3. 'I,

+

Valid logical formulas

To begin to make this clear, consider these two necessarily true statements: (1) If 6 < 7 and 7 < 10, then 6 < 7. (2) If the sun is shining and the temperature is lloa, then the sun is shining.

Let us try to analyze a common form that these two examples have. Let us begin by replacing 6 < 7 in (1) with P and 7 < 10 with Q. The result of such a replacement for (1) is (3) If P and Q, then P. Next let us regard P and Q as variables for statements. Thus as x stands' for numbers in, say, x 5 = 10, so P and Q stand for statements. As we have seen, we obtain (1) from (3) by replacing P with 6 < 7 and Q with 7 < 10. In turn, we can obtain (2) from (3) by replacing P with 'The sun is shining' and Q with 'The temperature is lloa,. We shall call (3) a conditionaljol'mula, since variables are employed. Both (1) and (2) will be spoken of as substitution instances or just instances of formula (3). A substitution instance of a formula of the sort we are considering is the statement that results from the given formula when the variables occurring in the formula are replaced throughout by statements, it being understood that the replacing is done uniformly. Thus if we replace both P and Q with 'This text is dull', we obtain an instance of (3), namely

+

(4) If this text is dull and this text is dull, then this text is dull. Here we have a statement that results from replacing each occurrence of P and Q with 'This text is dull', and this is done uniformly-the replacement we do for a variable in one of its occurrences we do for it in all of its

Validity

10

statement is not an instance of (3): occurrences. However tIle .collowing .L'

11

Valid Logical Formulas

(8) is necessarily true. However if we seek the sentential formula of which (8) is an instance we obtain, for example,

If this text is dull and the sun is shining, then the sun is shining.

since the substitution is not done uniformly-the same statement is not put in for P in each of its occurrences. With (3) we have a common form of (1), (2), and (4). Other common forms that (1), (2), and (4) have are (5) If P,

then Q.

P. Q.

An important feature of formula (3), which the other t.hree formulas k is that every instance of formula (3) is a necessanly true state~:n~. In other words, any statement of the form 'If P and Q, then P' is necessarily true. Formula (3) is not the only formula with this feature. Other examples of formulas that only have necessarily true statement

I

instances would be or Q and not P, then Q. P or not P. If P and if P then Q, then Q.

(6) If P

In contrast, the next formulas are such that not all instances are necessarily true: (7) If P or Q, then P.

P orQ. If Q and if P then Q, then P.

Let us now introduce a second use for the term 'valid'. Earlier we introduced 'valid' as an appraisal term for arguments. We. will ~0;V speak of some formulas in logic as valid. To say a formula III .10gIC IS l'd will mean that all instances of the formula are necessanly true ~:a:ements. The formulas in groups (6) and (3) are all examples of valid formulas, whereas those in groups (5) and (7) are all invalid. The formulas above are made up of variables such as P and Q and . of expreSSIOns suc h as 'or' " 'l'f then' , 'not' , and 'and'. Let us call these sentential formulas. Statements (1) and (2) may. now be .said to be necessarily true in virtue of being instances of a valId sententIal formula. It is easy to think of statements that are necessarily true but that are not instances of valid sentential formulas. Consider, for example, (8) If all bachelors are lonely and Phil is a bachelor, then Phil is lonely.

If P, then Q. \~ If P and Q, then R.

P. Q.

none of which are valid. To describe the logical form of (8) in virtue of which it is a necessarily true statement we need to introduce other kinds of variables. Let us use the letter x as a variable for names of individuals such as 'Phil'. And let us use F and G as variables for expressions such as 'men', 'mortal', 'bachelor', and 'lonely being'. With this new notation we may state the pattern of (8) relevant to its truth, as follows: (9) If every F

is G and x is F, then x is G.

(9) is a valid formula, for whatever individual is picked out for x, and no matter what property is chosen for F and G, the result will be a necessarily true statement. For example if we allow x to be 6 and F and G to be the properties respectively of being an even number and being divisible by 2, the instance of (9) below is obtained: (10) If every even

number is divisible by 2 and 6 is an even number, then 6 is divisible by 2.

(10), like (8), is a necessarily true statement. (8) and (10) are thus necessarily true because they are instances of the same valid logical formula: (9). Note that (9) is not a sentential formula since sentential formulas are formulas made up only of variables such as P and Q and expressions such as 'or', 'if, then', 'not', and 'and'. We may now bring together some of the elements of this discussion. Earlier it was said that an argument is valid iff its corresponding conditional is necessarily true. In this section the point is made that if a conditional is an instance of a valid logical formula, then it is necessarily true. Thus to show an argument to be valid it suffices to show that its corresponding conditional is an instance of a valid logical formula. If an argument corresponds in this way to a valid logical formula, we will say that the premises of the argument logically imply the conclusion. The immediate task before us is twofold. We wish to set down the structure of arguments relevant to their validity, thus obtaining our logical formulas. And we wish to set down exact techniques by which we can

Validity

12

determine whether a formula is valid or not. (We sh~uld not~ that it is not until we reach Section 7.2 that definitions can be gIVen for Important notions such as 'formula in logic' and 'logically imply'.)

1.4

Formal Language

13

Answers 1. Only (d) and (f) are not sentential formulas. Only (a), (b) and (e) are not valid. 4. Only (a) and (f) are invalid.

Exercises 1. Which of the following are sent en t'IaI f ormuI a s1. Which formulas are valid1 (a) P and not P. (b) If POI' Q, then P. (c) If if P then Q, then if not Q, .then not P. (d) If every x is F, then some x IS F. (e) If P or not P, then P. (f) If x is F, then something is F. (g) If P and Q, then Q and P. 2. Give instances of (c) and (d) that express necessarily true statements. Why

can you not do this for (a)1 3. Provide an instance of each of the argument patterns below. The premises

appear above the bar and the conclusion is below the bar. (a) If P then Q Q P (b) POI' Q Q or P (c) If P, then Q If Q, then R If P, then R (d) P and not P Q (e) Every x is F x is F (f) Some things are F Some things are G Some things are F and G (g) If P, then Q and not Q Not P (h) For any x if x is F then x is G If every x is F, then every x is G 4. An argument pattern is valid iff its corresponding conditional is valid. Which of the above arguments are valid 1 If a statement is necessary in virtue of being an instance ?f a valid logical 5. formula, then it is now generally referred to as an analyt~c statem~nt. On the basis of the results of 3 and 4, provide some ex~mples of ~nalytlC statements. Can you give some examples of nonanalytlC necessarIly true statements1 --,~

.,

5. If there are rules, then there must be human agreement.

1.4

Formallanguage

In order to study the exact conditions under which a logical formula is valid, and in order to display the logical form of arguments, a formal or artificial language is necessary. This means, in part, that we need to set down the expressions to be used in the language, give an exact interpretation of the expressions, and set down rules for when the expressions are correctly combined together. The formal language used to express sentential formulas-the sentential language-is a comparatively simple language, and it is introduced in the next chapter. The expressions used in this language are variables for statements P, Q, R, ... and are expressions corresponding to devices in the natural language we use to connect together statements, for example, 'or' and 'and'. However, exact interpretations are given for these latter expressions so that no ambiguity or vagueness can result in their use. Later additions will be made to this language. For example we will introduce variables for individuals x, y, z, ... and variables for properties F, G, H, .... The resulting language is called the predicate language and is taken up in Chapter 5. Having introduced this formula language, we can introduce techniques by which we can determine whether a formula expressed in this language is valid or not. We will introduce validation techniques first for sentential formulas and later for predicate formulas. If an argument is expressed using the formal languages, then we can say without hesitation that is is valid if its corresponding conditional exemplifies a valid logical formula. If, on the other hand, an argument is not expressed using the formal language, then before using the techniques oflogic one must first paraphrase the argument so that it employs the expressions of the formal language. This means that we must engage in three separate steps. First, layout the formal language. Second, develop the systematic methods to determine the validity of a formula expressed in the language. Third, develop the art of proper paraphrase of arguments into the language if they are not already

14

Validity

expressed using the language. It should be stressed at this point that today the preferred way to express statements in mathematics and the other sciences is to use the formal language presented in this book. In the next chapter the formal language needed for sentential logic is introduced. But before turning to it, one more matter needs to at least be touched on in this introductory chapter.

1.5

The use/mention distinction

It is a truism to say that an assertion about a thing contains the name of the thing and not the thing itself. When we speak about Lompoc we use the name 'Lompoc'; the Oalifornia city does not appear in our assertion. In turn we may have occasion to speak not of the city W. O. Fields made famous, but of the name of the city. For example we might wish to say that the name of the city is made up of six letters, or that it is disyllabic. The customary way of doing this is to use single quotation marks. Thus to say the name of the city has six letters, following this convention, we say

(1) 'Lompoc' has six letters. In turn to say something about the city rather than the name, we do not use any quotation marks at all, as the next sentence illustrates. (2) Lompoc is near the coast. A kind of nonsense is obtained if we replace thing with name of thing and vice versa, as the next two sentences illustrate. (3) Lompoc has six letters. (4) 'Lompoc' is near the coast. (1) is about a word; (2) is about a city; (3) is about a city but makes the queer statement that it has six letters, whatever that could mean; and (4) is about a word that is near the coast-imagine millions of pieces of paper with 'Lompoc' on them. In (1) the name is commonly said to be mentioned, while in (2) it is said to be used. Oonfusions can result from carelessness with the use/mention distinction. This is especially likely when we operate on a double single quqte level. But following the widely practiced custom in introductory

The Use/Mention Distinction

15

bool{s, we will ignore distinction unl ess confus " . . the. use/mention . IOn IS hkely to result from Ignormg It. We have up to this time b . £ . thO t· een, In act fol I owmg IS prac ICe. The reason for this is to av 01'd th e unSIghtly . ' . . . prolIferatIOn of smgle quotation marks It is hoped th t th b . . a e a sence of quotatIOn marks when terms are mentioned has not and will not undul upset the reader. y

2.1

The main interest in this chapter is with part of the formal language needed to display the structure of arguments relevant to questioning whether the premises logically imply the conclusion. The part of this formal language to be considered in this chapter is the part that relates whole statements. This language will be made up of two kinds of ingredients: variables for statements and symbols to relate statements. We will call this fragment of the formal language the sentential language.

sentential language

2.1

17

The first two columns list all possible assignments of truth-value to the pair P and Q, and the last column gives the corresponding truth-values for P A Q. This table indicates that when P is true (abbreviated T) and Q is T-first line-then P A Q is T; and when P is T and Q is false (abbreviated F)-second line-then P A Q is F; and so on. The table above is an example of a truth-table. It will be spoken of as a basic truth-table since it provides the interpretation for A. An example of a conjunction is 2

+3=

5

A

2

+3=

which is false since, even though the first conjunct is true, the second conjunct is false. The most familiar way to deny a statement is to prefix the words 'it is not the case that' to it. For example, we may deny 'Richard is a pale person' in this way: It is not the case that Richard is a pale person.

Other ways to deny this statement would be It is not true that Richard is a pale person. It is false that Richard is a pale person.

We are all familiar with the process of joining two statements together by using 'and'. For example 'Richard is a pale person' and 'Richard is a poet' may be so combined as follows: Richard is a pale person and Richard is a poet. or, simply, Richard is a pale person and a poet. According to the usual meaning of 'and', P and Q is true iff P along with Q is true. We will now introduce the symbol A and so understand it that it will correspond to this use of 'and'. P A Q is called a conjunction, and the components P and Q are called conjuncts. As indicated above, P A Q will be so understood that it is true iff P along with Q is true. This interpretation for the symbol A can be given

The next symbol, ""', will be understood to correspond to the words 'it is not the case that' in the natural language. We thus write the negation of P as ""P. "",P is called the negation of P. In the natural language the negation of P is true when P is false and is false when P is true. The same will be true with "",P, as the following basic truth-table indicates: P

"",P

T F

F T

If Pis T, then ,......,p is F, and if P is F, then "",P is T. Thus

"",Mao is a poet.

using the table below:

P

Q

T T

T

T

F

F F

F

F F F

T

4.

Richard is not a pale person.

Conjunction and denial

--,~

Conjunction and Denial

PAQ

is false, whereas "'" ""Mao is a poet. is true since Mao is a poet.

.,

16

18

Sentential Language

It is not difficult to build compound statements using us do this. Suppose R: P:

Richard is a pale person. Richard is a poet.

(1)

R A ,-..,p

A

and '--'. Let

then

is the compound statement: Both Richard is a pale person and it is not the case that Richard is a poet. Or by using parentheses as we use punctuation in our natural language, we can obtain

19

Exercises 1. F~rmulate the statements corresponding to each of the form mme the truth-value of the statement. ulas and deterP: 5 + 2 = 7. Q:2+4=7. (a) P A Q (b) ,--,(P A Q) (c) ,-...;P A ,......,Q (d) ~'"'-'P (e) ,-......, ,--,(,-...,p A ,-....,Q) (f) P A Q A P A Q

Answers (2)

,,-,(R A P)

1. Only (b) and (d) are T.

"it is not the case that both Rand P". Other examples of compound statements would be: 2.2 (3)

r-..>R A ,,-,P

and

(4) r-v(,--,R A ,....,P)

"it is not the case that R and it is not the case that P" and "it is not the case that both it is not the case that R and it is not the case that P." We may also inquire whether such compound statements are true or false. Given, say, that R is T and Pis F, is (1) T or F~ It is T. To figure this out, we first replace the capital letters with the indication of the truth-value of the statements they express. Thus with (1) we obtain

since ,,-,F is T (see the table for ,-..,) and since TAT is T (see the table for A), (1) is T. We may work out the truth-value of (2), (3), and (4) as follows, given that R is T and P is F. ,-,.,(T A F) ,.....,( F ) T

It will prove useful to introduce more stat . . emen t connectIves namel connectIves corresponding to 'or' 'if then' d "f d I" Y . . '" an 1 an on y If'. One wa to ~~ thIS IS by ~efining our new connectives using A and "-' Th~ addItIOnal connectIves we will use can be . t, d d' . . follows: m 10 uce m thIS way as (1)

P V Q =dr r_+",p A "-'Q).

(2) (3)

P-+Q=dr"'-'PVQ. P
In (1) V corresponds to 'or', in (2) -+ corresponds to 'if then' and' (3) <~> corresponds to 'if and only if'. ' , m

T A,.....,F

(2)

Other statement connectives

(3)

,.....,T

A

,-,.,F

FAT F

(4)

",-,(,-,.,T A ,--,F) ,-,.,(F AT) ,-.v ( F ) T

The symbols ,....., and A are both called stcttement connectives, even though '--' is a modifier. We might well think of statement connectives as operating on statements. ,-.v can be looked at as a one-place connective because it operates on one statement, and A is a two-place cOl!,~ective because it operates on two statements.

Each of the definitions indicates that h t . . ,_ , . " w a IS on one SIde of the -df SIgn IS Just another way of writing what·s th tl . 'I'h P Q' . 1 on e 0 leI' SIde us V IS Just a shorter way of writing r-..;(,-.....,P A ""Q) d' r-..J(,......,p A r-..-Q) h tl h ' an . as exac y t e same meaning as P V Q Wh t ' th mcamng of "-,(",,P A ,-.....,QP ( . . a IS e I h .,....., ",-,p A ,.....,Q) IS T whenever P is T or Q is T and' F ]S on y w en both P is F and Q' F Tl' . the truth-table for V is IS . lIS bemg the case,

PVQ'

P

Q

PVQ

T T F F

T F T F

T T T F

. IS called a disjtlnction. The comlJonents of P called disjuncts.

V

Q,

P and Q, are

20

Sentential Language

Proceeding along the same lines, the truth-table below gives the intended interpretation of ---->- and +->. P T. T. F F

Q T.

P---->-Q

T.

T..

F T. F

F·

F F

T. T.

2.3

Determining the Truth-Value of Compound Statements

2. Formulate the statement corresponding to each of the formulas and determ' the truth-value. me P: A triangle has three sides. Q: A triangle has four sides. (a) P ->-Q (b) Q ---->- P (c),--,P---->-Q (d) ,--,Q --+ P (e) ,--,P V,-...;Q (f) ,--,(P V Q) (g) P+-> Q (h) ,--,P+-> Q

T.

This table shows that P ---->- Q is false only when P is T. and Q is F, otherwise it is T; and P +-> Q is T iff P and Q have matching truth-values. P --+ Q is called a conditionctl, with P the antecedent and Q the consequent. P +-> Q is called a biconditional. As indicated above, ---->- corresponds to 'if, then'. The truth-table interpretation of --+ fails to capture the full meaning of 'if, then' as it is often used in the natural language. If this is not obvious, then the discussion in Section 2.4 will make this clear. ,Vhat is well to emphasize at this point is that, first, P ---->- Q is equivalent to ,-....;p V Q, which is equivalent to r--.;(P /\ ,-...;Q). This means that to assert P ---->- Q is to assert that it is not the case that P.is true and Q is false. Whenever we assert 'if P, then Q', we mean at least that it is not the case that the antecedent is true and the consequent false. Second, it is this feature of the use of 'if, then' that is sufficient for mathematical and scientific use of 'if, then'. All that we need to mean by 'if P, then Q' in standard mathematical usage is ,-....;p V Q or ,-...;(P /\ ,-...;Q). The above truth-table interpretation for these statement connectives also corresponds to the standard uses in the sciences. We 'will call a statement that uses no statement connectives a simple or (ttomic statement, and a statement formed by using connectives, a compound or molecular statement. Thus a more accurate name for the five symbols just introduced would be compound statement forme1's rather than statement connectives. Nevertheless we will continue to speak of these five symbols as statement connectives.

2. (a) F, (b) T, (c) T, (d) T, (e) T, (f) F, (g) F, and (h) T.

Exercises

3. Only (b) is simple.

1. Give the truth-value of each of the following statements. (a) 2 > 1 /\ 1 > 2. (b) ,,-,(2) 1 /\ 1 > 2). (c) 2 > 1 V 1 > 2. (d) ,-...;(2 > 1 V 1 > 2). i (e) '--'(2) 1 ---->- 1 > 2). (f) 2 > 1 --+ 1 > 2.

(g) 1 > 2 ---->- 2 > 1. (h) 1 > 2 +-> 2 > 1. (i) r-v(1 > 2 +-> 2> 1).

m (1 >

2 /\ 1 > 2)

--+

1 > 2.

21

3. Which. of the . (a) WIlson IS (b) Wilson is (c) If Wilson

following statements are simple and whl'ch are compoun1 d not on the job. . on the job. is on the job, then Bob is on the job.

4. The statement connectives introduced in this chapter are tmth-f1tnctional because· the compound statements formed by using tllem are t rue or £aIse depend Illg t N on the truth-value of the simple component t t 'k ( ) tl ' T ' s a emen s. ow It K e mean now slat. 0 express that 'a knows that', we write Ka. A compound sentence can now be formed using this s b I F Itt' P b 'M t' t I h ym o. 'or example c Illg e ar III s 0 e t e cat', we can write 'Dick knows th t M ~t' ' a al III stole the cat' as (1) KaP. If we allow => to read ' - - - is less probable than ____ " we can form compound sentences such as (2) P=> Q. Show that neither K nor => is truth-functional. Show this also for 'because'.

Answers I. (a) F, (b) T, (c) T, (d) F, (e) T, (f) F, (g) T, (h) F, (i) T, and (j) T.

2.3

Determ"mmg th e truth-value of compound statements

!:1 f~;'ming compound formulas or statements we can encounter am19l1l

y. For example how are we to understand this formula: (1)

P /\ Q V R

Sentential Language

22

Is it a conjunction with one conjunct a disjunction, or is it a disjunction with one disjunct a conjunction? In other words, is (1) to be read (2)

Both P and Q or R.

(3)

Either P and Q or R.

or

P, and Q or R.

2.3

Determining the Truth-Value of Compound Statements

23

Capital letters are here used as sentential constants; they are abbreviations, stand-ins, for actual statements. A little knowledge of U.S. political history at the beginning of the twentieth century will reveal that Rand B are true, whereas Wand L are false. The question can be raised, is the following compound statement true or false?

P and Q, or R. R A (W V r->L)

To remove this ambiguity we continue to use parentheses. To indicate To solve this, we first indicate the truth-values of the simple statements as follows:

(2) we will write: P A (Q V R)

T A (F V,....",F)

and to indicate (3): Next we consider the connective with the least scope. Each occurrence of a connective connects or operates on one or two parts of a statement, called the scope of that occurrence of the connective. Parentheses clearly mark out the scope of connectives when more than one connective is used. In the above case:

(P A Q) V R

Intuitively

""PVQ is understood as not P or Q. But if we wish to negate P

V

Q we will write:

connects to L. connects Wand ,...., L. A connects Rand (W V ""L).

r->

V

,,-,(P V Q)

(Note the difference between r-..;(P V Q) and ,...."P V ,-....,Q and ,.....,(P and ,-....JP A ,......,Q.) ·When we have strings of conjunctions

A

Q)

or disjunctions

Thus, in the above statement ,-..." has the least scope and A has the greatest scope since,......" connects the least number of parts, and A connects the greatest number. Starting with the operator with the least scope, we ask if the compound it forms is T or F. ,-....,F is T, so we have T A (F V T)

PvQVR we need not use parentheses (though we can if we like) since (P A Q) A R is equivalent to P A (Q A R) and (P V Q) V R is equivalent to P V (Q V R). For the purpose of becoming acquainted with using the five connectives, let us use the following capital letters for the indicated

statements. R

= Theodore Roosevelt was the only presidential nominee of the

Bull Moose Party. William Jennings Bryan at one time lent his name to real estate selling in Florida. W = ·Woodrow ·Wilson had a lovable personality. L = Senator Robert La Follette was a gracious loser. B

=

The V connective is now the operator with the least scope, and since F V Tis T, we obtain TAT

and since TAT is T, the answer to whether the statement we started with is true or false is that it is true. This problem could be solved as follows: T A (F V ,,-,F) T A (F V T ) T A (T )

l'

Sentential Language

24

2.4

Statement Connectives and the Natural Language

25

it follows that the next statement is false: To consider a second example, what is the truth-value of (2) r--'(E

-4-

((r--'W V B)H r--'E))

. Working this in succeSSIVe st eps, we find, as follows, that the answer is T: r--'(T -4- ((r--'F V T)H r--'T)) V T)H F )) ,-.J(T -4- ((T r--'(T -4- ( T H F ))

c-->(T -4r--'( F T

(

F)) )

Exercises 1. Determine the truth-value 0 f each 0 f tl1e f 0 11ow ing formulas, given the indicated truth-values of its components. P:T Q:F E:T (a) ((P -4- Q) /\ Q) -4- P (b) r--'(PH (Q V E)) (c) ((P-4-Q) /\ (Q-4-E))--+ (P-4-E) (d) (P --+ Q)H (r--'Q -4- r--'P) (e) ((P -4- Q) H (r--'Q -4- r--'P)) /\ E (f) ((P /\ Q) -4- E) -4- (P -4- (Q -4- E)) (g) r--'(r--'(Q --+ r--'P) --+ (P -4- E)) --+ r--'(P /\ Q)

Answers 1. Only (b) is F.

2.4

Part of what is implied, normally, by (1) is that John got well after he had an operation, which, if true, makes (2) false if we understand 'and' in (2) to be used like 'and' in (1) to express a temporal sequence. Now if 'and' as used in (1) and (2) were synonymous with /\, then (1) would be true iff (2) were true. This thus shows one use of English 'and' that differs from /\. To take another familiar example, from the truth ofthese statements: Tom came to the party. Jane came to the party. it does not follow that (3) Tom and Jane came to the party. if (3) is understood in one of its most popular senses-to express that Tom and Jane came to the party together. It is easy to see that (3) would follow if 'and' in (3) were replaced by /\. It may be that as more and more people take logic and mathematics courses, the use of 'and' and the other connectives in the natural language not in accordance with truthtables might someday come to be regarded as deviant or incorrect, but this is certainly not true today. 'Or' in the natural language can have what has become known as an exclusive use. Often we so use 'or' that the compound statement it forms is true iff one or the other disjunct is true, but not true when both are true. For example, if one said Hale Boggs is either a Senator or a Congressman.

Statement connectives and the natura/language

" an d' ,1,. '·f then' ' That the connectives of the natural language- ,not,"or, and 'if and only if' -are used in a variety of ways and often are not ll1terchangeable with r--', V, /\, --+, and H , respectiv~ly, is dis~uted b.y v.~ry £ For our purposes it counts for little to reVIew the wIdely dIffenng ues:~ of English connectives. However, it will help to il:umin~t~ the nature and importance of the formal language for sententIalloglC If we take notice of some of these uses. As 'and' is sometimes used, from the truth of this statement: (1)

John got well and had an operation.

John had an operation and got well.

one would be saying that Hale Boggs is either a Senator or a Congressman but not both. This use of 'or' is commonly called the exclusive use. The use of 'or' which would correspond to V is thus called the nonexclusive or inclusive use. A capital v, V, could be introduced and interpreted so as to correspond to the exclusive use of 'or'. Accordingly we would supply the truth-table below for V: P

Q

PVQ

T T F F

T F T F

F T T F

Sentential Language

26 Or we could introduce V by this definition: P V Q =df (P V Q) A ",-,(P A Q)

It is interesting to note also that 'or' in the natural language is often used to express an exhaustion of alternatives. This is best explained by an example. In one familiar context in which one would say that Rio Mundi is either a socialist country or a capitalist country, the statement would imply that these are the only alternatives. These two alternatives-being socialistic or capitalistic-exhaust the possibilities. When such exhaustion is implied, 'or' is not used truth-functionally since the truth-value of the compound statement is not an exclusive function of the truth-value of its components. Turning to 'if, then', it is now commonplace to say that there are many uses for this important connective in the natural langugage. Consider, for example, this list: (4) (5) (6) (7) (8) (9) (10)

If the sun shines, then I will swim. If it snows, Saul won't show up. If iron is heated sufficiently, it will glow red. If x is a body, then x does not travel faster than light. If x is red, then x is not blue. If there are rules, then there is agreement in a society. If Stevens is a doctor and all doctors are well paid, then

Stevens is well paid. (11) If 8> 7 and 7 > 6, then 8> 6. (12) If McCartliy had run, Nixon would have lost. These are not statements using --+, for if they were, then for each one, the fact that the antecedent is false or the consequent is true would alone be sufficient to establish the truth of each of the conditionals. In fact, no combination of true/false antecedents and consequents by itself would be sufficient to declare any of the above true. We must distinguish, then, the uses of 'if, then' in the above from --+, remembering that P --+ Q is equivalent to ,-....,(P A "'-'Q). In (10) and (11), at least, we can say that the antecedent entails or implies the consequent, these being analytic statements. With (9) we have a conceptual connection, and with (6) a casual one. Statement (12) provides us with an example of a counterfactual conditional, a statement that may be true or false, even though both the antecedent and the consequent are false. Statement (4) is used to inform someone of the speaker's intentions and (5) is used to predict the beh~vior of another person. However, it should not be overlooked that whenever we use, 'if, then', what we say is false if the -,-

2.4

Statement Connectives and the Natural Language

27

antecedent is true and the consequent false. The symbol for the conditional, --+, thus captures a common feature of all uses of 'if, then' , since P --+ Q is equivalent to ,-....,(P A r-->Q). And it is the maintaining of this feature, as indicated earlier, that is sufficient for the use of 'if, then' in mathematics and scientific contexts. The obvious gap that exists between the statement connectives and their counterparts in the natural language need be no cause for alarm', rather it makes the activity of this chapter all the more necessary. First, it is our desire or goal to form systems that will generate all the valid and only the valid sentential formulas. If the class of formulas were in part formed by connectives in the natural language, their multiple and changing uses would make such systems impossible. Second, the techniques that we will begin to develop in the next chapter for determining the validity of arguments are to be applied to arguments employing the statement connectives. The truth-table interpretation for the connectives conforms to standard mathematical and scientific usage. When, however, we come upon an argument in a nonscientific context, methods of paraphrase will be at hand and standards for correct paraphrasing will be given so that arguments in ordinary contexts can often be appraised by the forthcoming standard logical techniques. More will be said about this last point after the reader becomes acquainted to some extent with the terms of the formal language and with the structures of statements. Exercises 1. Show that each of the italicized connectives in the following statements, if replaced by the indicatedsta tement connective, may not preserve truth-value. (a) Ruth got married and had twins. (A) (b) I won't go unless she apologizes. (V) (c) I love you but I love your sister as well. (A) (d) LBJ would have won in '68 if he had run. (--+) (e) Either you drop that stick or I'll tell Mother. (V) (f) He read the logic manuscript and got apoplexy. (A) (g) There is coffee on the stove if you want some. (--+) 2. Provide a reading for P V Q. 3. Given P: ']' and Q: F, determine the truth-value of each of the following formulas: (a) ,-...,p V Q (b) P +--+ (P V P) (c) (P V Q) V (P V Q)

4. I~valuate the following claim: The statement 'If P then Q' asserts no more than r--..;(P A ,-....,Q), though one may say it because one knows or believes that some non-truth-functional connection exists between P and Q.

28

Sentential Language

2.5

Putting the Natural Language into Symbols

P if and only if Q P if Q, and Q if P P exactly if Q P is (a) necessary and sufficient (condition) for Q PiffQ If P then Q, and conversely

Answers (a) 'Ruth got married A Ruth had twins' is equivalent to 'Ruth had twins Ruth got married'. (g) If there is no coffee on the stove, then (g) is false. 3. (a) F, (b) F, and (c) T. 1.

2.5

A

If P, then Q When P, then Q In case P, Q Q provided that P P is (a) sufficient (condition) for Q Q is (a) necessary (condition) for P P implies Q Q if P Q when P Q in case P P only if Q P only when Q P only in case Q

Putting the natural language into symbols

In translating the natural language into the sentential language some confusion can arise because of the various ways in which we express biconditionals, conditionals, conjunctions, and disjunctions, and the various ways we have for denying statements. In paraphrasing the natural language into our notation it will prove useful to have a list that covers the most frequently encountered idioms that can be expressed using statement connectives. (In the chart below we assume that the use of the natural language conforms to standard mathematical and scientific usage.) Below is such a list. 1 In each section each statement on the right can be expressed using the statement on the left. For example, in the fourth section we find that P unless Q can be expressed as P V Q (which is equivalent to r-vP -+ Q). In the second section we see that P only if Q is equivalent to P -+ Q, whereas P if Q comes to Q -+ P. Sometimes 'or' is used in the inclusive sense and sometimes in the exclusive sense. When 'or' is used in the first way we have P V Q, and when it is used the other way we have (P V Q) A r-v(P t\ Q). These two ways are indicated in the chart by placing '[sometimes]' after P or Q. Since the use of 'unless' and 'except when' can also express the exclusive sense of 'or', [s]'s are placed after them in each section. The words 'necessary' and 'sufficient' are frequently used in various contexts. For example we may say that having a right angle is necessary for a figure to be a square, and we may say that a figure's being a square is sufficient for a figure to have a right angle. Vlfe can express these relations using -+ as follows:

PAQ

PandQ PbutQ P although Q Both P and Q Not only P but Q P despite Q PyetQ P while Q

PvQ

P or Q or both P or Q [s] P unless Q [s] P and/or Q [in legal documents] Either P or Q [s] P except when Q [s]

(P V Q) A ",-,(P A Q)

Either P or Q [s] P unless Q [s] P except when Q [s] P or else Q [s]

x is a square -+ x has a right angle.

Q, in other words, can be read "Q is a necessary condition for P" or "P is a sufficient condition of Q." P ~ Q, in turn, can be read 'P is a P

-+

1 This list is a modification of a list in Stephen Cole Kleene, ~Mathematical Logj:c (New York: John ~Wiley & Sons, Inc., 1967), pp. 63-64. Adapted by permission of the Pl1,blisher.

P or Q but not both P or Q [s]

",(P V Q)

Neither P nor Q Not P [or the result of transforming P by putting 'not' just after the verb or an auxiliary verb] P does not hold It is not the case that P P isn't so

29

so

Sentential Language

necessary and sufficient condition for Q'. (These relationships are all set down in the chart above.) To say P only if Q is to say Q is necessary for P, thus P -+ Q, whereas to say P if Q is to say Q is sufficient for P. To say P provided that Q is to say Q is a sufficient condition for P.

2.6

Answers 1. (a) P -+ E, (c) A 2. (a) P-+E ~E

~P

2. Paraphrase the following arguments into the sentential language. Allow letters to stand only for simple statements. (a) A student passes logic only if he does the exercises. But students never do exercises. Therefore a student never passes logic. (b) He will not come unless it rains. It is raining, so he will come after all. (c) If we do not draw and do not lose, then we win. Therefore if we do not draw then we win, or if we do not lose then we win. (d) P provided that neither Q nor R. T, if Rand S. If U, then S but not T. P and Q being true is sufficient for U being false. (e) If P implies Q, then R. Either not T or not U is necessary for both Rand S. S If Q, then neither P nor U. (f) P, if M; but if P, then either L or H. Now H if and only if A. On the

other hand, not L unless not P. Not P only if S and A. So a sufficient condition for H is M. 3. Try to determine whether the conditionals corresponding to each argument are analytic. If the conditionals are analytic, what does this imply about the argument? 4. For each pair below state all necessary and sufficient relationships that exist. (a) x is a normal man; x is a biped. (b) x is a nickel; x is worth five cents. (c) x is divisible by 2; x is an even number. (d) x is greater than 6; x is greater than 34 (e) x is a flaming match; x is struck against a match book. (f) x is a philosophy teacher; x is a radical. (g) x's action is freely done; x's action has a cause. (h) x thinks; x is a per~on. '"

-+

R, and (e) (E

A

D

A

I)

-+

(e) (P-+Q)-+R

(R AS) S

Exercises 1. Paraphrase the following statements into the sentential language using the suggested letters for the simple statements. (a) Only those who do exercises will pass logic. (E, P) (b) This litmus paper turns red if it is placed in acid. (R, A) (c) This litmus paper has been placed in acid only if it turns red. (A, R) (d) You won't pass the course unless you do the exercises. (P, E) (e) If you do the exercises you will pass the course provided that you are diligent and intelligent. (E, P, D, I)

31

Formation Rules for the Sentential Language

-+

(f) M-+P P -+ (L V H)

(,""T V ,,-,U)

Q-+~(P V

P.

H~A V "-' P ,....,P-+ (S A A)

"-' L

U)

M-+H

3. Only (a), (c), and (f) have analytic conditionals. The argument is valid, the premises logically imply the conclusion.

2.6 Formation rules for the sentential language

It is time that we made quite clear exactly what is being called the sentential language. Up to this time the reader has noticed that the sententiallallguage uses symbols for statement connectives and P, Q, ... as variables for statements. Parentheses are also employed to remove any ambiguity. The reader has also, more or less, been able to see when these elements are properly put together. But it is now time to make all this oxaot. What is being called the sentential language essentially consists of three elements: First, a collection of symbols; these are: (a) Statement variables: P, Q, R, ... (b) Statement connectives: r-v, A, V, -+, (c) Parentheses: (, )

and~

(The three dots, ' ... ' indicate that we suppose we have an infinite number of statement variables available. When we reach the end of the alphabet we can continue the series by using subsoripts PI' Qv ... and then P 2, Q2' ... , and so on.) Second, a truth-table interpretation for the statement connectives. Third, rules for when we have a proper string of symbols. The reader has been able to rely on his intuition to determine when one has a proper string of symbols. For example, we intuitively reject the following: PQ

vP

Pv (Q-;.-+Q)

,...,yp

,,-,(PlY Q)

as proper strings of symbols. And on reflection we see that the criterion that controls what we regard as a proper string is this: Proper strings of symbols can have statements as instances, but improper strings in general do not.

32

Sentential Language

In this section we wish to set down rules by which we can determine in all cases whether or not a string of symbols, a formula, is a proper string, whether it is what is called a well-formed formula. We wish Our notion of a well-formed formula to be an effective notion. We have an effective notion of a well-formed formula when we can settle in a routine fashion whether any possible formula is well-formed or not. To be more specific, the notion of a well-formed formula is effective when a digital computer can be programmed to check whether any formula is wellformed or not. If no intelligence beyond the ability to follow clerical instructions is necessary to check whether a formula is well-formed or not, we have an effective notion of a well-formed formula. The rules we set down will insure that the notion of a well-formed formula is effective. The rules that provide the procedure for determining whether a formula is well-formed are called the formation Tules. The formation rules for the sentential language may be given as follows (abbreviating 'well-formed formula' as 'wff'): 1. Any statement variable is a wff. 2. Any wff preceded by r - ; is a wff. 3. Any wff followed by Y followed by a wff, the whole enclosed in parentheses, is a wff. 4. (3), replacing Y with A. 5. (3), replacing Y with -->-. 6. (3), replacing Y with~.

To complete the definition of a wff we need add that a formula is wf iff its being so follows from the formation rules. (This condition will henceforth be supposed in formation rules for wffs.) A useful device in stating these formation rules is to use a term that can range over wffs. We will use the capital letters A, B, and 0 for this purpose. A, B, and 0 are thus variables for wffs and are called metalogical variables. The prefix 'meta' ('about') indicates that they are not part of the language in question-the sentential language-but are used to talk about the language in question. P, Q, ... have statements as instances; A, B, and 0 have well-formed formulas in the sentential language as values. Using these metaterms we may restate the formation rules in this manner: 1. 2. 3. 4. 5. 6.

Any statement variable is a wff. is a wff, then r-;A is a wff. and Bare wffs, then (A Y B) is a wff. and Bare wffs, then (A A B) is a wff. and Bare wffs, then (A -->- B) is a wff. and B i1re wffs, then (A ~ B) is a wff.

If A If A If A If A If A

Formation Rules for the Sentential Language

33

To see how these rules provide us with a procedure to determine whether any formula is a wff we show that (( (P -->- Q) A C"-'P) -->- Q)

is a wff in the following way: P is a wff by rule 1. Q is a wff by rule 1. (P -->- Q) is a wff by rule 5. r-;p is a wff by rule 2. ((P -->- Q) A r-;P) is a wff by rule 4. ( (( P -->- Q) A r - ; P) -->- Q) is a wff by rule 5.

On the other hand, since we cannot show, for example, that (Y P

-->-

Q)

is a wff by using rules 1-6, it is not a wff. It is important to note that by these rules

for example, is not a wff. The reason for this is that given rule 5, A -->- B must be surrounded by parentheses if it is to be a wff. Thus (1) must be surrounded by parentheses to be a wff. This is necessary to remove the possible ambiguity that was discussed in Section 2.3. We do not wish to admit, for example, (2) P A Q -->- P

as a wff.s~nce we cannot tell if instances of this formula are conjunctions or condItlOnals. The requirement to insert parentheses when using the two-place connectives eliminates the possibility of such ambiguous formulas. However, no ambiguity results when the outer parentheses are dropped, so to reduce the number of parentheses it is convenient to adopt the convention of dropping the outer parentheses. Thus (1) will be regarded as a wff though (2), of course, is still not a wff. . There is another way to reduce the number of parentheses that appear m form~las. We can Tank the connectives in relation to scope. The connectIves may be ranked in this order: First ~ Second -->Third A, Y Fourth

Sentential Language

34

f-* has the highest rank, with A, V having the same rank. To illustrate this convention, formulas with and without parentheses appear below:

(P A Q) ---+ P P f-* (Q ---+ P) r-.;P ---+ (r-.-P V Q)

(P A Q) ---+ (P P ---+ (Q f-* P) P V (Q A R)

---+

Q)

PAQ---+P Pf-*Q---+P ,,-,p ---+,....,P V Q P A Q ---+ (P ---+ Q)

no change no change

"

.

eCISIOn

ro edu

It should be stressed that these conventions are practical moves to make it easier to write formulas and to see the structure of formulas. Formally speaking, what counts as a wff in the sentential language is what satisfies the formation rules. With the sentential language in hand, we are now in position to introduce the effective procedures by which it can be determined whether a well-formed formula is valid or not. Such procedures are introduced in the next chapter.

PIIQ->-P

Examples of invalid would be those below: P

2. Select a formula from this chapter, rewrite it so that it is a wff, and then demonstrate that it is a wff. 3. State precisely why each of the following formulas is well-formed or why it is not well-formed. (a) r--.lP(---+ (Q V R)) (b) ,......,p V Q A r-.JR (c) (P ---+ A) t"'-'Q (d) (Q ---+ R) ---+ ((P V Q) ->- (P V R))

4. Use conventions to eliminate as many parentheses as possible.

---+

(P A Q))

Answers 1. No, V is neither a statement variable nor one of the statement connectives appearing at the beginning of Section 2.6. 3. (d) is not a wff since A

->-

4. (a) (P ->- Q) A r-->Q ->- ,......,P, ---+

PA,.....,P

formulas

P->-Q

(P->-Q) AQ->-P

1. Is V a term of the sentential language?

(a) (((P ---+ Q) A r--IQ) ---+ ,......,P) (b) ((,-....;p ---+ P) ---+ P) (c) (P ---+ (Q ->- (P A Q))) (d) ,.....,(P ---+ (P ->- Q)) (e) (P ---+ (Q V (R A Q))) (f) ((P V Q) ->- (P V Q)) f-* ((P V Q)

P->-P

(P->-Q) A P->-Q

Exercises

(d} no change, (e) P

Earlier, in Ohapter I, the point was made that some well-formed formu!as in the sentential language are valId (have only necessarily true statements as instances). Examples of such formulas would be the following:

B is not a wff. Outer parentheses are needed.

(b) (,......,P ->- P) ---+ P, (c) P ---+ (Q ->- P A Q), Q V (R A Q), (f) P V Q ---+ P A Q +--+ P V Q ->- P II Q.

An important question we may now ask is whether there is some effective or mechanical decision procedure by which we can tell whether ~ form?la in the sentential language IS valId or not. A decision procedure is mechanical for determining whether a formula is valid or not when we could programme a comp~ting machine in such a way that WIth respect to any formula it would at some point give a 'yes' answer if the formula Were valid or a 'no' anSWer if the formula were not valid. If there is a mechanical procedure for determining whether wellformed formulas in the sentential language are valid, then the notion of a valid sentential formula is an effective one. It turns out that there are several mechanical decision procedures for determining whether a sentential well-formed formula is valid or not. Two of these methods

"

35

36

Decision Procedures

are discussed in this chapter: truth-table methods and transformation into what is called conjunctive normal form.

3.1

Truth-tables

Our first procedure for determining whether a well-formed formula in the sentential language is valid or not is the truth-table method. To introduce this method we must first consider how one can construct a truth-table for any well-formed formula of the sentential language. The procedure is quite simple. Suppose we wish to construct a truth-table for this formula:

A truth-table for a formula is a table in which the possible combinations of truth-value for the component variables of the formula are listed and the resulting truth-value of the formula for each possible combination is indicated. To make this clea~ we observe that formula (1) contains two different statement variables. This means that there are four possible ways in which truth-value may be assigned to these statement variables, which may be set down as follows:

P T

Q T

T F

F T

F

F

If the formula under consideration had three variables, rather than two there would be eight possible ways in which truth-value may be assigned to these statement variables, namely: P

Q

T T T T F F F F

T T F F T T F F

R T F T F T F T F

3.1

Truth-Tables

37

in what follows is a variable for the positive integers 1,2, 3, ... ). One term m~ans ~1 or t.wo lines, two terms means 22 or four lines, three terms mean 2 or eIght hnes, four terms mean 24 or 16 lines, and so forth. The truth-table for (P -+ Q) A P -+ Q is constructed as follows: P

Q

T T F F

T F T F

T F T T

T F F F

T T T T

Each line is obtained by following procedures discussed in the last chapter. On the first line we are given Pas T and Q as T. Starting with the connective with the narrowest scope, the first -+, we enter a T under it since T -+ T is T. We next enter a value for the connective which now has the narrowest scope, namely A. Since TAT is T we place a T under A. The last -+ is the connective with the widest sc ope. The value of the antecedent is given under the A , namely T . S'Ince T -+ T is T we place a T under -+. The second -+ is the main connective (the connective with the greatest scope). The first line thus shows that if Pis T and Q F, then the formula is T. A truth-table can be constructed for any sentential formula. However, given the rule that for a formula with n variables we need 2" lines to provide for all its truth possibilities, truth-tables become cumbersome when we have more than three variables. Exercises 1. Construct a truth-table for the following formulas. (a) P V ""P (b) P A ""P (c) (P -+ Q) -+ (Q -+ P) (d) ""Q -+ "-'P +--t P -+ Q (e) P V (Q V R) +--t (P V Q) V R (f) ",,(P A P +--t P) (g) (P -+ Q) A (Q -+ R) -+ (P -+ R) (h) ",,(P A Q) +--t '"'-'P A '""-'Q (i) ""(""(Q -+ ",-,P) -+ (P -+ R)) -+ ",,(P A Q)

2. Suggest some ways to reduce the number of T's and F's in a truth-table. Answers

In general if a formula contains n number of variables, then the possible ways in which truth-value may be assigned to the components can be calculated in this way: 2" = number of lines on truth-table (n here and

1. You ~hould get all ~'s under the main connective of (a), (d), (e) and (g), all!, s under ~he mam connectives of (b) and (f), and T's and F's under the roam connectIve of (c), (h) and (i).

38 3.2

Decision Procedures

Valid, inconsistent, and contingent formulas

In Chapter 1 (Section 1.2) we noted that statements can be divi~ed ~nto those which are necessarily true, e.g., 'It is false that Amenca IS a republic and is not a republic'; those which are necessa~'ily false,. e.g., 'America is a republic and is not a republic'; and those winch are neIther necessarily true nor false but are true or false depending on e~isting states of affairs, e.g., 'America is a republic'. The first group of statements we may speak of as necessarily true, the second as inconsistent, and the third as contingent. Somewhat analogous to this last classification, well-formed formulas of the sentential language can be classified as valid, inconsistent, or contingent. This classification may be explained as follows: A formula is valid if and only if all instances of the formula are true statements.

Valid, Inconsistent, and Contingent Formulas

or contingent. If a formula always comes out F no matter what truthvalue is assigned to its component variables, then it will always come out false no matter what the instance. Thus if a formula always has the value F under its main connective of its truth-table, it must be inconsistent. By the same reasoning it follows that if a formula has a mixture of T's and F's under its main connective of its truth-table, if it has neither all F's nor all T's, then it is a contingent formula. In other words, if we wish to know whether A is valid, inconsistent, or contingent, we prepare a truth-table for A. If all T's are found under its main connective, it is valid. If all F's are found under its main connective , it is inconsistent. If neither is the case, the formula is contingent. This procedure may be applied to the first examples above of valid, inconsistent, and contingent formulas. In the tables we will place the possible truth-values for P and Q directly under the variables in the formula:

A formula is inconsistent if and only if all instances are false statements.

(I) P

y

T F

T T

A formula is contingent if and only if by suitable choice of substitution the formula can have true instances or false instances. Examples of valid formulas are found in the first group below, inconsistent formulas in the second group, and contingent formulas in the last group.

py,.......,p P A Q-+ P (P-+Q)AP-+Q

P A,.....,P ,....",r--.J(PA,......,P)

P

,,-,(P A Q -->- P)

P Y

PYQ

C. . ..JP A P)

If a formula in sentential logic is such that it always has the value T 10 matter what truth-value we assign to its component sentential I variables then it must be a valid formula. We need not conS1'der instances' of a sentential formula in determining whether or not it is valid. For if a sentential formula always comes out true no matter what truth-value we assign to its component variables, then all instances of the formula must be true statements since the truth-value of the statement instance is solely a function of the truth-value of the component statements. Truth-tables provide a way of determining whether or not a formula comes out T no matter what truth-value is assigned to its component statement variables. Consequently we can use truth-tables to determine whether a formula in sentential logic is valid or not. In turn we can use the tables to determine whether a formula is inconsistent

39

r-JP F T

(2) P

A

r-Jp

T F

F F

F T

(3)

P T T F F

y

Q

T T T F

T F T F

We observe all T's under the main connective of (I), all F's under the main connective of (2), and a mixture of T's and F's under the main connective of (3). As noted above, contingent formulas can have true or false statements as instances. To illustrate this, note that the following necessarily true statements, If this object is red, then it is colored. 2 2 = 4. If all men are mortal and Socrates is a man, then Socrates is mortal.

+

are instances, respectively, of the following contingent formulas:

P P P

In addition, the following inconsistent statement is an instance of P and P A Q: Some men are wise, and no men are wise. A formula is a ta1ltology iff it takes the value T for every assignment

40

Decision Procedures

of truth-values to its components. All valid sentential formulas are tautologies. However, when the sentential language is expanded to include other kinds of symbols we will find logical formulas that are valid but that are not tautologies. Consequently it is useful to have classification for formulas that take the value T for every assignment of truth-value, and these are designated to be tautologies. (The explanation for exactly what is a component in any kind of logical formula-later to be called a basic component-is given in Section 6.1.) The notion of a tautology in the sentential language is effective since there is an effective, mechanical procedure for deciding whether or not a formula is a tautology-construct the truth-table for the formula. Exercises 1. Construct a truth-table for each formula and state whether the formula is tautologous, inconsistent, or contingent. (a) P --+ "" ""p (b) ",,(P V Q) --+ ""p A ""Q (c) (P --+ Q) --+,....,P V Q (d) ",,(P --+ P) (e) (P --+ Q) A (R --+ Q) --+ (P --+ R) (f) P A (Q V R) ~ (P A Q) V (P A R)

(g) (P --+ ""P) --+ ""p (h) (P A ""Q --+ R A ""R) --+ (P --+ Q) (i) P V F~ P 2. In exercise 3.1.1 sort the formulas out in terms of being tautologous, inconsistent or contingent. (3.1.1 means Chapter 3, Section 1, exercise 1.) Answers 1. (d) is inconsistent, (e) is contingent, and the rest are tautologous.

2. See answers for exercise 3.1.1.

3.3

3.3

The argument pattern of which (1) is an instance is (2) P --+ Q

""Q ""p (1) is obtained from (2) by substituting simple statements for P and Q, namely P: George needs a vacation. Q: Classes should end.

In Chapter 1 (Section 1.3) the point is made that an argument is valid iff its corresponding conditional (conjunction of premises--+ conclusion) is necessarily true. And a conditional is necessarily true if it is an instance of a valid logical formula. This being the case, the validity of (1) can be established by showing that (3) (P --+ Q) A

(1) If George needs a vacation, then classes should be ended. Classes should not be ended. Therefore George does not need a vacation.

~Q

--+ ""P

is a tautology. By using truth-tables we can quickly verify that it is a tautology, thus establishing the validity of (1). It is useful to put this in slightly different terms. Let us now speak of valid argument patterns. An argument pattern is valid iff it cannot have an instance such that the premises are true and the conclusion is false. If (3) is a tautology, then it immediately follows that (2) is a valid argument pattern. And if an argument pattern is valid, then any instance of the pattern is valid. Showing (3) is a tautology thus establishes that (1) is valid. To take a second example, suppose we wish to establish the validity of the following argument: If Tom is guilty, then if Robert is guilty, Jones is guilty. If Tom and Robert are guilty, then Jones is guilty.

Arguments and truth-tables

Truth-tables can be used to establish the validity of certain kinds of arguments, namely, those arguments whose validity is a matter merely of how statements are related by statement connectives. Suppose we were asked to consider the validity of the following (simple) argument:

41

Arguments and Truth-Tables

Let P: Tom is guilty. Q: Robert is guilty. R: Jones is guilty.

The argument pattern is P --+ (Q --+ R) PAQ--+R

Decision Procedures

43

Using truth-tables one may confirm that its corresponding conditional is valid; thus the argument is valid. We should note that if an argument pattern is invalid, it does not follow that an instance of the pattern is invalid. For example

And since it is valid, the argument is valid. But in symbolizing the argument in this way we did more than is needed if we follow the principle of correct paraphrase. No mistake is made in symbolizing the argument in the manner above. But following the principle, we could just as well have made these assignments:

42

P Q

is clearly an invalid argument pattern. However, the instance of this pattern below is a valid argument: Some men are lovable. Some lovable things are men. In the application of sentential logic to arguments (and to what is to come, predicate logic), the principle for correct paraphrase or correct translation must be observed. This principle may be stated as follows: Principle for correct paraphrase: The correct paraphrasing of a statement into the formal language is relative to a given argument context. The statement is correctly paraphrased when the structure of the argument relevant to validity is exposed. The force of this principle will only be appreciated as we build the formal language so that we are in position to expose the logical structure of a simple statement. At this time the language is rather limited since all we have are sentential variables and statement connectives. However, a few simple examples will help to illustrate the application of this principle. Suppose we wish to establish the validity of the following argument: If Miller's statement is meaningful, then it is either true or false.

His statement is neither true nor false. His statement is meaningless. If we follow the procedures used up to this time, we will let sentential variables replace simple statements, for example

P: Miller's statement is meaningful. Q: Miller's statement is either true or false.

and in so doing we would have expressed the logical structure relevant to the validity of the argument. For the argument is valid if and only if

is valid. It is not too difficult to so imagine the context that the following would be a valid argument, (4) Either we will disarm or there will be nuclear war. So since we will disarm, there will be no nuclear war.

If the first premise were symbolized P V Q, this would violate the principle for correct paraphrase. Insofar as (4) is valid, the first sentence must be understood as follows: Either we will disarm or there will be nuclear war and it is false that we will disarm and there will be nuclear war. Thus to expose the structure of (4) relative to the validity of (4), the first sentence may be paraphrased as; (P V Q) A r-.J(P A Q)

or, simply, r--..;(P AQ)

Earlier we noted that P: Miller's statement is meaningful. Q: Miller's statement is true. R: Miller's statement is false.

The argument pattern obtained would thus be

-"

Some men are lovable. Some lovable things are men. is a valid argument and it is an instance of

P-+QVR ,--,(Q V R)

P

r-.,/p

Q

44

Decision Procedures

3.4

A Short-Cut Truth-Table Method

If, for purposes of applying validating techniques of logic to this argu-

3.4

A short-cut truth-table method

ment, it were symbolized in this way, then the principle for correct paraphrase would be violated. The validity of this argument does not rest merely on how statements are related by statement connectives. Rather it rests on the meaning of 'some' and on how the expressions 'men' and 'lovable things' are related. We are not at this time in position to correctly paraphrase this argument. We must wait until additions are made to the sentential language that will allow us to expose the inner logical structure of simple sentences. (These additions are made in Chapter 5.)

45

Truth-tables enable us to decide in a mechanical way whether a formula is a tautology. But with formulas having more than three different sentential variables the tables pass beyond what is reasonable for a person to do. For example, to determine the validity of this moderately simple (valid) argument pattern (1) P

-+

C--'Q -+ R)

Q-+SAT

,....,T

Exercises 1. Use truth-tables to test the validity of the arguments in Section 2.5.2.

2. Let us call a sentential argument an argument whose validity is merely a matter of how statements are related by statement connectives. Symbolize and test the validity of those arguments below that are sentential arguments. If the argument is sentential, be sure that letters are used only for simple statements. (a) If either Nixon or Rockefeller wins the nomination, then McCarthy will be nominated and a Democratic victory will be assured. Therefore either Nixon will not win the nomination or McCarthy will be nominated. (b) Only servants of the Lord are missionaries. Not all missionaries are Catholics. Therefore at least one servant of the Lord is not a Catholic. (c) Either D or both Rand F. If D, then F. Therefore F. (d) P provided that Q if P. If P, then Q. Therefore P. (e) Not Q. Therefore not P unless if Q then R. (f) If P then Q and if ,....,p then R, so it is false that both Q and R. (g) P and not-Q imply Rand not-R, so P implies Q. (h) If you love her and give her gifts, then she will fall in love; but if you love her and do not bring her gifts, then she will not love you. Hence if you love her, she will love you if and only if you bring her gifts. (i) Anderson did not marry a beautiful woman because he is happy. For if he is happy, then he is not jealous, and we know that if someone marries either a beautiful woman or a very popular woman, then he will be jealous and disappointed.

we must have an unwieldy table with 32 rows. A short-cut method of determining validity is much in order. There are several short-cut truth-table methods. The one selected here may be outlined as follows: Let A be a well-formed formula. We suppose that A is not a tautology and proceed to fill in truth-values for the components of A until either a contradiction results or we can consistently assign a truth-value to each of its components. If a contradiction results, then our initial assumption that A is not a tautology is false. Hence A is a tautology. If we can consistently assign a truth-value to each of the components, then A is not a tautology. This method is called the reductio ad absurdum truth-table test. Often in mathematics and philosophy one proves P true by assuming not P, and from this assumption deriving a contradiction. If not P does imply a contradiction, then P must be true. The procedure in employing this useful technique to test the validity of sentential formulas consists of two steps: 1. Assume the formula is not a tautology by placing an F under the

main connective. 2. Follow out the consequences of this. If we arrive at a contradiction, the formula is a tautology; if we do not, it is not a tautology.

3. Prove the following: (a) If A (b) If

-+

B is a tautology, then

A

A

13 is a valid argument pattern. A

13 is a valid argument pattern, then any instance of 13 is a valid

argument.

]'01' example, we take this formula and assume it is not a tautology

(step 1): (P

-+

Q)

-+

(,--...,Q -+ ,....,P)

F

Answers 2. Only (b) and (i) are not sentential arguments. All the sentential arguments are valid except (f).

,.

An F can appear under the -+ in the above formula only if the antecedent is true and the consequent is false. P -+ Q must be T and

Equivalence and Transformations

Decision Procedures

46 ~Q -+

,-....;p must be F, which we will now indicate below: (P

-+

'1'

Q) -+ C--Q -+ r--.;P) F F

If "'-'Q -+ r--.>P is F then its antecedent must be T and its consequent F, and this can only occur if we place F for Q and T for P as follows: (P

-+

'1'

Q) -+ C""'Q -+ r-...;P)

47

F for P is consistent. So the formula is invalid. Note that had putting F for Q -+ R led to a contradiction, we could not have concluded from that cdonc that the formula is valid. In such a case we must also try T for Q -+ R, which might yield a consistent assignment to the whole formula and hence show it to be invalid. So whenever a choice is open, (tU alternatives must lead to contradictions if the formula is valid. The validity of the argument pattern (1) that appeared at the beginning of this section may be demonstrated by using the rcductio test on its corresponding conditional as follows:

F '1'F F F'J' -+ (,-....,Q -+ R)) A (Q -+ SA '1') A "",,'1' -+ (P -+ R) '1''1' '1'FFF F'1' FF '1'FF'1'FF

(P

All the occurrences of Q must have the value F and those of P, T, so: (P -+ Q) -+ C.....,Q -+ ,-....,;P)

Exercises

'1' '1' F F ,...., F F ,....,'1'

But now we have a contradiction-which may be indicated by a linefor P -+ Q cannot be T if P is T and Q is F. Thus our original assumption that this formula is F must be false. It is T and is valid. All this can easily be wri.tten on one line. For example, the proof of the invalidity by the rcductio method of

1. Use the red1lCtio method and state whether the following formulas are tautologous. (a) P V P-+ P

(Q -+ R) -+ (P V Q -+ P V R) (P <-+ Q) -+ (P -+ Q) A (Q -+ P) (,-....;p -+ R) A ",-,R -+ P ,....,(P A Q) -+ ",-,P (P A "-'Q -+ Q) -+ (P -+ Q) (g) (P V Q) A (P -+ R) A (Q -+ R) -+ R (h) (P<-+Q)<-+ (P A Q) V (,,-,P A ",-,Q) (i) (P -+ Q) V (P -+ R) -+ (P -+ Q A R) (b) (0) (d) (e) (f)

would proceed as follows: (P-+Q) A (R-+S) A (r-..;R VS)-+ (Q-+S)

'1' '1' '1' F '1' F '1' '1' F '1' F F '1' F F

We can consistently supply either Tor F for P in completing the assignment of truth-values. Thus following out the consequence of an F under the main connective has not resulted in a contradiction; the formula is not valid. In using the rcdnctio test one will occasionally encounter choicepoints. For example, the J'cdnctio test proceeds mechanically on the formula below up to the point indicated: (P V S) V (Q -+ R) -+ (P V Q -+ (S -+ R))

'1' '1' '1'

F F

'1'

F '1' F F

No further assignment of values is determined. We can, for example, put Tor F for Q -+ R. Let us put F, then Q is T and putting either Tor

"

Answers 1. All but (e) and (i) are tautologous.

3.5

Logical equivalence and transformations

Two formulas in logic, A and B, are said to be logically cqnivalent if and only if A <-+ B is a valid formula. With respect to sentential formulas, A and B are logically equivalent if and only if A <-+ B is tautologous. For example, the following pairs of formulas are logically equivalent: P AQ and Q A P "'-' "-' P and P

(P V Q) V Rand

P

-+

P V (Q V R)

Q and ",-,P V Q

48

3.5

since connecting them with a biconditional produces a formula that is tautologous. For example, that P -+ Q is logically equivalent to r-..Jp V Q is established in the table below.

can be transformed into

P

-+

Q

T T F F

T F T T

T F T F

~r-..JP

T T T T

F F T T

V

Q

T F T T

T F T F

Law 12 Law Law Law Law

13 14 15 16

(P

-+

Q)

~

(,-..,Q

P~r-..Jr-..JP

P

-+

r-vP)

P}

Q~ Q V A Q~Q A P P V Q ~ "-'(r-..JP A ,-...,Q)} V

P

P A Q~r-..J(r-..JP v,-...,Q) (P V Q) V R ~ P V (Q V R)} (P A Q) A R ~ P A (Q A R) P V (P A Q) ~ P} P A (P V Q) ~ P P A (Q V R) } ~ (P A Q) V (P A R) P V (Q A R) ~ (P V Q) A (P V R) P -+ Q ~ r-..Jp V Q (P ~ Q) ~ (P -+ Q) A (Q -+ P) P V P~ P} P A P~P

49

r-..J(P A Q) V R

since law 13 states that a conditional P -+ Q is equivalent to r-..Jp V Q. Similarly, the last formula can itself be transformed into

by using law 5, DM. And erasing double denial using law 2, DN, we can obtain:

Truth-table methods can be used to verify that each of the biconditional formulas below is valid. These are well-known and useful equivalences that have been given names (and abbreviations), as has been indicated. We will make frequent use of these equivalences in what follows. Law 1 Law2 Law 3 Law 4 Law 5 Law 6 Law 7 Law 8 Law 9 Law 10 Law 11

J,ogical Equivalence and Transformations

Contraposition (Con) Double Negation (DN) Commutation (Comm) de Morgan's Theorems (DM)l

Such transformations can continue indefinitely. In transforming A into B we use biconditional formulas that are valid. B obtained in this way is equivalent to A since we come upon B by successively replacing formulas with formulas that are their equivalents. If we made all the steps involved in these transformations explicit, we would see that we make use of two rules, a rule for substitution and a rule for replacing formulas with their logical equivalents. These rules may be set down as follows:

Association (Assoc)

Rl. Rule of substitution: A simple or compound sentential formula may be substituted uniformly for a single sentential variable. R2. Rule of replacement: If A is equivalent to B then A may be replaced by B whether A is alone or part of a formula.

Absorption (Abs)

Distribution (Dist) Implication (Imp) Equivalence (Equiv)

Returning to our first transformation, we are justified in moving from

to

Idempotent (Idem) given the rule of replacement and the logical equivalence below:

The first use of these laws will be to transform formulas. If A is logically equivalent to B we may transform A into B. For example, the conditional

1

After the English logician Augustus de Morgan (1806-78).

P

A

Q -+ R ~ r-..J(P

A

Q)

V

R

This equivalence is obtained by substituting P A Q for P and R for Q in law 13, implication. The next transformation involves substituting ,,-,(~ A Q) for P, and R for Q in law 5, and using the rule of replacement. Erasmg double denial uses law 2, the substitution of P A Q for P and again, the rule of replacement accounts for the final transformation:

Decision Procedures

50

The justification for the above transformations may be explicitly set out as follows-allowing AI B to signify that A is substituted for B. 1. PAQ--+R 2. P --+ Qf-+ r-IP Y Q 3. P A Q --+ Rf-+ r-I(P

given Imp A

Q)

Y

4. r-I(P A Q) Y R 5. P Y Qf-+ r-I(r-IP A ,.......,Q) 6. ",,(P A Q) Y R f-+ ,.......,("" ",,(P

7. r-I("""'" r-I(P A Q) A ,.......,R) 8. P f-+ r-I "",p 9. P A Qf-+ r-I ,.......,(P A Q) 10. t".I((P A Q) A t".IR)

P AQIP, R/Q, RI

R

1,3, R2 DM A

Q)

A

r-.JR)

",,(P A Q)/P, R/Q, RI 4,6, R2

DN P A Q/P, RI 7,9, R2

On the right we have indicated the justification for each line. A justification consists of the given formula, a law, or a use of RIoI' R2. In other words, a transformation is a sequence of lines, each line either being given, being a law, a substitution for a law (RI) or a line obtained by using R2. It should be noted that the first two times R2 is used, B replaces A where A is the whole formula, while the last time R2 is used, A is part of a formula. In transforming formulas using our equivalence formulas, it will prove convenient if we take certain short-cuts. To move from (I)PAQ--+R

to (2) r-I(P A Q) Y R

involves, as we have seen, using law 13, substitution, and the replacement of an equivalent by its equivalent (R2). We may, for the sake of brevity, now omit setting out the lines that involve law 13 and making substitutions, and move directly from (1) to (2), giving as our justification law 13, or Imp. In such a context we can regard Imp as indicating the collapsing of four lines into two. Similarly, to move from, say, RYR--+Q

to

involves law 15, RI (RIP), and replacing equals with equals (R2). But we will collapse these steps and move directly from the first to the second, giving as our justification law 15, Idem. Consequently, neither ,.

3.5

Logical Equivalence and Transformations

51

RI nor R2 nor substitution will any longer be cited in making transformations. Let us now employ this short-cut in making some transformations and reduce further the amount of writing by omitting line numbers. (P --+ Q) --+,.......,R r-IP Y Q --+ ,.......,R r-I(r-IP Y Q) Y r-IR r-I('---P Y r-I r-IQ) Y r-IR (P A ,-.."Q) Y r-IR ,......,R Y (P A r-IQ) (r-IR Y P) A (('-...IR Y "",Q)

Imp Imp DN DM (law 6) Camm (law 3) Dist (law 12)

Note that to use DM, to change a negated disjunction into a conjunction, we must have a formula that is an instance of ,.......,(r-.JP Y ,.......,Q). ,-...;( ""p V Q), the first disjunct of the third line, cannot be transformed using DM. However, using DN on ,.......,(""P V Q) we obtain '--'C--'P V "",......,Q), whichis a substitution instance of ""(",,P V ,.......,Q)(""QIQ) and thus we transform ",,(,.......,P V Q) into P A ""Q using first DN and then DM. These transformation techniques can be usefully employed to prove that one formula is logically equivalent to another. Suppose we wish to show that A is logically equivalent to B. If we can start with

and successively transform the A on the right into B we will have shown that A f-+ B is a tautology. For example, we can show that (P --+ Q) --+ (Q --+ P) is logically equivalent to (P A Q) Y ",,(Q A ""P) as follows: (P

--+

Q)

--+

(Q

f-+

--+

P)

(P --+ Q) --+ (Q --+ P)

f-+",,(P--+Q) Y (Q--+ P) ,.......,(""P V Q) Y (""Q Y P) f-+ ""-,(",,P Y "" ""Q) Y (""Q V P) f-+ (P A ""Q) Y ,....",("",....",Q A ""P) f-+ (P A ""Q) Y ",,(Q A ,....",P) f-+

Imp Imp (2) DN DM (laws 6 and 5) DN

Every tautology is logically equivalent and every inconsistent formula is logically equivalent. Let us represent a tautology by T and an inconsistent formula by F. Now what can be said about a disjunction that has an inconsistent formula as a disjunct? That is, what Can be said about: PYF

52

Decision Procedures

The disjunction is true if P is true; otherwise it is false. It therefore is logically equivalent to just P. Similarly, if one of the conjuncts of a conjunction is a tautological formula, if we have PAT

then the conjunction is true if P is true and false if P is false. Therefore PAT is logically equivalent to just P. These truths are expressed in the next two laws below. Law 17 P A T+-+ P Law 18 P V F +-+ P

Dropping tautologies (DT) Dropping inconsistencies (DI)

One of these laws-DT-is used in the following proof of equivalence by transformation. We wish to prove that

3.5

Logical Equivalence and Transformations

Exercises 1. Use truth-tables to show the 20 laws are tautologous.

2. State the ways the following uses of DM, Con or Imp are erroneous, though each pair is equivalent. (a) ,.....,(P V Q) ,-....; ,--.,(,.....,P A ,.....,Q) DM (b) P V Q r--.JP-+Q Imp (c) r--.J(P V r--.J ,-.....Q) ,-.....,--.,(,--.,p A ,-...,,.....,,.....,Q)

DM

(d) "-'P V f'-'Q ,.....,(P AQ) (e) P -+,.....,Q

DM Con

,.....,Q-+ P PVQ-+PAQ

53

(f) ,.....,(P V Q) V P ,.....,,.....,(,.....,P A ,.....,Q) V P

DM

is logically equivalent to P +-+ Q. P V Q -+ P A Q+-+ P V Q -+ P A Q +-+ ,.....,(P V Q) V (P A Q) +-+ ,.....,(,.....,,.....,P V ,....., ,.....,Q) V (P A Q) +-+ (,.....,P A ,.....,Q) V (P A Q) +-+ «,.....,P A ,.....,Q) V P) A «,.....,P A,.....,Q)VQ) +-+ (P V (,.....,P A ,-..."Q)) A (Q V (,.....,P A ,.....,Q» +-+ «P V ,.....,P) A (P v,.....,Q» A «Q V ",-,P) A (Q v,-...,Q)) +-+ (P V "'-'Q) A (Q v,.....,P) +-+ (,....",.....,p V ,.....,Q) A (,.....,,.....,Q V ,.....,P) +-+ (,.....,P -+ ,.....,Q) A (,-....;Q -+ ,.....,P) +-+ (Q -+ P) A (P -+ Q) +-+ (P-+Q) A (Q-+ P) +-+P+-+Q

3. By transformation prove the following pairs of formulas are equivalent. For

Imp DN (2) DM Dist Comm (2) Dist (2) Comm,DT(2) DN (2) Imp (2) Con (2) Comm Equiv

Be sure to see why the uses of DN are necessary. If any disjunct of a disjunction is a tautology, the whole disjunction is a tautology, and if any conjunct of a conjunction is inconsistent, so is the entire formula. Again, letting T stand for a formula that is always true, a tautology, and F stand for a formula that is always false, an inconsistent formula, these truths may be expressed in the following laws: Law 19 P V T+-+ T Law 20 P A F +-+ F

at least one transformation take no short-cuts. (a) P+-+,.....,Q

Q+-+ ,.....,P (b) ,.....,(R -+ ,.....,S)

RAS (c) P A (P-+Q)

PAQ (d) ,.....,(P -+ ,.....,Q) (P A Q) V (P A Q) (e) P-+Q V,.....,Q

T (f) (P -+ Q) V (P -+ R) P-+QVR (g) P A (P V Q) P (h) (P -+ Q V ,.....,Q) A W W (i) (P A (Q A S» V (,.....,P A (Q AS» QAS (j) ,.....,(P A ,.....,(P A ,.....,(P A Q») ,.....,p V "-'Q (k) (P -+ Q+-+ "-'Q

-+

,.....,P)

R (1) (,.....,p V (Q V "-'Q» A R

R

A

R

Decision Procedures

54

Answers DN,DM DN DN,DM Dist Comm, DT Assoc Idem

r-JP V r-J r-J(P /\ r-;(P /\ Q)) r-JP V (P /\ r-J(P /\ Q)) r-JP V (P /\ (r-JP V r-JQ)) (r-JP V P) /\ (r-JP V (r-JP V r-JQ)) r-JP V (r-->P V,--.,Q) (r--'P V r-JP) V r-JQ r-->P V r-JQ

The four symbols V, /\, --+, and +--+ are constants that act on pairs of things. We may speak of them as operators and say that they act on pairs of truth-values. For example, T V F acts on the pair T and F such that the result is T. Furthermore we may speak of these symbols as two-place truth-value operators since they relate pai1'8 of truthvalues. So far only four two-place truth-value operators have been considered. But there are other possible two-place truth-value operators, and the table below sets out all the possible combinations for a two-place truth-value operator: v

T T

T F'

F' T F' F'

T T T T (I)

T T T F'

(2)

+--+

->-

T T

T T

F'

F' F'

T (3)

(4)

55

A (1) B =dfA--+A A (2) B =df r-JA --+ B A (3) B =df B --+ A A (4) B =dfA A (6) B =df B A (7) B =df r-J( (A --+ B) --+ r-J(B --+ A)) A (8) B =df r-J(A --+ r-JB)

Other two-place operators

P Q

Other Two-Place Operators

exception of --+, we will allow the column numbers to be symbols for the possible two-place operators.

3. (j) r-J(P /\ r-;(P /\ r-J(P /\ Q)))

3.6

3.6

A

t

-1--+

V

T

T

T

T

F'

F'

F'

F'

F'

F'

T

T

T

T T T

T T

F'

(6)

(7)

F' F' F' (8)

T

T T (5)

F' F'

F' F'

T F' F' T T (9) (IO) (Il) (I2) (I3) F'

F'

F' F'

T F'

(I4)

F' F' F'

T (I5)

F' F' F' F'

(I6)

Each of these columns would allow us to introduce a two-place connective. For example, we might introduce V and define it by column 10. This symbol so defined would roughly correspond to English 'or' when we so use it that P or Q is true iff either one or the other disjunct is true, but not true when both are true. Or we could introduce the symbol-t--+ for column 12. P -t--+ Q could be read "P does not imply Q" since P -t--+ Q is equivalent to P /\ ~Q or r-J(P --+ Q). Using denial with --+, V, or /\, one can define the remaining 15 twoplace operators indicated in the above table. If, for example, a formula using only r-J and --+ can be found equivalent to a formula using only V, then we could define V using r-J and --+. Let us use r-J and --+ and define the r.j:lmaining 15 two-place operators. For convenience, but with the

A (9) B A (10) B A (11) B

=df

A (12) B

=df

A A A A

r-JA =df r-J(B --+ A) =df r--'(,-....,A --+ B) =df ,-....,(A --+ A)

(13) B (14) B (15) B (16) B

A --+,,-,B (A --+ B) --+ ",-,(B =df r-JB =df

,-....,(A

--+

--+

A)

B)

=df

Of special interest is the ability to select one of the columns in the above table and define denial and the remaining connectives in terms of this selected column. This is possible by selecting column 9 or column 15. Using the sym~ol I-called the Sheffer stroke-for the two-place operator correspondmg to column 9, let us see how we can define,-...., V and /\ in terms of /. ' , If we again look at the table for possible combinations for a twoplace operator, we see that P / Q has this matrix: P

T T F F

Q F T T T

T F T F

If a. formula can be found using only this connective and it is logically eqUIvalent to a formula using only r-J, then we could define r-J using /. P I Q naturally reads 'not both P and Q'. This indicates that P / P is equivalent to r-J(P /\ P). ,-....,(P /\ P), by Idem, reduces to ",-,P. Thus we may use this definition for "'-' in terms of /:

Further~ore we can obtain such definitions for the four familiar two-place connectr;es an~ .indeed for the 12 other connectives. One way to proceed III obtammg these is as follows: Take V for example. P / Q, as w~ have seen, can be written as ",-,(P /\ Q). This can be transformed by usmg DM into a formula only using V and r-J: ",-,P V "'-'Q. From "",p V ,-....,Q we can obtain just P V Q by negating each disjunct; thus

"""Normal Forms and Testing for Validity

Decision Procedures

56

if we do this to P

I Q-negate each constituent-we should obtain this

equivalence:

57

Answers 1. P! Q may be read "neither P nor Q." 2. (b) F, (f) T, (g) T, and (h) T. (f),(g), (h), and (i) are tautologies.

Now we can remove the ,-..., using the above definition, thus obtaining a definition for

V

A V B

using only I: =df

(A

I A) I (B I B)

3.7

Since PIP is equivalent to ,-...,P, (P I Q) I (P I Q) is equivalent to ,-...,(P I Q). And since PI Q means 'at least one of P and Q is false', ,-...,(P I Q) means 'it is false that at least one of P and Q is false' or 'both P and Q are true'. Thus A II B

=df

(A

I B) I (A I B)

Exercises 1. Give a reading for P the above table.

! Q and for

the other unfamiliar operators indicated in

2. Determine the truth-value of the following formulas given P: T and Q: F. Which of the formulas are tautologies? (a) PIQ (b) (P ~ P) I ,-...,Q (c)

P!,-...,Q

(d) (P I Q)

I (P I Q)

,-...,(P! Q) I (,-...,p

! ,-...,Q) (f) (,-....;p<,---} P) I P (g) (P ~ Q) ~ ,-...,(P-t-+ Q) (h) (P (16) Q)<,---} P II'-""P (i) (P (14) Q) ~ Q II ,-...,P

(e)

4. There will be 2 2n truth functions for n variables. A three-place truthfunctional operator will thus have columns for 256 operators.

There is a second procedure by which one can decide if a sentential formula is valid. We can use the equivalence laws to reduce any formula to a normal form. There are several normal forms customarily treated in beginning logic books. In this section our interest is in the normal form called the conjunctive normal form (abbreviated CNF). Using CNF to determine whether a formula is valid is tedious business, especially when a method such as the reductio test is available. However, reducing formulas to normal forms provides good practice in using the equivalence laws and, as we will see in Part 2, CNF's can be highly important for theoretical purposes in logic. To define the notion of a conjunctive normal form it is well to start with what we will take to be a single formula. By a single formula we will understand either a sentential variable or a negated sentential variable. Double negated formulas will not count as single formulas. If we let AI> ... , An be a list of n single formulas, then we will call an elementary disjunction a formula of the form

Examples of elementary disjunctions would be

3. For two- place truth-value operators there are 16 possible operators or connectors, as the table above shows. Construct a table for possible one-place truth-value operators. 4. How many possible truth functions are there for n variables? 5. Use only'-"" and V and then use only'-"" and II and define the remaining 15 connectives. 6. Define the remaining 15 connectives by only using using !.

Normal forms and testing for validity

I

and then by only

7. Show why one cannot define the remaining two-place connectives by using either II and V, ~ and II, 01'"-* and v.

PvQ

P vQ v,-...,p P

,-...,p 'l'he last two examples above are the limiting cases where n = 1. Thus a single formula alone will count as an elementary disjunction. We can now say a formula is in conjunctive normal form if and only if it has the form

Decision Procedures

58

where AI> ... , An are elementary disjunctions. Where n = 1, all of the above examples will count as being in conjunctive normal form. Other examples of CNF are

transformations: (P -+ P V Q) A (P V Q -+ P) (,-...;p V (P V Q)) A (,-...;(P V Q) V P) (,-...;p V (P V Q)) A (,-...;(,-...;,-...;P V ,-...; ,-...;Q) V P) (,-...;p V (P V Q)) A «",-,P A r-;Q) V P) (",-,P V (P V Q)) A (P V C---P A ",-,Q)) (",-,P V P V Q) A (P V ",-,P) A (P V ,-...,Q)

(P V r-.;Q) A (P V ,-...;Q) ,-...;P A (Q V R) (R V ,-...;Q) A (P V ,-...;R V S)

Ignoring the cases where n = 1, a formula is in CNF if it is a conjunction of disjunctions and all negation signs negate single variables. It would be well to note that the following are not in CNF: (,-...;p A Q) v,-...;R V Q)

,-...;(,-...;p

(P A Q) V (P A R) r-.;(R V Q) A (P V ,-...;P)

Any propositional formula can be put into CNF by following these steps in this order: Use Imp and Equiv to obtain a formula containing only the connectives,-...;, V, and A. (2) Use DM and DN to remove all negations outside parentheses. (3) Use DN to remove all double negations. (4) Use distributive laws as many times as necessary to produce a conjunction of disjunctions of single formulas. (1)

It should be remembered that since (P V P) +-+ P and (P A P) +-+ P, we may replace P V P by P at any time and PAP by P. To put Q -+ (P -+ (Q -+ P)) into CNF we proceed as follows, first removing -+.

Imp

59

Normal Forms and Testing for Validity

Equiv Imp DN DM Comm Dist

The last line is the CNF of the formula we began with. We should note that some formulas have more than one CNF. If a formula is valid, then its CNF is valid. In order for a CNF to be valid, each of the conjuncts must be valid. Each conjunct of a CNF will be valid if and only if a variable and its negation appear in the conjunct. (The conjuncts, we remember, are disjunctions, and a disjunction is valid iff a variable and its negation appear as disjuncts.) The first transformed formula above is valid, but the second is not valid because in its CNF, though a variable and its negation appear in the first two conjuncts, such a combination does not appear in the last conjunct(P V "'-'Q). If a formula is in ONF, we can tell by simple inspection whether it is valid. To test P -+ (Q -+ P A Q) by reduction to ONF we proceed as follows: P -+ (Q -+ P A Q) ,-...;P V ("'-'Q V (P A Q)) ,-...;P V «,-...,Q V P) A (,-...;Q V Q)) (",-,P V ,-...;Q V P) A (",-,P V ,-...;Q

At a glance we see the ONF of P this formula is a tautology.

-+

V

Q)

(Q

-+

Imp Dist Dist

P

A

Q) is a tautology; thus

Exercises 1. Determine the validity of each of the following by putting it into CNF.

Since we have a string of disjunctions, we can drop parentheses, which gives us

This is an elementary disjunction. Since in the limiting case where n = 1 an elementary disjunction is in CNF, we can stop, the formula is in CNF. The next example below uses all four steps. The problem is to transform P +-+ P V Q into CNF. It can be put into CNF by these -./

..

(a) P-+ (Q-+ P)

(b) (0) (d) (e) (f)

Q -+ (P -+ (Q -+ P))

P A (P -+ Q) -+ Q (P V Q -+ ,....."R) -+ (R -+ P) V (R -+ Q) (P -+ Q) -+ (P A R -+ Q A R) ,-...,(P A Q -+ «P -+ R) -+ Q)) (g) (P +-+ Q) -+ ,,-,(P A ",-,Q)

2. One can obtain a disjunctive normal form (DNF) of a formula by expressing the disjunction of the formula that describes its true cases. To apply this method to, say, (P -+ Q) +-+ (Q -+ P)

Decision Procedures

60

consider its truth-table and the cases in which the formula comes out true. It comes out true only in two cases, namely

P

Q

T F

T F

Now express each case as a conjunction with the variable for true and the negation of the variable for false. Thus in this example we have

Normal Forms and Testing for Validity

61

(b) How could you show that any of (1) to (5) in (a) above could be used to express all formulas using one-place, two-place, three-place, or any truth-functional connectives?

Answers 1. All are valid except (d) and (f). 2. (a) (b) (P II Q) V (P II ~Q) V (~P II Q) V (~P II ~ Q)

PIIQ ~P II~Q

Finally, link the resulting formulas by using as many disjunctions as necessary. Doing this with the example, we obtain (P II Q) V (~P II ~Q)

which is the DNF of the original formula. To take another example, the true cases for

(b) A DNF is valid iff it has 2 n disjuncts where n equals the number of component variables in the formula. 3. (b) Any formula employing truth-functional connectives has a truth-table. Thus every formula has a DNF. And thus any formula using one-place, two-place, three-place, and so forth, truth-functional connectives can be expressed using ~, V, and II. Since V can be defined by using only f"oo.J and II, any formula can be expressed using the pair ~ and II. Since II can be defined using ~ and V, any formula can be expressed using (2), and so on, through (5).

are

P T

Q

T F

F T

F

F

T

thus the DNF for the formula is (P II Q) V (P II ~Q) V (~P II Q) V (~P II ~Q)

(When a formula has only one true case, the resulting conjunction is counted as the DNF, and when a formula has no true cases, it is equivalent to P II ~P, which is counted as being in DNF.) (a) Put the formulas found in Exercise 3.7.1 into DNF. (b) When and only when is a formula in DNF valid? 3. With the use of certain equivalence laws any formula can be put in CNF, and with the use of truth-tables any formula can be put in DNF. (a) How can this fact be used to show that any formula using twoplace statement connectives can be expressed using (1), (2), (3), (4), or (5) below: (1)

~,

(2)~,

(3) (4) (5) ""'/-,

II V

~,-*

I t

Supplement: Logical networks

Today a great deal of calculation is done by computing machines. A modern digital computer is to a large extent a complicated logical machine. Certain of the components of computing machines can be regarded as performing logical operations. We shall not be interested in the electronic details of these components. Rather we will see how sentential logic can be applied to simplifying a circuit. Electronic computing machines can be built of the following switches among others:

(1) Inverter switch: This is a device with one input terminal and one output terminal. When the input is hot (voltage), the output will be cold (no voltage), and vice versa. This component acts like the connective ~. (2) Or-switch: The or-switch has two or more inputs and one output. The output is hot, unless all inputs are cold. This component acts like the connective V. (3) And-switch: There are two or more inputs and one output. The output is hot when all inputs are hot; otherwise it is cold. This component acts like the connective II.

Decision Procedures

62

These three components can be represented by the following figures:

P--[J --p

p-G-"'p

v

Q-

.

to the above network is (P A '""-'Q) V R

vQ

Q--

Given the truth-value for the variables above, we can calculate that the output is T (hot). To take another example, what is the output of the following when P is F and Q is T?

G

P--+--•

AND-SWITCH

A

--p

I

AQ

]IIi

Q-----!III

If more than two inputs go into an

63

Normal Forms and Testing for Validity

OR-SWITCH

INVERTER SWITCH

p-D

3.7

box, then we have a string of conjuncts, and if more than two inputs go into a V box we have a string of disjuncts. For example A

With

A

!O'

D

..

A

II!

as the main connective, we see that the formula will be ,-....;P A (

V

)

The V switch connects P and Q, so the formula is A

_

A A B A 0

0-

Now if we allow hot to correspond to T and cold to F, we can see one application of logic to these components. For example, suppose we have this arrangement:

,.....,.,p

A

(P V Q)

which, given the above truth-values, shows that the output is T (hot). Sometimes the formula corresponding to one network may be logically equivalent to a formula that is simpler. ,-...,,(P V ,-....;Q) is equivalent to ,-....;P A Q. Networks corresponding to these formulas would be

(1) P - - - , " , , " 1 1 0 ' ' _

:---D-~D

v

R------------------------------~·.-

where Pis T, Q is F and R is F. Will the output be hot or cold? To answer this we need first to figure out the formula analogous to this network. This is easily done. The last box gives us the main connective v. This V connects R and a conjunction, as follows: A

) V

R

The conjunction is made up of P and ,-....;Q. Thus the formula analogous ~-I

,

Clearly, other things being equal, (2) is a simpler network than (1). When networks have the same formula, they will have the same hot or cold output. Thus network (2) may be substituted for network (1) in the circuit with a reduction in cost and complexity, if, again, other things are equal. Let us measure simplicity of a circuit in terms of the number of statement connectives in circuit formula. Thus A is simpler than B iff A has fewer connectives than B. To illustrate, ,-....;(P A Q) is simpler than ",P A "-'Q since the former has two connectives, while the latter has three connectives.

Generally, to find a simpler circuit formula one will find useful most of the 20 laws set down in Section 3.5. For an illustration of how these laws can be used in simplifying formulas, consider the circuit below:

(e) P

,

p--+---0

o

Q-D -- O~

Dist Comm Assoc Idem Dist DI

G Q - -..- G

(e)P

Q

(b)

FLO Q

101

'"

D

I

-_..-E]--

""

II

.----__lBIIIr_

G ----

R--ojOlIO-G

1. Indicate the formula of each of the following circuits, simplify the formula, and draw the simplified circuit.

v

___

R-

Exercises

>0 [0

v

(d) P ------------.~

The simplified circuit will thus be merely P -

___

[J-

R ---------------------------------------.. -

It may be simplified in this way:

(a) P

-It-'-- B

Q ------------------------~..-

Q........--LG~_[J-[J-D(P V (Q A ,.....,Q)) A (P V Q) «P V Q) A (P V ,.....,Q)) A (P V Q) «P V ,.....,Q) A (P V Q)) A (P V Q) (P V ,.....,Q) A «P V Q) A (P V Q)) (P V ,.....,Q) A (P V Q) P V ("-'Q A Q) P

65

Normal Forms and Testing for Validity

Decision Procedures

64

o

(f) P

II

A

Q

[3-

... s

II

.

(g)

P

Q

D

-->

, [J-

t

III"

v 10'

--

~-----'f

c]---Gt

-c]--G--8-G

Decision Procedures

66 R

(h) P

L-------r-----'A -

Dv

EJ -- CJ ______

-1110'__

'-_---I

Answers 1. (a) (P V ,-...,Q) A Q is equivalent to P A Q. (b) ~P V (P A Q) is equivalent to ~P V Q. (e) ,,-,(~P A ~Q A ~R) is equivalent to P V Q V R. (g) ~(P A ~(P A ~(P A Q))) is equivalent to ~(P A Q). (h) (~P A (Q A R)) V (P A (Q A R)) is equivalent to Q A R.

natural deduction system sc

Truth-table techniques provide us with an effective procedure for deciding whether a well-formed formula of the sentential language is valid or not. In this chapter we wish to set down a system in which proofs are constructed for theorems. The theorems will all be valid sentential formulas, and all valid sentential formulas will be theorems of the system. Such a system, which collects together all valid sentential formulas, is called the sentential calculus. Actually, the name 'sentential calculus' is given to anyone of various equivalent systems in which theorems are tautologies and all tautologies are theorems. The system developed in this chapter wiII be called system SO (after sentential calculus). The system SO is a natural deduction system since it makes use of rules without any axioms and since derivations roughly correspond to how we "naturally" construct them. Why introduce a system that has tautologies as theorems-that constructs proofs for tautological formulas-when we have truthtable methods for determining whether a sentential formula is or is not a tautology? One reason for doing this at this time is that we wish to make additions to SO. There are well-formed formulas in logic other than sentential formulas, namely, predicate formulas. And these also make up two classes-valid and invalid formulas.

/.

67

68

Natural Deduction System SO

However, there are no mechanical procedures like truth-tables that enable us to tell whether any predicate formula is or is not valid. To establish the validity of such formulas we must use a system that is constructed by making additions to the sentential calculus. These additions to system SC will be made in Chapter 6.

4.1

Rules of inference and proofs

Much of the system to establish formulas as tautologies, which will be given later (Section 4.5), makes use of rules of inference and proofs using rules of inference. What we wish to do now is to acquaint the reader with these rules and the notion of a proof. The procedure we will follow is, first, to informally discuss the rules of inference used in the forthcoming systems and proofs using these rules. Then, later, we will incorporate the main body of this discussion into the formation of a system that has tautologies as theorems. Suppose the question is raised whether the formula ,......,R follows from the following set of formulas:

69

formulas, would go like this: 1. If P ---+ Q and P, then Q. 2. If Q and if Q ---+ ,......,R /\ S, then ,......,R /\ S. 3. If,.....,R /\ S, then ""R, so ,......,R must follow from the set.

In figuring out that ,......,R follows from the set, what we do is to assume that each formula is true and we follow out the consequences of this assumption. In doing this we see why ,......,R must also be true if we assume the formulas in the set are true. In figuring out that ""R is a consequence from the set we in effect show that an argument pattern is valid, namely P

Q, Q ---+,......,R /\ S, P :. ,......,R

where the premises are separated by commas before the therefore sign, and the conclusion follows this sign. Whether or not one fully realizes it, rules of inference are employed in figuring out that ,......,R follows from or is a tautological consequence of the set of formulas. Returning to the numbered lines above, the rule justifying the inference of Q given P ---+ Q and P is called modus ponens (abbreviated MP) and may be formulated as follows: RUI,E OF

That is, suppose the question is raised whether, if this set of formulas is true, ,......,R must be true. Supposing each of these formulas is true, must ,......,R be true~ Can we have this set true and yet have ,......,R false~ In the last chapter two procedures were introduced by which one can determine whether ,......,R follows from the set of formulas above. If we are dealing with sentential formulas, then B follows from A iff A ---+ B is a tautology. B may be said to be a tautological consequence of A iff A ---+ B is a tautology. Thus in asking if ,......,R follows from the above set of formulas we are asking if ,......,R is a tautological consequence of the set of formulas. So one could use one of the decision procedures introduced in the last chapter, say the reductio truth-table test, and calculate whether or not ,......,R follows. However, the steps we would naturally employ in figuring out whether ,......,R follows or is a tautological consequence of the above set of

---+

Modus Ponens (MP): Given A ---+ B and A, we may infer B.

On line 2 what warrants going from Q and Q ---+ ,......,R /\ S to "-'R /\ Sis also the rule of inference, modus ponens. With the activity on line 3, a different rule of inference is supposed. It is the rule that allows the inference from ,......,R /\ S to ,......,R. This rule is often called the rule oj simplification and is stated below: RULE OF SIMPLIFICATION

(S):

Given A /\ B, we may infer A or we may infer B. The capital letters A and B occur in the statement of these two rules. 'fhese will again be taken as variables for well-formed formulas of the sentential language (see Section 2.6), as has been our practice from the beginning. Let us illustrate some simpler applications of these two rules of inference. Below, the last line of each sequence can be inferred from the

Natural Deduction System

70

se

The validation of this argument pattern will be written out as folloWS, and what appears here is a proof:

previous lines by the indicated rule of inference: (P V Q) /\ R pvQ

w /\,.....,0 ,....,.,0 (P /\ Q) /\ (Q V R) P/\Q

S

P/\Q-+R p/\Q R

MP

A A A

1. P-+Q 2. Q -+,....,.,R /\ S

3. P S

,....,.,P-+R V S ,....,.,P RvS

71

Rules of Inference and Proofs

from lines 1 and 3 by MP from lines 4 and 2 by MP from lines 5 by S

4. Q 5. ,-..,R /\ S

MP

6. ,....,.,R

S

A rule of inference is something that allows us to go from some given assumption to some other statement. A rule of inference can be valid or not valid. If a rule of inference is one that allows only true or acceptable statements to follow from true or acceptable assumptions, it is a valid rule. In other words, a valid rule can never lead us from a true assumption to a false conclusion. In the matter at hand, a rule of inference is valid if and only if from assumed well-formed formulas of the sentential language it allows the inference of formulas that are tautological consequences of the assumed formula. Both modus ponens and simplification are such valid rules. Consider modus ponens. If A -+ B and A are assumed to be true we have A-+B

TT in which case B can only be true. In turn, if modus ponens is applied, the conclusion must be a tautological consequence of the assumptions since ((A -+ B) /\ A) -+ B is a tautology. Since we are building the elements for a system in which only tautological inferences will be allowed, we wish each of our rules to be valid. Having introduced the two rules of inference used in figuring out that ,....,.,R follows from the above set of formulas, we can noW more formally write out the "figuring out" of ,....,.,R from the set of assumptions. We will call what we do giving a proof. What we wish to do is to establish the validity of the following argument pattern:

This is the pattern to be shown to be a valid one in the proof. It is made up of three premises separated by commas. The premises are separated from the conclusion ,....,.,R by the sign ~, called the turnstile sign. ~ can be treated like: .. Av ... , An ~ B can be read "Av ... , An therefore B" or "B follows from Av ... , An" or "There is a proof of B from Av ... , An. "

This is an example of how proofs will be written. Each line is numbered and at the right we find a justification for each line. For the premises we write A in the justification position. We justify lines that are premises by the rule of assumption, which we now state: RULE OF ASSUMPTION

(A):

An assumption may be introduced at any point in a proof. ~he. deri:ed !ines in the ab~ve proof are the non-A lines and are given JustIficatIOn m terms of theIr place of origin and rule of inference use d . Generally before the numbered sequences of lines we will give the argument patter~ to be proved (as we do in the next proof). Finally we note.that .each hne of the proof is either an assumption or comes from prevIOus hnes by an application of one of the inference rules. Since each of the rules is valid and is properly applied, the above sequence of lines shows t~at ,....,.,R follows from the premise assumption, that ,....,.,R is a tautologICal consequence of the premises. We thus have a proof that ,-vR follows from the premises-in other words, we have established th validity of the argument pattern above. e The rule of assumption no doubt seems to some readers to be ~xcessively liberal. Can anything enter as an assumption ~ The answer IS yes, for in logic t.he. concern is not with the nature of the assumptions but wheth~r what IS mferred from assumptions logically follows or not. In settmg out the rules for SC it will be convenient to dispense with phrases such as 'given' and 'we may infer-'. Let us regard all such rule talk as collapsed in a line, such as . The two rules so far considered can thus be conveniently written:

MP A-+B A -B-

S

A /\ B

A/\B B

~'

Thus the first will be read: Given A

-+

B and A we may infer B.

72

Natural Deduction System SO

Let us construct another proof for an argument pattern. What We wish to show is that, given as assumptions P -+ (Q -+ R), P -+ Q, and P, we may validly conclude R. R is a tautological consequence of this set of assumptions. The proof: Prove: P-+ (Q-+ R), P-+Q, P f- R 1. 2. 3. 4. 5. 6.

P-+(Q-+R) P-+Q

P Q -+ R Q

A A A 1,3 MP

2,3 MP 4,5 MP

R

One final note of clarification. When a line is inferred from assumption lines by using one or more rules we will call this a deduction of the line in question from the assumptions. The last proof thus establishes that there is a deduction of R from P -+ (Q -+ R), P -+ Q, and P. The characterizations given for the notion of a proof and the notion of a deduction do not involve much precision. This is saved for later, when system SO is set down.

Exercises 1. Construct a proof for each of the following argument patterns. (a) (b) (c) (d)

P-+Q, PAR f-Q P -+ Q, Q -+ R, P f- R ,.....,P -+ Q A R, ,.....,P f- R

A (Q-+ R), P f- R (Q -+ R), ,.....,P -+ Q, ,.....,p f- R (f) P-+ (P-+Q), P f-Q

(P-+Q)

(e) ,.....,P -+

2. Is the following principle true: If A -+ B is a tautology then 'Given A we may infer B' is a valid rule. If it is true what are its consequences?

Answers 1. (d) 1. (P-+Q) A (Q-+R) 2. P-+Q 3. Q-+ R

4. P 5. Q 6. R

A 1,8 1,8 A

2,4MP 3,5MP

4.2

Four Additional Rules of Inference

73

to introduce formulas using A into a proof. Using the convention adopted at the end of the last section, this rule is: RULE OF CONJUNCTION (OONJ):

A,B AAB

Its use is illustrated in the proof that follows: Prove: P A Q f- Q A P 1. P A Q 2. Q 3. P 4. QA P

A 1,8 1,8

2,3 Conj

The proof that follows illustrates a special case. Pf-PAP loP 2. PA P

A

1, Conj

We will regard a rule that has metavariables separated by commas, as is the case with Oonj, as applicable when the same line is an instance of both variables. This is the case with the proof above. Line 1 is taken as an instance of A and B in the application of Oonj. The next rule to be introduced allows the introduction and removal of negation signs. It is the rule of double negation and is set down below: RULE OF DOUBLE NEGATION

A '"" ,.....,A '

(DN):

,-....;,....,A -A-

The rule of double negation allows us to go from any wff A to the formula with r-J ,....., prefixed, and vice versa. Thus DN justifies an inference of the second line below from the first, and vice versa.

2. Yes. It generates an infinite number of valid rules of inference. The use of this rule is illustrated in the following proof: 4.2

Four additional rules of inference

The third rule of inference concerns the use of the conjunction. The rule says that given A and given B we may infer A A B. This rule allows us

Prove: ,.,It""p -+ Q, P f- Q 1. '"" '"" P -+ Q 2. P 3. f'"Oo..),.....,p 4.Q

A A 2,DN 1,3 MP

Natural Deduction System SO

74

To use MP with ,-.., ,-..,p ~ Q and P we cannot take off the ,-.., '"'-' in the antecedent of ,-.., ,-..,p ~ Q. To apply DN the I"-' ,-.., must have the scope of the whole formula. For example we could use DN to remove the "'" ,-.., from ,-.., ,-..,(P ~ Q). The next rule allows us to enter disjunctions into proofs and is RULE OF ADDITION (ADD):

A

A

AvB'

BvA

75

Four Additional Rules of Inference

An argument pattern is valid if it has no instance that is made up of true premises and a false conclusion. We can now construct proofs for some argument patterns by using the rules just introduced and thus demonstrate that they are valid. In addition we can show that any invalid argument pattern is invalid by supplying statements for its components such that an argument with true premises and a false conclusion results. To illustrate, consider these patterns: P~Q,Qf-P

pvQf-P

From A this rule tells us we may derive A vB, and from A we may derive B V A. A may be any wff of the propositional language and B may be any such formula-this being the case, of course, with all of the rules. To illustrate Add's use, consider the next proof:

Neither is valid. To demonstrate the invalidity of the first pattern we can make these substitutions: P: Rockefeller is a conservative (F) Q: Rockefeller favors a growing economy (T)

P f- P A (P V Q)

A 2, Add

l.P 2. PVQ 3. P A (P V Q)

1, 2 Conj

The next rule again concerns the use of ~ and "-'. Given a conditional formula and the negation of its consequent, the rule permits us to infer the negation of the antecedent of the conditional. Here is the rule: RULE OF

modus tollens (MT): A~B

,-..,B .......,A

(P V Q)

This is the other side of the coin to MP. Its use is illustrated below: Prove: ,-..,p ~ ""'Q, Q f- P l. ,-..,P ~ "",Q 2. Q 3. ,-..,.......,Q 4. I"-' I"-'P 5. P

A A

2,DN 3,1 MT 4,DN

Note that P cannot be inferred from 1 and 2 by MT. For MT or any of the rules to apply, the same shape must obtain. With respect to MT we must have a wff A connected by ~ to a wff B and have another line made up of B with a before it. In the above proof where MT is properly applied, I"-'P is an instance of A and """Q is an instance of B; thus "",Q is an instance of f"Ooo./B. I"-'

I"-'

obtaining a true premise and a false conclusion. The same statements substituted in the same way will demonstrate that the second pattern is invalid. If there is an instance of an argument pattern in which all the premises are true but the conclusion is false, such a case is called a counterexample. Thus each of the above substitutions provides a counterexample for each of the above argument patterns. An argument pattern is valid if it has no counterexamples. With respect to argument patterns expressed in the sentential language, the quickest way to produce a counterexample is by using the reductio truth-table test. The pattern ~

R f- R V P

is invalid. Treating the turnstile like a conditional that is the main connective, the reductio test would reveal that the premise is true and the conclusion false when P is F, Q is F, and R is F. Thus we obtain a counterexample to this pattern by supplying false statements for P, Q, and R. We can, in fact, regard the reductio truth-table test as indicating whether there is a possible counterexample. If we obtain a contradiction, this shows that there cannot be a counterexample.

Exercises 1. Construct a proof for each of the following argument patterns. (a) ,,-,p ~ '"'-'Q, '"'-' "-'Q f- P (e) I"-'P, ,,-,(Q A R) ~ P f- Q (f) P A Q f- P V Q (b) (P A Q) A R f- P A (Q A R) (c) (P ~ Q) A (P ~ R), P f- Q A R (g) Q ~ (P ~ R), I"-'R, Q f(d) P V Q ~ '"'-'R, P f- ,,-,R (h) Q, P ~ '"'-'(Q A Q) f- ,,-,p

I"-' "-'

"",p

76

Natural Deduction System SO

2. Show that the following patterns are invalid by supplying counter_ examples. (a) P -+ Q, r-IP I-,......,Q (c) P 1\ Q -+ R I- P -+ R (b) P -+ Q, R -+ Q I- P -+ R (d) P-+QI-Q-+P Answers 1. (d) 1. P V Q -+ r-IR

2. 3. 4. (g) 1. 2. 3. 4. 5. 6.

P PVQ r-IR Q -+ (P -+ R) r-IR

A A

2, Add 1,3 MP

r-.;p

A A A 1,3 MP 2,4MT

r-.J'-""!"'-.Ip

5,DN

Q P-+ R

Assumption Discharging Rules

77

assumption in addition to the premise assumptions-and then attempt to derive B from A and the set of assumptions that make up the premises. If we succeed in deriving B from A and 6, then our new rule declares a deduction of A -+ B from 6. In passing from 6, A I- B to 6 I- A -+ B we will say that A has been discharged. Thus having assumed A, to obtain a conclusion of the form A -+ B, when we enter A -+ B on the last line, A is said to be discharged. The assumptions, if any, found in 6 are not affected; they remain. Let us illustrate a simple application of this rule. Suppose we wish to establish the following argument pattern: P-+QI-PI\R-+Q

2. (d) P: Reagan is an Arizonan (F), Q: Reagan is an American (T).

4.3

4.3

Since the conclusion is a conditional, we may use the new rule. To use it we not only take P -+ Q as an assumption, but we take the antecedent of the conclusion, P 1\ R, as an additional assumption. Then we attempt to derive the consequent of the conclusion, Q. We thus proceed as follows:

Assumption discharging rules

In this section three rules will be introduced that involve the removal of assumption lines in a proof. The rules in the previous section do not affect the number of assumptions in a proof. The rules in this section always result in the reduction of the number of assumptions in a proof. To set out these rules we will find it useful to introduce the symbol 6. Let 6 be a set of zero or more assumptions. Later we will encounter situations in which 6 is a set of zero assumptions, an empty set. The first assumption discharging rule incorporates this valid inference: If from a set of assumptions 6 and A we can derive B, then A -+ B is derivable from 6. This is a valid rule. 6 1\ A -+ Bis equivalent to 6 -+ (A -+ B). Therefore the formula A -+ B can be established as following from 6 by showing that B is deductible from the assumptions 6 and A. By using the turnstile sign this rule may be conveniently expressed as follows: Given 6 and A I- B, we may infer 6 I- A

-+

B.

This rule will be used in constructing proofs for argument patterns that have conclusions in conditional form. What we may now do when the conclusion has a conditional form A -+ B is to assume A-enter A as an -,.

1. 2. 3. 4. 5.

P-+Q P 1\ R

P Q P

1\

R-+Q

A A 2,S 1,3 MP

2--4 by new ru}e

In using the new rule to justify line 5, we discharge the assumption on line 2 and remove it from the assumptions of the proof. We wish to indicate this in some way. We will do this by writing the last proof as follows:

. U

1. P-+Q PI\R 3. P 4. Q 5. P 1\ R-+Q

A A 2, S 1,3 MP 2-4 by new rule

The line from 2 to between 4 and 5 indicates that the assumption on line 2 has been discharged on reaching line 5. Only the assumption on line 1 remains; thus 5 is derived from 1; the above argument pattern has been established. To set out this rule, which will be called rule of conditional proof, in the form of the previous rules we may do the following:

78

Natural Deduction System SO

RULE OF CONDITIONAL PROOF (Rep):

p

D, Af-B A-+B

II Q -+ R f- P 1. PIIQ-+R 2' P 3. Q P II Q

~

r::liR

Assumption Discharging Rules -+

-+

6. Q -+ R 7. P -+ (Q -+ R)

(Q

-+

R)

A A A 2,3 Conj 1,4 MP

RCP RCP

With assumption lines 2 and 3 discharged, 7 follows from 1 alone, thus proving the validity of the argument pattern. Note that this argument pattern corresponds to the Rep rule. Argument patterns corresponding to each rule, with the exception of the rule of assumption, can easily be constructed. Our drawing the lines to the left of the proof serves several functions. First, and foremost, it shows when an assumption has been discharged and where it has been discharged. Second, it shows the scope of an assumption. In the last proof the scope of the line 3 assumption is 3 to 5, while the scope of the line 2 assumption is 2 to 6. The scope of the assumption on line 3 is found within the scope of the assumption on line 2. This is permissible. Scopes of different assumptions may also follow each other with no overlap. But what is not allowed is any overlap of the scopes of two or more assumptions. Any such overlap will show up in crossing lines. To avoid crossing lines we will lay down this simple proviso for when Rep is used in a proof: One cannot use a line in the scope of an Rep assumption after the assumption has been discharged, unless the line can be gotten from other assumptions. This proviso will be followed and will apply to all the rules of inference. That is, in using any of the rules of inference we must not use a previous line that is in the scope of an assumption already discharged unless the line can be gotten from other assumptions. Observing this restriction on the use of rules of inference when assumption discharging rules are employed will prevent erroneous "proofs" such as the "proof" of the invalid pattern

79

Q II R f- Q below: 1. P-+QIIR 2. P

In this rule A f- B indicates that B has been derived from A. 6, A f- B thus reads that B has been derived from 6 and A. The angled line indicates that A has been discharged. A -+ B is thus said to be derived from the set of assumptions, if any, that make up 6. Rep can be used once, twice, three times, and so on in a proof, if necessary. For example, Rep is used twice in the following proof: Prove: P

4.3

A A

[i Q

3. Q II R

1,2MP

4. 5. P-+Q

3,S

RCP

6. Q

2,5 MP (erroneous)

Line 6 is erroneous since it is obtained in part by using a rule on a line within the scope of a discharged assumption. The next assumption discharging rule is relatively the most difficult to grasp. The rule states that given A V B if we assume A and derive 0 and if we assume B and derive 0, then 0 follows from A V B. If in deriving 0 other assumptions 6 are employed, then 0 follows from A V Band 6. This rule will be called rule of disjunction and may be indicated in this way: RULE OF DISJUNCTION (Drs):

6,Av B 6lA f-C 6,Bf-C C

The rule reads that given 6 and a disjunction A V B, if we assume A and obtain 0 (from A alone or in conjunction with 6) and if having assumed the other disjunct B we obtain 0 (from B alone or in conjunction with 6), then 0 follows from A V B and any other assumptions used in the derivation. In this rule, as with all assumption discharging rules, the proviso against using a line in the scope of an assumption after the assumption has been discharged must be observed. The use of this rather complicated rule is illustrated in two proofs below: Prove: P

V Q f- Q 1. P V Q 2. P 3. Q V P

U

V

P

A A

2, Add

4. Q

A

5. Qv P 6. Qv P

4, Add

V P f- P 1. P V P

Dis

Prove: P

r~' P

~P

4. P

A

A A Dis

80

Natural Deduction System S()

This last example reveals a limiting case of the assumption discharging rules. Having assumed A we will count A as itself following from A. Also in both uses of Dis, 6. is empty. That is, there are no assumptions other than the disjunction A V B. Thus 0, Q V P in the first case and P in the second, is derived from the disjunction alone in each proof. The last assumption discharging rule is the rule of reductio ad absurdum. The method of proof by reductio ad absurdum is already familiar to us. It was used as a short-cut truth-table test to determine whether or not a formula is tautologous (Section 3.4). This is the familiar pattern of argument in geometry, mathematics, and philosophy. For example in deriving his theorems, Euclid often begins by assuming the opposite of what he wants to prove. If that assumption leads to a contradiction, or "reduces to an absurdity," then that assumption must be false, and so its negation-the theorem to be proved-must be true. In using l'eductio for proofs of argument patterns we show that from 6. and the denial of the conclusion, ",,0, a contradiction follows. A contradiction must be false; thus by MT it must be false that 6. and ",0 are true. If it cannot be the case that 6. and ",,0 then 6. ~ must be valid, since we cannot produce a counterexample for the pattern 6. ~o. To state this last assumption discharging rule we must first define a contradiction. By a contradiction we will understand a conjunction in which the second conjunct is the negation of the first conjunct. Thus P /\ f"'.JP, f"'.Jp /\ ,-...., ,"",P, (P ---+ Q) /\ ,.....,,(P ---+ Q) are all contradictions. Now the last rule states that given a derivation of a contradiction B /\ ,....."B from an assumption A and any other assumptions 6., we may derive ,....."A from 6.. Here is the statement of the rule:

°

RULE

OF

reductio ad absurdum (RAA): 6.,rA~B/\,....."B

,......,A

A A

1,2 Conj RAA 4,DN

81

Assumption Discharging Rules

Since we count any line as following from itself, the shortest possible proof can be constructed for P ~ P, namely

A

1.P

We illustrate a second use of RAA in proving the validity of an argument pattern.

·,. . ",. . " U

Prove: P ---+,....."p 1. P

~,....."p

---+ ,....."P

P

3. P

4. ,....."p

5. P /\,....."p 6. ,....." ,......,P 7. ,....."P r--J

A A

2,DN 1,3 MP 3, 4 Conj RAA 6, DN

This proof can be shortened by assuming P rather than "" ",-,P. To assume either,....." ,....."p or P is to assume the contradiction of the conclusion. To end this section, the ten rules for constructing proofs are summarized below. 1.

(A): An assumption may be introduced at any point in a derivation.

ASSUMPTION

2. Mod7ls ponens (MP): A---+B A B

3. Modus tollens (MT): A---+B ,....."B ,....."A 4.

In using RAA for constructing proofs for argument patterns the procedure is to enter the denial of the conclusion as an assumption in addition to the premise assumption, 6., and to derive a contradiction. Having derived the contradiction we can enter the negation of the additional assumption, discharging the additional assumption. The use of RAA is illustrated in this proof of the simplest possible argument pattern: Prove: P ~ P l.P I~' ,....."P L..! P /\ ,,-,P 4. ,.....",......,p 5. P

4.3

SIMPLIFICATION (S) :

A/\B

---:::t

A/\B B

5, DOUBLE NEGATION (DN):

A ,....." ,....."A '

6.

,....." ,....."A A

OONJUNCTION (CONJ):

A,B A/\B

7.

ADDITION (ADD):

A AvB'

A BvA

Natural Deduction System SC

82 8. DISJUNCTION (DIS): 6, A vB 6'bA f-O 6,

4.4

o

A-+B 10. Reductio ad absurdum (RAA): 6,c A f-BA,-.,B

"",A Exercises 1. Construct a proof for each of the following argument patterns. P V (P A Q) f- P P -+ Q, Q -+ R f- P -+ R P -+ Q f-,-.,Q ---+ ",-,p P f- ,-.,(Q A ,-.,Q) (,-.,p -+ Q) A r-;Q f- P

(f) (g) (h) (i) (j)

P -+ (P -+ Q) f- P -+ Q P -+ (Q -+ R) f- P A Q ---+ R P V P f- ,-.,(,-.,p A ,-.,P) ,-.,P -+ P f- P P ---+ Q A R f- (P ---+ Q) A (P ---+ R)

2. Why does the proviso: "One cannot use a line in the scope of a discharged assumption unless the line can be gotten from other assumptions" prevent crossing lines?

p a

Answers

1. (a) 1. P V (P A Q) 3. P AQ 4. P 5. P (i) 1. ,-.,P -+ P

a~p

3. P 4. P A,-.,P 5. ,-.,,-.,P 6. P (j) 1. P-+Q A R

~p 3. Q A R 4. Q 5. P ---+ Q

~p 7. Q A R 8. R 9. P-+ R 10. (P ---+ Q) A (P ---+ R)

"

Useful strategies

Bf-O

9. CONDITIONAL PROOF (RCP): 6,CA f-B

(a) (b) (c) (d) (e)

83

Useful Strategies

A A A 3,8 Dis A A 1,2 MP 2, 3 Oonj RAA 5,DN A A 1,2MP 3,8 RCP A 1,6 MP 7,8 ROP 5,900nj

A computer could be programmed that would produce a proof for all valid argument patterns expressed in the sentential language, and would, after a time, give up, if the pattern is invalid. But we do not have access to such a machine, and even if we did, we ought to ignore it at this stage. We wish to acquire the ability to construct proofs on our own. This is a creative act in most cases and thus requires insight, diligence, and, often, luck. However, there are certain strategies that will keep one from blind alleys and that are generally successful. Here are some of these strategies. STRATEGY ONE: If the desired conclusion is of the form A ---+ B, add A as a further assumption and deduce B. Then use RCP to obtain A ---+ B. In turn, if the consequent of a conditional conclusion is itself a conditional, then assume the antecedent of that conditional, and so on. '1'0 illustrate: P ---+ (Q ---+ R) f- Q ---+ (P ---+ R) 1. P ---+ (Q -+ R)

~

'Q 3. P Q ---+ R

r::liR

6. P ---+ R 7. Q ---+ (P ---+ R)

A A A 1,3 MP 2,4MP ROP ROP

We should note that if the conclusion had the form (A ---+ B) ---+ C, to use this strategy one must assume the antecedent that is A ---+ B, not A. STRATEGY TWO: If one of the premises or assumptions is of the form A V B, then try Dis unless there is some other easier procedure. To illustrate: (P ---+ R) A (Q ---+ R) f- P V Q ---+ R ---+ R)

1. (P ---+ R) A (Q 2. P V Q 'P 4. P ---+ R 5. R 6. Q [ 7.Q---+R

I~R

UR

10. P V Q ---+ R

A A A

1,8 3,4MP A 1,8 6,7 MP Dis RCP

Natural Deduction System SO

84

Line 2 was introduced using strategy one. Since line 2 is a disjunction, we engage in Dis strategy assuming P and Q and obtain R from each. STRATEGY THREE: When the assumptions seem inadequate to get the desired conclusion, then use RAA. RAA will always work. To prove ,",-,(P A '"'-'Q) f- P ~ Q makes an interesting proof, for we can assume P for an RCP proof and also engage in RAA by assuming "-'Q. To illustrate:

85

Useful Strategies

same procedure with Q. The proof is thus: A A A 3,S 2,4 Conj

RAA A A 8,S 7,9 Conj

A A A 2,3 Conj 1,4 Conj

RAA Dis

RAA 6,DN RCP

Rather than write out lines 1 and 4 on line 5, we have indicated them by line number. STRATEGY FOUR: Conj, Add, RCP, and DN (one way) are rules for introducing connectives, whereas MP, MT, S, DN (the other way), and Dis are rules for eliminating connectives. Generally speaking, in a proof use at first as many of the elimination rules as possible to get as many component variables separated on lines as possible. Then, if necessary, use the introduction rule corresponding to the main connective in the conclusion. Thus one uses elimination rules on the premises, introduction rules on the conclusion, and one works towards the middle. Such considerations often lead to adopting one of the strategies above and, with good fortune, can make clear a successful proof procedure. To illustrate, suppose we are called upon to validate the following argument pattern:

Working from the premise, we see we can use the elimination rule Dis. The conclusion is not in conditional form and we are unable to get it from P alone and then from Q alone in a Dis strategy; but we could use RAA along with a Dis strategy. In using RAA within the Dis strategy we need to obtain the conclusion twice, once from P and once from Q. To obtain it from P using RAA we must assume its negation '"'-'P A '"'-'Q, and in obtaining it from Q we must again assume its negation. Having assumed P and '"'-'P A "-'Q, we can use an elimination rule, S, on "",P A '"'-'Q to obtain ,",-,P, which contradicts P. And we may follow the

,.

With line 1 being the only A line left we have derived 12 from 1, thus establishing the above argument pattern.

Exercises I. Construct proofs for the following patterns. (a) (b) (c) (d) (e)

P --+ Q f- ,",-,(P A ,,-,Q) P--+Qf-(Q--+R)~(P~R) 8 --+ (P ~ Q) f- (8 ~ P) ~ (8 ~ P A Q ~ R f- P --+ (Q ~ R) "",p f- P --+ Q

Q)

P f- Q --+ P P V Q f- (P ~ Q) ~ Q Q f- f',JR --+ f',J(Q ~ R) ",(P V Q) f- f',JP A f',JQ P V Q, '"'-'P f- Q "",p V '"'-'Q f- ,",-,(P A Q) """'(f',JP A f',JQ) f- P V Q P V (Q A R) f- (P V Q) A (P V R) (n) P A (Q V R) f- (P A Q) V (P A R) (f) (g) (h) (i) (j) (k) (1) (m)

Answers

.r:: hlLi

(Note: There are other ways to prove these patterns.) I. (b) 1.

P~Q

Q~R

A A A

3. P Q R

1,3 lVIP 2,4lVIP

6. P~R 7. (Q~ R) ~ (P--+ R)

RCP

RCP

86

A A

(e) 1. ,......,P 2' P

~

3. ,......,Q P A t'o.'P 5. ,......,,......,Q

[i;

6. Q

7. (i) 1. 2. 3. [} 4.

V

Q)

. [i

Q 7. P vQ

8. (7) A (1) 9. "",Q 10. ,...."p V ""Q

(j) 1. P V Q 2. t".IP

~

'P

4. ,......,Q PA""""P

[i

6. Q 7. Q 8. Q (k) 1. ",-,P

2. 3. . 4. 5. 6. 7. 8.

[1

A 1,2 Conj RAA 5,DN

RCP

P-+Q

,......,(P

P PVQ (3) A (1) 5. f"o-/P

4.5

87

Natural Deduction System SO

A A 2, Add 1,3 Conj RAA A 6, Add 1,7 Conj RAA 5,9 Conj A A A A 2,3 Conj

RM V

,....""Q

PAQ P

,..,.,,......,P ,......,Q

Q Q A,......,Q ,......,(P A Q)

A Dis A A 2, S 3,DN 1, 4 (by following proof of (j)) 2, S 5,6 Conj RAA

follow the practice of assuming A, deriving B, and thus by Rep obtaining A -+ B. In using ROP in this way,/'., is empty. If B is derived from A without any other assumptions by using valid sentential rules, then B is a tautological consequence of A alone, and thus A -+ B is a tautology-what we desire. Earlier it was announced that the system will be called SO and that it is a system for the sentential calculus. The sentential calculus, as indicated at that time, can be identified with an infinite number of systems that have the feature of having theorems which are only tautologies and having all tautologies as theorems. The system for the sentential calculus, system SO, is made up of four elements. First, SO contains rules for a well-formed formula in SO. These rules are the same as those for a well-formed formula in the sentential language set down in Section 2.6. Second, SO contains ten basic or primitive rules. The ten basic rules for derivation in SO are those summarized at the end of Section 4.3. Third, in SO we wish to have theorems that include the biconditional sign, +-0. As the inference rules are set up, though, ,......" A, v, and -+ can be introduced in a proof, +-0 is not introducible. So we will provide a definition for this connective in terms of connectives that can be introduced in proofs. \iVe thus introduce the following definition for the biconditional: A

+-0

B

=df

(A

-+

B) A (B

-+

A)

The effect of such a definition is to permit us to replace instances of what is on the left side of the =df sign with instances of what is on the right side, and vice versa. When this definition is applied we shall cite 'df'. Some uses of this definition are illustrated below:

System SC

In the next three sections we will ignore argument patterns and turn to our primary interest, which is a system that has only tautologies and all tautologies as theorems. Since a tautology is not an argument pattern, proofs will not begin by taking premises as assumptions. Rather proofs will begin by using either ROP or RAA. That is, the assumptions will not be premise assumptions but will be the assumptions obtained from using ROP or RAA that are later discharged. As we will see, most of the theorems of the system will be conditional formulas. With a formula of the form A -+ B we will generally

(P-+Q) P+-0Q

A

P +-0,......,R (P -+ ,......,R)

(Q-+ P)

df (,......,R

-+

P)

(P A Q -+ P) A (P PAQ+-0P

-+

P A Q)

A

df df

Ji'inally, we need an effective procedure for determining when we have a proof and a theorem in SO. vVe need a defillition for a proof and a

theorem in SC such that a computer could be programmed to check whether what is written down is a proof or a theorem. A proof in SC will be such that every step can be checked to see if it is justified, with the checking relying only on the forms of the steps, not on the interpretation of the symbols. To this end we first define a deduction in SC. A decl~lction is a finite sequence of formulas such that for each formula at least one of the following holds: 1. It is an assumption line. 2. It is a line inferred from previous lines that are not in the scopc

of assumptions already discharged using one of the ten basic inference rules.

A proof is a deduction from the empty set. That is, when in a deduction all the assumptions have been discharged, we have a proof. A theorem is the last line of a proof. Theorems in SC will only be tautologies. In the next two sections a number of theorems will be derived and two derived rules will be introduced. Many of the theorems will be conditionals corresponding to previously validated argument patterns, and the strategies followed in validating the argument patterns are and should be used with the theorems. We will continue to number the lines of a proof and to indicate origin and rules on the right of each line, though such procedures are not part of the definition of a proof in SC. But since this discussion is intended for human beings rather than machines, we shall take this liberty.

Exercises 1. How many theorems are there in SC? 2. 'Vhy are lines 1, 2, and 4 in this sequence not sequences in a proof? 1. "-' P A "-'Q 2. ,,-,(P A Q) 3. P A 4. "-'Q

3. Provide an inference rule that will allow the introduction of +-t into proofs. Would this alone be adequate for all tautologies using +-t?

89

Some Theorems of SO

Natural Deduction System SO

88

Answers 1. An infinite number. Why? 2. 1 is not an A line, 2 does not follow from 1 and is not an A line, and 4 is not

the last line in a sequence in which there are no undischarged assumptions.

4.6

Some theorems of SC

Only two rules, RCP and RAA, allow us to introduce assumptions so that proofs can get started. How this is done is abundantly illustrated below. The collection of theorems is chosen for special purposes, namely they are the major tautologies that should be studied until they become selfevident. Where a theorem is listed without proof it is expected that the reader will demonstrate it as an exercise (if its corresponding argument pattern has not been proven in previous exercises). The reader is urged to work carefully through each of these proofs, making sure that he sees what is being done at each step and why it is a reasonable move to make. Some discussion will follow some of the proofs, indicating abbreviations and reminding the reader of aspects of the strategy. Tl

I- P-+P l.P 2. Tl

Law of identity A

RCP

Though this proof is quite simple, it illustrates a basic strategy. In proving a theorem (abbreviated T) ofthe form A -+ B, the form of most of our theorems, the antecedent is taken as an assumption and the consequent is derived. Having done this, RCP gives us the theorem. 'When the theorem to be proven has the form A -+ B, as Tl has, we assume A, derive B, and use RCP to obtain the theorem. If the theorem to be proven has the form A -+ (B -+ 0) we assume A and B and derive 0, and so on, just as we did with argument patterns. We note that in the above proof all assumptions have been discharged when we come to line 2; thus, given the statement of a proof in SC, 2 is a theorem. It should again be pointed out that we will regard any line as derivable from itself. Thus the appropriateness of applying RCP on line 2. The sign I- is placed between P -+ P and T1. This indicates that P -+ P is a theorem or is provable in SC. From this point on we will regard the theorem numbers, 'Tl' in this case, as doing the job of 'Tn 1-' when they appear before a formula.

T2

P

---+

P

vQ

Addition

Tl7

A

I~' P

L2

PVQ

3. T2

1, Add ROP

· U

P V ""p 1. ",,(P V ""P)

P

3. P

V

""P

4. (3) A (1)

5. ""p 6. P V ""P 7. (6) A (1) 8. r....;r....;(P V r....;P) 9. P V ""p

Since T6 is an instance of A ---+ (B ---+ 0), we have taken A and B as assumptions in our attempt to obtain O.

"'-I.

Dis Dis ROP

T18 T19 T20

(P V Q) V R ---+ P V (Q P A (Q A R) ---+ Q A (P Q A (P A R) ---+ P A (Q (P V Q) A I'.JP ---+ Q

V R) A R) A R)

Disjunctive syllogism (DS)

'['21 may be proven by following the RAA strategy found in the answer for Exercise 4.4.1 (j). As indicated earlier, any strategies used to prove an argument pattern that corresponds to a theorem can be adopted to a proof for the theorem. It will prove useful to abbreviate the name of T21 as DS and indicate the theorem by name as we will have occasion to do with other theorems.

1,2 MT ROP ROP

TIl T12 T13 T14 T15 T16

A 9, Add

In this proof we first assume the antecedent of the theorem, line 1. '['hen from line 1 we take each disjunct and try to derive (P V Q) V R, the consequent of T17, by Dis. We can easily get this from the first disjunct P. But to get it from Q V R, line 5, we must again use Dis assuming Q, line 6, and R, line 9.

A A

TlO

6, Add 7, Add

13. T17

2, Add 1,300nj RAA 5, Add 6, 1 Oonj RAA 8,DN

---+

2, Add 3, Add A A

~

A A

T21

(""Q ---+ ""P) ---+ (P P V Q ---+ Q V P Q V P ---+ P V Q P A Q ---+ Q A P Q A P ---+ P A Q

3. PVQ

9. R 10. (P V Q) V R 11. (PV Q) V R 12. (P V Q) V R

Excluded middle

A A

4. (P V Q) V R

5. Q V R 'Q 7. P VQ 8. (P V Q) V R

In this interesting proof we cannot use ROP since the theorem is not a conditional.

T7 T8 T9

Pv(QvR)---+(PvQ)vR 1. P V (Q V R)

2. P

T2 corresponds to the rule of addition. If SO is to have all tautologies as theorems, then a theorem corresponding to each rule, excluding A, must be provable. T3

91

Derived Rules

Natural Deduction System SO

90

Exercises 1. Oonstruct a proof for each of the underived theorems. It is permissible to omit proofs for those theorems whose corresponding argument patterns have been proven in previous sections or in the exercises.

Q)

P V P ---+ P (P ---+ (Q ---+ R» ---+ (P A Q ---+ R) (P A Q ---+ R) ---+ (P ---+ (Q ---+ R» (P ---+ Q) ---+ ((Q ---+ R) ---+ (P ---+ R» (P ---+ Q) A (Q ---+ R) ---+ (P ---+ R)

4.7

Importation Exportation Transitivity of implication } (Trans)

Two derived rules

To simplify the construction of proofs we will introduce two derived 1'ules. A derived rule allows us to do no more than what we can do with the ten primitive rules. The usefulness of derived rules is that they allow us to do a proof in fewer steps, or they allow easier proofs. Whenever a line in a proof is introduced by a derived rule, this will mean that if we had chosen, we could delete the derived rule and use some combination of the ten primitive rules.

l'latUral uelluctlOn System

The first derived rule may be stated as follows:

this proof T2I-disjunctive syllogism-is used to separate Q from Q given ,""P. No substitution is needed in using T21. If we had to separate P --+ R from P V (P --+ R) given ,""P, then we would need to make a substitution in T21. Again, since TI is a derived rule the proof T24 can be done using only the primitive rules. One way is to inject into the above proof the proof for T2I, obtaining line 4. Another way is illustrated in the proof for the argument pattern P V Q f- "'-'( '""P A ",-,Q) found in Section 4.4.

p

(TI): A theorem or substitution instance of a theorem may be entered on any line of a proof.

RULE OF THEOREM INTRODUCTION

It will be recalled from the previous chapter that one obtains a substitution instance of a formula when one uniformly substitutes a well-formed formula for a single sentential variable. The use of TI is illustrated in the next proof. T22

P ---+ (Q 'P

---+

V

T25 ",-,(,,-,P V Q)

A

2. Q

A

3. P ---+ P 4. P 5. T22

TI (TI) 1,3, MP RCP (2)

1. "'-' "'-'P

~

P

A

2. Q

A A

cl P 4. P 5. P

---+

P

6. T22

P A "-'Q

To shorten proofs, we will collapse introducing a theorem (with or without substitution) and MP into a single step. Thus we may make this mOve, for example:

No rule so far allows us just to repeat line 1, so TI along with TI plus MP is used at this point. The use of TI in proving T22 can be abandoned if we like. This is a necessary feature of a derived rule: it is dispensable: In effect TI (T 1) at the justification place on line 3 means 'at this point insert the proof for TI'. Thus we have not a proof, but a proof outline. A proof of T22, one using only the }}asic rules, would be: .

--+

Law of Clavius

P)

O

93

Derived Rules

--+

P

--+

Q

2. P--+Q

3. "'-' "-'P

1,2 Trans

rrrans here collapses several steps, which may be explicitly unpacked as follows: 1. "'-' "-'P --+ P 2. P--+Q 3. (P --+ Q) --+ ((Q --+ R) 4. (1) --+ ((2) --+ (5)) 5. "'-' "-'P --+ Q

RCP I,4MP RCP (2)

T26

(P ---+ Q)

--+

,,-,p

--+

1. PVQ 2. r--;p II "'-'Q 3. 4. 5. 6.

'""-'P (P V Q) II ",-,p ---+ Q (P V Q) A ,,-,P

Q

7. "-'Q 8. Q II "-'Q 9. "-' ("-' P A ,,-,Q)

10. T24 """/-,

6. "'-'Q

A

A 2, S

TI (DS) 1, 3 Conj 4,5 MP 2, S

6, 7 Conj RAA RCP

T27

(,,-,P ---+ Q)

--+

TI(TI5)

V Q

4. P 5. Q 7. Q A "'-'Q 8. "-' ,,-,(,,-,P 9. ",-,p V Q 10. T26

R))

1,2,4 MP (2)

~3. ~(~PvQ) P A "-'Q

T23 ",-,p ---+ (P ---+ Q) Law of Duns Scotus T24 P V Q ---+ ",-,(",-,P II '"'-'Q)

---+

"'-' "-'PIP, P/Q, QIR in 3

1. P--+Q As can be seen, the earlier proof of TI has merely been injected at the point where TI(TI) appears.

(P

V Q)

A A 2, T25 3, S I,4MP 3, S 5,6 Conj RAA 8,DN RCP

P V Q

The definition for ~ can be used to obtain biconditional formulas from previously proven theorems. The strategy when one has a theorem of the form A ~ B is to construct a proof for A ---+ B and a proof for

Natural Deduction System SC

95

B. This strategy is illustrated in

introduction of the JYI +-t N line. Thus we may move directly from 2 to 4 with the justification being, simply, T28. And we may move from 4 to 6 with the justification being, simply, Idem. Placing 'Idem' at the 'ustification spot in such a move leaves it ambiguous whether TI and ~IP or R is being used, but this is no matter of concern.

94

B -+ A and then use df to obtain A the proof of the next theorem. T28

P +-t "'-' ",-,P 1. P -+ "'-' ",-,p 2. "'-' ",-,p -+ P

3. 4. T29 T30 T31 T32 T33 T34 T35 T36

(1) 1\ (2) T28

P

V

Q +-t Q V

P}

+-t

Double negation T4 T5 1, 2, Conj 3, df

P 1\ Q +-t Q 1\ P P V P+-t P} P 1\ P+-t P (P V Q) V R +-t P V (Q V R)} (P 1\ Q) 1\ R+-t P 1\ (Q 1\ R) (P +-t Q) +-t (P -+ Q) 1\ (Q -+ P) P -+ Q+-t~P V Q

T38 T39 T40 T41 T42 T43 T44 T45

Commutation Idempotent Association Equivalence Implication

T46

T4S T49 T50

Nand f- AM, then f- AN.

In using this rule we must have previously proven biconditional theorems or substitution instances of them so as to obtain M +-t N. Let us illustrate the use of this rule.

· U

T37

(",-,P -+ P) -+ P ~P-+P

2. ~,-.."PV P 3. "-' r-..Jp +-t P

4. Pv P 5. P V P+-t P

6. P 7. T37

Conseq1lentia M imbilis A 1, T26 T28 3,2, R Idem 4,5, R RCP

Line 3 is the double negation theorem. In the first application of R, M +-t N is line 3. AM is line 2 and AN is line 4. In the second application of Equiv, M +-t N is line 5. AM is line 4 (here A is M) and AN is line 6 (here A is N). A much shorter proof of T37 can be constructed using RAA (see exercise 4.3.1 (i)), but this longer proof usefully illustrates two uses of R. In using R we may shorten the proofs further by omitting the

V

-+

(P V Q) 1\ (P V R)

Q+-t r_+_"P 1\ ",-,Q)}

"'-' (P V Q) +-t "'-' P 1\ "'-'Q ,,-,(P 1\ Q) +-t ",-,P V ,......,Q P 1\ Q +-t r_+'_' P V ",-,Q) P -+ (Q -+ R) +-t P 1\ Q -+ R "'-'( P 1\ ",-,P)

P

1\

de Morgan's Theorems Export-Import (E-I) La w of contradiction

(Q V R)

}

P V (Q

Q) V (P

1\

R)

Distribution

R)

1\

+-t (P V

T47

RULE OF REPLACEMENT(R): +-t

P

+-t (P 1\

The second derived rule will make use of biconditional theorems. The rule will permit the replacement of A with B (and vice versa) whether A and B are alone or in a formula if A +-t B is a theorem. To state the rule, let M and N be well-formed formulas. Let AM be a formula such that M is A or part of A. Let AN be the result of replacing one or more, but not necessarily all, occurrences of M in AM by occurrences of N, then

If f- M

P V (Q 1\ R)

Q) 1\ (P V R) (P -+ R) 1\ (Q -+ R) -+ R P -+ Q +-t "'-'Q -+,.......P (P-+Q) 1\ (R-+S) 1\ ("'-'Q V ",-,S) -+,......,P v,.......R (P-+Q) 1\ (R-+S) 1\ (P V R) -+Q V S (P V Q)

1\

T51 T52 T53

P V (P 1\ Q) +-t P P 1\ (P V Q)+-tP (P -+ Q) 1\ (P -+ R)

T54 T55

("",P -+ R 1\ ",-,R) -+ P } (P 1\ "'-'Q -+ R 1\ ",-,R) -+ (P -+ Q)

Proof by cases Contraposition Destructive dilemma (DD) Constructive dilemma (CD)

}

Absorption

+-tP-+QI\R

. . Proof by contradIctIOn

Exercises 1. Construct a proof for each of the underived theorems. 2. Since any line is taken as derivable from itself, shorten the proof of T22. 3. The ten primitive rules are not all independent. We can dispense with some and still obtain the same set of theorems. If one can, say, derive a theorem corresponding to one of the rules not using this rule, then this theorem plus theorem introduction, substitution, and MP allows one to do all you can do with the rule. Thus the rule is a dependent rule relative to the other rules. For some of the rules derive a theorem corresponding to the rule not using the rule or derived rules.

4. Give a proof using only the ten primitive rules for one of the proofs in which you used derived rules. 5. ]<"'or cach argument derive the conclusion from the premises, using any of the rules, listed theorems, and derived rules of SC. See answer to (g) below for procedure.

Natural Deduction System SC

96

(a) If there is no test, then the students will be happy only if the professor stays in bed. The professor is in class every day. If the students are happy, then there is no test. Therefore the students will not be happy. (b) Either the students will be happy and there will be no exam or the students will be happy and class will be dismissed. If there is no exam, then the students will be unhappy. Therefore class will be dismissed. (c) If either the students are happy or classes are dismissed then Roger loses his bet. If Roger loses his bet, then Stan gains. Classes are to be dismissed and Stan will not gain anything. So the students are not going to be happy. (d) If you are interested in history or in scenery, then you will leave the main road. If you are not interested in history, then you won't leave the main road. Therefore if you are interested in scenery, you are interested in history. (e) Logic makes no sense. Therefore either I give it up or logic makes sense iff I give it up. (f) The instructor will be pleased if Anderson takes the test and passes. But if Anderson takes the test and writes illegibly, the instructor will not be particularly pleased. In fact Anderson will pass the test only if he writes illegibly. (Then the instructor won't be able to see how bad his exam is.) So Anderson won't both take the test and pass. (g) If JB has his way and KF is defeated, then the Fitzwigg dynasty will come to an end. But it is not true that the Fitzwigg dynasty will come to an end if KF is defeated. Therefore if KF is defeated, JB will not have his way. 6. For each valid argument found in exercise 2.5 and exercise 3.3, derive the conclusion from the premises, using SC.

Answers 3. For example, MT can be gotten from the other rules minus MT. So can MP, Add, and half of DN.

5. (g) Let L: JB has his way R: KF is defeated F: Fitzwigg dynasty will come to an end The proof of the validity of this argument may be given as follows: Prove: L 1\ R --+ F, ",,(R --+ F) I- R --+ ""L 1.LI\R--+F A 2. ",,(R --+ F) A 3. R A 4. ",,(,-.....;R V F) 2, Imp 5. "" ""R /\ ""F 4, DM 6. ""F 5, S 7. ,-.....;(L 1\ R) 6, 1 MT 8. ""L V ""R 7, DM 9. """"R 5, S 10. ,-.....;L 8,9 DS 11. R --+ r--I L RCP

4.8

4.8

Soundness, Consistency, and Completeness of SC

97

Soundness, consistency, and completeness of SC

However confident we may be with the version of the sentential calculus we are calling system SC, we need to raise and provide answers to these questions: l. Is the system consistent? 2. Are the only formulas provable in the system tautologous? 3. Are all tautologous formulas provable in the system?

Such questions are questions about system se and are spoken of as metalogical questions. The theses in se are exhausted in the theorems of se. Thus the statement 'se is consistent' is a truth that goes beyond se and is about se and is not a thesis in se. There are several senses of the term 'consistent' when applied to a system, but the basic notion of consistency is what is called negation consistency or simple consistency. A system is negation consistent if and only if the system does not permit the proof of a theorem of the form A 1\ ,.....,A. When a system has as theorems only acceptable formulas, the system is said to be sound. If it can be proven that only tautologies are theorems of se, se will be proven sound. A system is said to be complete, in one of the senses of this term, if all the acceptable well-formed formulas ofthe system are theorems of the system. To say se is complete is to say that all tautologous formulas are theorems. Thus to prove se complete is to provide a decisive answer to the third important metalogical question above. Why is negation consistency a desired feature for system Se1 If se were inconsistent, then it would not serve the purpose for which it was constructed, viz., providing or establishing tautologies. For if se lacks consistency, then any formula would be a theorem. One way to show this is to suppose that an inconsistency, say P 1\ ""P, is a theorem of se. Since disjunctive syllogism is a theorem of se, we could produce this proof: 1. P I\,....,P

2. 3. 4. 5.

P PvQ ""P

Q

TI 1, S 2, Add 1, S 3,4 DS

And if Q is a theorem, then given theorem introduction, any wellformed formula is a theorem of se. If this obtained, se would be a useless theory.

98

Natural Deduction System SC

How are we to go about showing that SO is consistent? Olearly, our not having already derived P /\ ",-,P is not sufficient to establish the consistency of SO. We need some systematic method. One such procedure is to show that every theorem of SO must be a tautology. In other words if we can show that SO is sound, then it must be consistent, since no formula which is equivalent to F can be a tautology. A precise proof of the soundness of SO will be taken up in Part II of this book. At this time let us consider some of the notions relevant to a proof of the soundness of SO. We note that when a theorem is obtained in SO, then either ROP or RAA is used and all assumptions are discharged. If either rule is so applied, then the result, a line from no assumptions, is a tautology if the derived line-the B from A or B /\ ",-,B from A-is a tautological consequence from the final discharged assumption. Oonsider ROP and RAA in turn: ROP Supposing from assumption A there is obtained the line Band B is a tautological consequence from A, then A --'>- B is a tautology. Note that if B is not a tautological consequence from A, then A --'>- B is not tautologous. RAA Supposing from A there is obtained the line B /\ ",-,B as a tautological consequence, then A --'>- B /\ ",-,B is a tautology. If A --'>- B /\ r-->B is a tautology, A must always be F; thus r-->A is a tautology.

4.8

Soundness, Consistency, and Completeness of SC

Exercises 1. Suppose a nontautologous formula to be a theorem and show how this would make SO inconsistent. 2. Supposing SO to be sound, show that SO is absolutely consistent (a system is said to b~ abso~utely c?nsistent if not eve~y wff ?f the system is a theorem) and consIstent m Post s sense (a system IS consIstent in this sense if a wff consisting of a sentential variable alone is not a theorem). 3. If SO is complete,. sh~w why the premises of a sentential argument logically imply the conclUSIOn rEf the corresponding conditional of the argument is an instance of a theorem of SO. 4. Show for each rule in SO that if it is applied to a line or lines, the result is a tautological consequence from these lines. 5. There is an interesting alternative method for demonstrating validity using rules called the method of semantic tableaux, 01' the tree method. The method is quite simple. Begin with premises and the denial of the conclusion. For each compound statement (now we treat denials of simple statements as noncompound statements) we indicate below all the conditions under which the statemer:t is true. If there a.r~ more than two conditions, we employ a fork; otherWIse we stack the condItIons. An illustration will show how the t~'ee is built. The tree for

P

--'>-

Q I- """Q --'>- ",-,P

would be as follows:

If when ROP or RAA is used, all assumptions have been discharged, and if the derived line is a tautological consequence from the final discharged assumption, and if a theorem can only be obtained by using ROP or RAA in this way, then all theorems must be tautologies. If all theorems are tautologies, then SO is sound. And if all theorems are tautologies, no theorem equivalent to F, no inconsistency, is a theorem. Proving that SO is complete is also a task that will be saved for Part II. We note that neither SO's soundness nor its consistency implies that SO is complete. However if SO were inconsistent, then SO would be complete. For if SO were inconsistent, any well-formed formula would be a theorem and thus all tautologies would be theorems. Why is the completeness of SO desirable? SO is constructed to establish tautologies. If SO were incomplete, then it could not be used to establish all tautologies. Furthermore we wish to say that the premises of a sentential argument logically imply the conclusion iff the corresponding conditional of the argument is an instance of a theorem of SO. This claim would be false if SO were not complete.

99

P --'>- Q "'-' (r-->Q -+ "'-' P)

1. 2. 3. 4.

"'-'Q

P

AQ 5. ",-,p x

x

Line 1 is the premise. Line 2 is the denial of the conclusion. Lines 3 and 4 are the. conditions in which line 2 would be true-both r---;Q and P must be true. Smce both """Q and P must obtain, "'-'Q and P are stacked. We now t~rn to the only remaining compound statement, line 1. Line 1 is true iff eIther ",-,p or Q; thus we use a fork. 'Ve find two paths on the tree. If we had another compound sentence to break up, the truth conditions would have ~o be ~dded to each path of the fork. Going up each path, we encounter an ~n~?nsI~tency-thus the use of 'x' at the end of each path. A path with an x IS saId to be closed. An argument is valid iff all of the paths of its tree are closed. All compound sentences in paths must be broken down unless all the paths have closed. Ajinished tree is one with all closed paths, or one in

Natural Deduction System SO

100

which all compound sentences have been broken down (or both). Consider these next examples of finished trees.

P -+ Q, P V Q I- P P-+Q PvQ

PI\QI-PvR PI\Q r-.I(P V R) ,--,P

Q

predicate language

,--,(P 1\ Q)

,--,R P

,--,p

Q x

1\

P x

~

"-'Q

~

Q

A

P

~

/\

,--,p Q ,.....,P Q x

Q

Not all valid arguments are valid because the conclusion is a tautological consequence of the premises. To illustrate consider the two following arguments:

x

All men are mortal. All Greeks are men. Therefore all Greeks are mortal. All circles are figures. Therefore whoever draws a circle draws a figure.

x

The first is valid; the second is invalid. (a) Use the tree method to test the validity of some of the exercises in 4.3 or 4.4. (b) Use the tree method to establish the validity of some theorems of SO. Note A -+ B is valid iff ,,-,(A -+ B) has a tree with each path closed. (c) If a path does not close in a finished tree, this path provides a counterexample. Read off some counterexamples from trees for invalid argument patterns. (d) Why does the tree method work? Answers 2. If A is a single sentential variable, then A is a nontautologous wff. Since all theorems of SC are tautologous, A is not a theorem; thus SC is consistent in both senses.

Each argument is valid, for it is evident that the conditionals corresponding to each argument are necessarily true. However, if we replace each simple statement with sentential variables and supply the appropriate statement connectives, the result would not be valid argument patterns. Neither

P,QI-R PI-Q nor their corresponding conditionals can be established using the system SC for the sentential calculus .. What this indicates to us is that the validity of the above arguments is not merely a matter of how simple statements are related by statement connectives, but depends on the inner structure of the simple statements. In the first argument, for example, the validity depends on the sense of 'all' as well as the way the words 'men' and 'mortal' are related. The object of this chapter is to make additions to the sentential

,.

101

102

Predicates

language so that the inner structure of simple statements can be reflected. The result will be called the prediwte language. Later (Ohapter 6) the system for establishing valid formulas that reflect the inner structure of simple statements is introduced.

5. 1

Predicates

Several kinds of symbols are needed for the predicate language. One such symbol is what in this book will be called a predicate term. To begin the explanation of the notion of a predicate term, let us consider the following expressions:

By singula-r term will be meant any expression that names or picks out a single thing-whether or not that individual exists in the realm we have under consideration. The above examples indicate that proper names, demonstrative expressions, and what are called definite descriptions can all be used as singular terms. Abstract nouns are also singular terms. ]l'or example, the subject term of the next sentence is a singular term. Courage is a virtue. vVhen we have expressions that contain one or more individual variables and that produce a statement when a singular term (or terms) replaces the variable(s), we have what are called open sentences. Thus 'x is a person' and 'x > 7' are open sentences as well as 'x is married to y' and 'x = y'. One way to obtain open sentences is to replace the occurrences of names with individual variables. Thus from the sentences

x is a person. 7.

George Washington is alive in Argentina. The number 6 > 5.

x>

These are neither true nor false. But one could obtain true or false statements from these expressions in this way: Substitute a name for one of the things the x ranges over in each of these expressions. It is safe to say that in this first expression x ranges over individuals such as Barry Goldwater, Bertrand Russell, and Tiny Tim. Thus if we supply a proper name for a person we obtain a statement from the first expression. For example, if we supply 'Shirley Temple Black' for x in the first expression we obtain the true statement: Shirley Temple Black is a person. Similarly, if we supply numerals for x in the second expression, we can obtain a statement, for the x here is naturally taken to range over numbers. For example, if we supply 8 we obtain a true statement, but if we supply 7, we obtain a false statement. The letters x, y, z, ... used as variables for names, as x is used above, are called individual variables. Individual variables will be one kind of symbol that is necessary for the predicate langugage. Individual variables may take singular terms as values. Thus fDr the first expression, x is a person, we may obtain the following instances: Strawson is a person. That fellow is a person. The philosopher who wrote The Bounds of Sense is a person. ~I·,

103

we may obtain open sentences as follows: x is alive in Argentina. The number x > 5. The number x > y.

x is alive in y. The number y > 5. George vVashington is alive in x.

or from Straws on knows Jones is a fink. For all x, x > O. we may obtain the open sentences x knows Jones is a fink. For all x, x > y.

Open sentences are represented in a special way in the predicate language. An open sentence is indicated by a capital letter followed by the individual symbols involved. Thus if the open sentences are x is a little girl.

x melts.

they will be represented in this way: Lx

Mx

104

105

Predicates

where L is taken to stand for 'is a little girl' and M is taken to stand for 'melts'. Similarly, if we allow Nto stand for' is next to , x is next to y becomes Nxy. To indicate the open sentence that, say, Fx represents, we wiII use the following form:

Fx:xis--Thus to indicate that Gxyz is the open sentence 'x gave y to z', we would write:

Gxyz: x gave y to z. It makes no difference, we should note, what individual variables we use to represent open sentences. Thus the last may be written:

Gzyx: z gave y to x. An expression such as Lx is also known as a one-place predicate. Nxy is a two-place predicate, and Gxyz is a three-place predicate. A predicate may be defined as what occupies the F position in

F( where ( indicates the positions to be occupied by individual variables. An n-place predicate is an open sentence containing n different kinds of individual variables. In Lx, n = 1; thus Lx is, as indicated above, a one-place predicate. In Nxy, n = 2; thus Nxy is a two-place predicate. But if we had Nxx, we would have a one-place predicate since only one kind of individual variable is involved. When a predicate contains more than one occurrence of the same kind of variable, it is understood that the same name of an individual is to be supplied for each occurrence. To illustrate, x

+x =

10.

is a one place predicate from which we can get statements such as 2 + 2 = 10 and 10 + 10 = 10. However, x

+y =

10.

is a two-place predicate from which we can get statements such as 2 + 3 = 10, 10 + 7 = 10, and 2 + 2 = 10. It is permissible to use the same singular term for different kinds of individual variables. What one

cannot do is use different singular terms for the same kind of individual variable. It wiII prove convenient from time to time for us to indicate explicitly what the individual variable ranges over in an open sentence by using the sign for the universe of discourse or domain of discourse in set theory, V. If the x ranges over cities, this may be indicated in this way: V: cities. But if the x ranges over physical objects or persons, this may be indicated, respectively, in this manner: V: physical objects; and V: persons. If we restricted V to the set {I, 2, 3}, that is, if our universe of discourse is the set made up of only three individuals, the numbers 1, 2, and 3, then the open sentence Ox: x is odd.

would yield three statements, when we replace x with a name of one of the individuals in {I, 2, 3}, only one of the statements being false.

Exercises 1. Obtain open sentences, and thus n-place predicates, from the following sentences. (a) 10 is greater than 5. (b) 7

+5=

12.

(c) Americans prefer Democrats to Republicans. (d) Chicago is larger than Phoenix. (e) This book = this book (f) It is raining. 2. Indicate if the expressions below are one-, two-, or three-place predicates.


(a) x

(b) (c) (d) (e) (f)

x x

+x +6

(g) For any x, x + y = 10 3. If possible, obtain true propositions from the following n-place predicates by supplying singular terms. (a) V: everything: x is smaller than y, x exists, x is red and green all over (b) V: positive integers: x + 2 = 5,x + x = x,x + y < x + 6,x(y + 2) = xy + xz (c) V: human beings: x is the present President of the United States, x is older than y and y is older than z, everyone loves x, x is wise and is a United States Senator

Answers 1. (d) x is larger than y, Chicago is larger than y, x is larger than Phoenix. 2. (a) 2, (b) 1, (c) 2, (d) 2, (e) 1, (f) 3, and (g) 1. 3. (c) J. William Fulbright is wise and is a U.S. Senator.

5.2

107

Predicate Language

106

where, naturally, the universe of discourse for 'rIxPx is numbers and the universe of discourse for 'rIx,....,Hx is persons. In other words, in saying, for example, 'rIxPx, we are saying

Quantifiers

There is a second way to obtain a statement from an open sentence besides supplying a singular term for the individual variable. We can prefix expressions such as 'for all x' and 'for some x'. For example, supposing the universe of discourse is persons, the open sentence

For all x's (in the class of numbers), Px. The quantifier 'for some x' is called the existential quantifier and will be denoted by this symbol: 3x. Each of the following statements may be paraphrased as 3xMx:

x is mortal.

turns into a statement in either of these two ways: For all x, x is mortal. For some x, x is mortal.

For somc x, x is mortal. } Some x is such that x is mortal. There exists an x such that x is mortal. There is an x such that x is mortal. There is at least one x such that x is mortal.

3xMx

where, it seems likely, the universe of discourse is persons. Similarly The expressions 'for all x' and 'for some x' are called quantifiers, since they indicate the quantity of the individuals being considered in the universe of discourse. Since the universe of discourse for the above sentences is persons, the last statement is, in effect:

There is at least one number that is prime. At least one person is not happy. may be paraphrased into

For some x (in the class of persons), x is mortal.

If, say, the universe of discourse were the gods, then we would have: For some x (in the class of gods), x is mortal. The quantifier 'for all x' is called the unive1'sal quantifier' and for this quantifier we shall use this symbol: 'rIx. Using the one-place predicate 'x is mortal' (Mx) and the new symbol, each of the statements below may be paraphrased as 'rIxMx: For all x, x is mortal. } For every x, x is mortal. For each x, x is mortal. For any x, x is mortal.

'rIxMx

Again using predicates and the universal quantifier, we obtain these paraphrases: All numbers are prime numbers. Everyone is unhappy.

'rI xPx where Px: x is a prime number. 'rIx,...., Hx where Hx: x is happy.

3xPx 3x"",Hx

where Px: x is prime where Hx: x is happy

where the universe of discourse of the first is numbers and of the second , persons. Syllogistics, traditional Aristotelian logic, treats four types of propositions, which are called categorical propositions. Categorical propositions were regarded in traditional logic as sUbject-predicate statements and were classified as affirmative or negative depending on whether the subject was affirmed or denied of the predicate, and universa~ or particular. depending on whether all or some of the things des~l'lbed by the subject were under consideration. Allowing S to be the subject and P the predicate, the four forms of a categorical proposition are: Universal Particular

Affirmative A: All Sis P I: Some S is P

Negative E:NoSisP 0: Some S is not P

where A, E, I, and 0 are used t~ identify the four forms. If. we Suppose the universe of discourse to be everything and let x IS Sand Px: x is P, then the particular affirmative propositions can

108

Quantifiers

109

of Bx, and the second quantifier, naturally, only has the scope of ,,-,Bx. The parentheses thus serve as punctuation marks to indicate what the quantifier has within its scope. If we wished to symbolize:

be paraphrased 3x(Sx A Px)

read "there is an x such that x is S and x is P." The particular negative categorical proposition, 0, would be rendered:

(5) If anything is digestible, it is eaten. we would do the following symbolic paraphrasing:

3x(Sx A ,",-,Px)

"there is an x such that x is S and it is false that x is P." If our universe of discourse is everything, then to say 'All S is P' may be construed as saying of anything if it is S then it is P. The A proposition thus goes over in our notation to

\:Ix(Dx -+ Ex)

rather than \:IxDx -+ Ex

\:Ix(Sx -+ Px)

"for any x if x is S then x is P" and the E naturally becomes

"for any x if x is S then it is false that x is P." After some further preliminaries, we will return to categorical propositions and raise the question concerning the logical relationships between A, E, I, and propositions. In using quantifiers the scope of the quantifier is of prime importance. The scope of a quantifier is the formula to which the quantifier applies, and it is indicated by parentheses. To illustrate, in ordinary discourse we can easily see the disjunction between saying

°

(1) Everything is blue or not blue. which is true, if we are talking about visible objects, and (2) Everything is blue or everything is not blue. which is clearly false. In the notation just introduced, (1) and (2) are symbolized respectively: (3) \:Ix(Bx v,......,Bx) (4) \:IxBx V \:Ix "-' Bx

In (3) the quantifier has the scope ofthe area marked by the parentheses. The quantifier in (3) covers the whole compound or has the scope of the entire compound. However, in (4) the first quantifier only has the scope ""'/-,

for the 'it' in (5) clearly refers back to the 'anything', so the quantifier must be made to cover the whole compound. Also note that (5) is a statement, whereas \:I xDx -+ Ex is an open sentence and is neither true nor false. We will adopt the convention-which we have in fact been following-of not using parentheses when the scope of a quantifier is a single predicate expression. Thus to indicate \:Ix(Fx) we will merely write \:IxFx. Exercises 1. Symbolize each of the following, assuming that the universe of discourse is everything. (a) Only idiots drink paint thinner. (b) Some numbers are prime and odd. (c) All that glitters is not gold. (d) Nothing is a horse unless it is a mammal. (e) If something is a horse, then it is a mammal. (f) A thing is a pen only if it holds ink. (g) Only nonspiders are insects. (h) At least one plant is edible. (i) Not all philosophers are logicians. (j) All Americans except New Yorkers are friendly. (k) None but the brave deserve the fair. (1) If there is a frustated dean, then something is frustrated and something is a dean. (m) Everyone is good and kind if and only if everyone is good and everyone is kind. 2. Distinguish (a) \:Ix,,-,Fx from l"o.'\:IxFx (b) 3x,-.....,Fx from '"'-'3xFx

IlO

Predicate Language

3. An individual variable can occur either free or bound in a formula. This distinction may be put as follows: An occurrence of an individual variable is bound if and only if it is within the scope of a quantificational expression that contains an occurrence of that individual variable. An occurrence of a variable is free if and only if it is not bound. For example in Vx((Fx /\ Oy) -+ 3y(Gy /\ (Hx V Mz))) the first y and the z are free, but the rest of the individual variables are bound. Which variables are bound and which are free in the following formulas? (a) VxFx -+ Gx (b) 3x(3yFxy -+ Gyx) (c) Fx -+ (VxFx -+ VyJ/x) (d) 3x(Fx /\ VyFy -+ x = Y /\ Gx) 4. When in the natural language we use the form 'all Sis P', is this equivalent in meaning to Vx(Sx -+ Px)? Note that Vx(Sx -+ Px) has a truth-value whether or not there are x's that are S. Also evaluate the following claim: The statement-making job done by 'all S is P' in the natural language is done adequately by Vx(Sx -+ Px) and when the two have different meanings the statement in the natural language has failed in its statement-making function. Answers 1. (j) Ax: x is an American, Nx: x is a New Yorker, Fx: x is friendly. Vx(Ax /\ "",Nx -+ Fx).

2. (a) Vx,,-,Fx says "nothing is F" whereas "",VxFx says "something is not F."

r~t~rpretation of Quantifiers

III

be understood to mean that at least one ofthe statements Fa, Fb, Fc, ... is true. In other words, 3xFx can be interpreted as a disjunction of singular sentences as follows: 3xFx

Fa

V

Fb

V

Fe

V '"

Disjunctions and conjunctions can only be finite in length. Often we wish our universe of discourse to be made up of an infinite number of individuals. So, in general, the interpretation for the two quantifiers may be indicated in this way. Let Ax be any predicate that at least has x not under the scope of a quantifier. And let us say that an individual, say a, in a universe of discourse, V, satisfies Ax iff Aa is a true statement. With this in mind we may indicate the interpretation for the two quantifiers as follows: V xAx is true in V iff all the individuals in V satisfy Ax. 3xAx is true in V iff at least one of the individuals in V satisfies Ax.

To see how this interpretation for the quantifiers works out, let Lx:

x is less than 5, and Suppose that our universe of discourse is {l, 5, lO}. 'rhus to say VxLx would be to say Ll /\ L5 /\ L10. Since L10 is false the conjunction is false and thus V xLx is false. In short, since all th~ individuals in V do not satisfy Lx, L10 being false, VxLx is false. But to say 3xLx would be to say something true since to say 3xLx is to say L1 V L5 V L10 and L1 is true. In other words, 3xFx is true since there is at least one individual in V that satisfies Lx, namely 1. To say Vx Lx or 3x "'" Lx would amount to: ,-..0

5.3

Interpretation of quantifiers

In sentential logic an interpretation was given for the statement connectives. This interpretation removed the possibility of ambiguity arising from their use. We wish to provide an interpretation for the quantifiers that will similarly result in their unequivocal use. We desire a fixed meaning for the quantifiers. Suppose our universe of discourse consisted of a finite set of individuals a, b, c, ... To say VxFx, everything is F, will be so understood to mean that Fa and Fb and Fc and .... In other words VxFx can be interpreted as a conjunction of singular sentences as follows: VxFx

Fa /\ Fb /\ Fc /\ ...

By a singular sentence is meant a predicate followed by n number of names for individuals. In turn, the statement 3xFx, something is F, will

Vx"",Lx ~x""" Lx

"",L1 /\ "",L5 /\ ,...."LlO ,....,L1 V "",L5 V "",LlO

'I'hus Vx "'" Lx is false, but 3x "'-' Lx is true. In turn, we obtain the relationships below in this example: "",(L1 /\ L5 /\ LlO) "",(L1 V L5 V LlO)

Thus """VxLx is true, but ,....,3xLx is false. For the universe of discourse {China, Cuba, U.S.S.R.} letting Ox, . t country, we can produce statements such as ' . ,x is a commums VxOx, 3xOx, ,....,Vx "'" Ox, "",3x "'" Ox

113

112

whose truth-value, given the interpretations for the quantifiers, would be T for each. Finally, if Fy: 3x(x + y = 10) and V: {O, 2, 5, 10}, then the following statements would have the indicated truth-value: F2 A F5 "IyFy 3yFy

-+

FlO

(T)

1. Let Fx: x is less than 10. Below are three universes of discourse. In each universe indicate the truth-value of "IxFx and 3xFx, and "Ix,,-, Px and 3x"-' Px. (a) V: {,,-,2, 0, 2, 5.5} (b) V: {O, 5, 10} (c) V: {15, 23, 10}

2. For each open sentence and its associated universe of discourse give four true statements using quantifiers. V: {,-.....,1, 0,1, 2} (a) Fx: x is odd. V: {Cuba, U.S.A., New Zealand} (b) Fx: x is a free nation. V: {,,-,1, 0,5, ,-.....,3} 2 is less than x 3. (c) Fx: x V: {.....,2, 0,.2} (d) Fx: 2x - 5 + x 2 = 3. V: {Nixon, Ted Kennedy, Lindsay, (e) Fx: x is a great leader and McCarthy} a moral man.

+

3. Put the following into symbolic form for each of the given universes. Every male American is ambitious. No male American is ambitious. There is a male American who is ambitious. There is a male American who is not ambitious. (a) V: All male Americans (b) V: All Americans (c) V: All males (d) V: Everything 4. Let the universe consist of all positive integers and let Ex: x is even. Ox: x is odd. Px: x is a prime. Gx: x is greater than 7. Paraphrase each of the following into symbolic form. (a) Some positive integers are odd. (b ) No integer is both even and odd. (c) Some primes are odd. (d) 1 is even. (e) Some integers are greater than 7. (f) If any x is a prime then x is greater than 7. ~I

,

Fa! V Fa 2 V ... V Fan
(T) (F)

Exercises

+

5. We know by DM that the following biconditionals are valid:

Answers 1. (c) Both "IxFx and 3xFx are false. 2. (d) ?xFx,""'" "IxFx, ,.....,"Ix,-....., Fx, 3x,....., Fx. (e) ? 3. (d) "Ix(Mx A Ax -+ Bx) "Ix(Mx A Ax -+ ,-.....,Bx) 3x(Mx A Ax A Bx) 3x(Mx A Ax A ,-.....,Bx)

Mx: x is a male Ax: x is an American Bx: x is ambitious.

4. (f) "Ix(Px -+ Gx).

5.4

Valid predicate formulas

Formulas in sentential logic can be classified as valid, inconsistent, or contingent. A similar classification can be made for predicate formulas. Below are examples of each kind of predicate formula. valid: "IxFx -+ 3xFx inconsistent: 3x(Fx A ,.....,Fx) contingent: 3xFx

It is necessary to give a precise account of what it means to speak of a predicate formula as valid. It is also useful to have an account for the notion of validity so that it can apply to specimens of both sentential formulas and predicate formulas. To this end, we first modify slightly the notion of a valid sentential formula. To assign a truth-value to the components of a sentential formula will be to interpret a sentential formula. Valid formulas will be those formulas that have the truth-value true no matter what interpretation is supplied. A predicate formula can be said to be valid iff it is true on all interpretations. And one obtains an interpretation of a predicate formula, if the formula is composed out of quantifiers, n-place predicates with bound variables, and sentential variables, in the following way: First, select a universe of discourse that has at least one member (the reason for this proviso is examined at the end of this section). Second, assign to each n-place predicate symbol an

1I5

Predicate Language

n-place predicate. Third, assign a truth-value to the sentential variables , if any. The formula VxFx -+ 3xFx is declared above to be a valid predicate formula. (Soon techniques to establish the validity of predicate formulas will be introduced.) Let us interpret the formula as follows: V: living Americans and Fx: x is a long distance runner. We see that the statement resulting from VxFx -+ 3xFx on this interpretation is true. If VxFx -+ 3xFx is valid, as it is, a true statement will result no matter what (nonempty) universe of discourse is selected and no matter what one-place predicate we supply for Fx. In explaining the notion of a valid predicate formula it is said that if a formula is valid, then it will be true no matter what universe of discourse is selected~so long as there is at least one individual in the universe. vVhat is the reason for this proviso? All formulas in predicate logic that are classed as valid yield true statements whether or not an empty term is assigned to a predicate variable. By an empty tel'm is meant a predicate that is such that no individual in the selected universe of discourse satisfies the predicate. For example: (1) VxFx

-+

3xFx

comes out true even if Fx is an empty one-place predicate. For if no individual in V satisfies Fx, then VxFx is false and thus (1) is true. However, what happens if the universe of discourse itself is empty? For example, what if we have a statement of the form (1) and the V has no members~there are no individuals in V? The situation that develops is simply this. If V is empty, then to say either 3xFx or 3x ~ Fx is to say something false. Since ~VxFx is equivalent to 3x ~ Fx, it follows that if 3x ~ Fx is false, then VxFx is true. If VxFx is true and 3xFx is false, then (1) is false. Not all the predicate formulas that we wish to classify as valid break down in this way if the domain of discourse is empty. For example, VxFx -+ ~3x ~ Fx is true on interpretation even if the universe of discourse selected is empty. It is customary in predicate logic to designate as valid any formula that is valid in any domain excluding an empty domain. This restriction is not unreasonable, for a moment's reflection shows that we do not have many occasions to make assertions that have domains of discourse with no members. It is a simple matter to determine whether a formula holds in an empty universe of discourse. Merely replace all universal quantified predicate expressions with T and the existential ones with F, being sure ~I·,

leave

. any'-"'" that is found in front of the quantifiers. Thus

111

=

,-....,F -+ F

VxFx

bccomcS

T '

-+

-+

F

=

F

,-...,Vx,-...., Fx

,-....,T = T

1

vve 0 btaIll al

-+

T

-+

F

=

F

F , the formula is false in an empty domain of discourse.

Exercises I. 'rry to classify the following as valid, inconsistent, or contingent predicate formulas: (a) Vx,-...., Fx -+ 3x,-...., Fx (b) Vx(Fx /\ Gx) -+ VxFx /\ VxGx (c) 3xFx /\ 3xGx -+ 3x(Fx /\ Gx) (d) Vx(Fx -+ Gx) -+ (VxFx -+ VxGx) (e) converse of (c) (f) converse of (d) 2. Give an interpretation for each of the predicate formulas in exercise 1. 3. Which of the following are valid even in an empty domain? (a) Vx(Fx /\ Gx) -+ VxFx /\ VxGx (b) ,-....,3xFx -+ ,.....,Vx,-...., Fx (c) Vx(Fx -+ Gx) -+ (VxFx -+ 3xGx) (d) 3x(Fx /\ Gx) -+ 3xFx /\ 3xGx (e) VxFx /\ Vx(Gx V Hx) -+ 3x(Fx /\ Gx) V 3x(Fx /\ Hx) 4. A model for a predicate formula A is an interpretation M for A such that A is true in M. A formula is inconsistent if it has no model. (a) Give a model for some invalid consistent predicat~ formulas. (b) Prove that the following sets of formulas are conSIstent: ,-....,VxFx Vx(Fx -+ Gx) 3xFx Vx(Fx -+ ,-....,Gx) 5. What must the nature of V be if we make an assertion that has an empty V? Answers 1. Only (c) and (f) are invalid. 3. Only (a) and (d).

·I..lU

4. (a) If M is V: {1,2}, Fx: x is even, Gx: x 3x(Fx A Gx) is satisfied in M.

Predicate Language

117

< 3, then 3xFx A 3xGx __

the pattern accordingly, remembering again that universal ts expand into conjunctions, whereas existential statements statemen . . _",~"r'" into disjunctions. If V: {a, b} then (3) IS eqmvalent to

5. An inconsistent predicate determines V.

Fa V Fb ~ Fa A Fb 5.5

Proving the invalidity of predicate formulas

From the definition of a valid predicate formula it immediately follows that a predicate formula is invalid if we can give an interpretation that results in a false statement. For example a proof of the invalidity of (1) 3xFx A 3xGx --* 3x(Fx A Gx)

is obtained when V: drawn figures, Fx: x is a square, and Gx: x is a circle. For under this interpretation (1) becomes the false statement: If some drawn figures are squares and some figures are circles, then some figures are squares and circles. Or if V: {I, 2}, Fx: x is even, and Ox: xis odd, then (1) is a false statement. To show that an argument pattern made up of predicate formulas is invalid we can produce a counterexample. A counterexample for an argument pattern is an interpretation that results in true premises and a false conclusion. To illustrate, we demonstrate the invalidity of the following argument pattern (2) Vx(Fx --* Gx)

~

3x(Fx

A

Gx)

with this interpretation: V: living animals, Fx: x is a saber-toothed tiger, Ox: x has claws. For given this interpretation 3x(Fx A Ox) is falsethere are no saber-toothed tigers among living animals, whereas Vx(Fx --* Ox) is true. To see that Vx(Fx --* Ox) is true let a, b, e, ... be names for living animals. Since universally quantified statements expand into conjunctions in a finite universe, we obtain this relationship: Vx(Fx --* Gx)<.---+ (Fa --* Ga) A (Fb --* Gb) A (Fe --* Ge) A ... and since FCl, Fb, Fe, . .. are all false-no living animal is a sabertoothed tiger-each conditional is true, and thus the conjunction is true. We have successfully produced a counterexample to (2). Truth-table procedures can be usefully employed to find abstract counterexamples. Suppose we wish to demonstrate the invalidity of the next pattern: (3) 3xFx

~

VxFx

We may do this by supposing a universe of two individuals {a, b} and /-,

'i'reating the '~' as a '--*', w~ may use th; redueti,o truth-table. test to determine its validity. We wIll find that T sand F s can be consIstently . ed so that the premise is T and the conclusion F. If Fa is T and ~p . . Fb is F (or vice versa), then the premISes are true and the conclUSIOn Thus V: {a, b}, Fa is T and Fb is F provides an abstract counterexample to (3). A concrete counterexample is obtained by selecting two individuals for a and b and supplying an interpretation for Fx so that Fa is true and Fb is F. For example: {I, 2} and Fx: x is odd. To prove the invalidity of predicate formulas, we may also give abstract interpretations. To illustrate: Vx«Fx

--*

Gx) A Gx --* Fx)

is proven invalid when V: {a} and Fa is F and Ga is T. To prove the invalidity of the next formula 3x(Hx

A

Gx)

A

3x(,-.....,Hx

A

Gx)

AV

x(Fx --* Hx)

--*

Vx(Fx

--*

Gx)

using the abstract method, demands that we have three individuals in the domain of discourse. Exercises I. Demonstrate the invalidity of the following formulas or patterns. Provide a concrete counterexample for at least one. (a) Vx(Fx --* ,...,Gx) --* 3x(Fx A ,-.....,Gx) (b) Vx(Fx V Gx) --* VxFx V VxGx (c) (VxFx --* VxGx) --* Vx(Fx --* Gx) (d) ""VxFx ~ Vx,-....., Fx (e) 3x(Fx --* P) ~ Vx(Fx --* P) (f) VxFx --* P ~ Vx(Fx --* P) (g) ",,(Vx(Fx --* Gx) A Vx(Fx --* ,-.....,Gx» (h) 3x(Fx A Gx) V 3x(Fx A ,.......,Gx) 2. If the categorical propositions A, E, I, 0 are represented as VxFx, V x Fx, 3xFx, and 3x,-....., Fx, respectively, then the following are valid formulas: (A implies I) A --* I E --* 0 (E implies 0) ,-.....,(A A E) (A and E are contraries) I V 0 (I and 0 are sub contraries) ,-.....,(E <.---+ I) (I and E are contradictions) ,-.....,(A <.---+ 0) (A and 0 are contradictions) r-.J

lIS

Predicate Language

If A, E, I, and 0 are represented as Vx(Fx -+ Gx), Vx(Fx -+ '"'-'Gx), 3x(Fx A Gx), and 3x(Fx A ,,-,Gx) , respectively, then only the last two formulas

are valid. Demonstrate the invalidity of the first four formulas in this latter case. (Note (a), (g), and (h) in the first exercise.) Answers 1. (c) Abstract counterexample: V: {a, b}, Fa is T, Fb is F, Ga is F, and Gb is 1'. Concrete counterexample: V: {I, 2}, Gx: x is even and Fx: x is odd. (f) V: {a, b}, P is F and Fa is true and Fb is false. V: {I, 2}, P: 1 2 = 2, Fx: x is less than 2.

+

5.6

Proving the validity of predicate formulas

If a predicate formula has the form of a tautology, then it is valid. An indefinite number of valid predicate formulas can thus be obtained from sentential logic by uniform substitution of quantified formulas for sentential variables. This may be illustrated by our making the following substitu.tions for the tautology (P -+ Q) A P -+ Q: VxFxjP, VxFxjP, 3xFxjP,

(VxFx -+ Q) A VxFx -+ Q VxGxjQ (VxFx -+ VxGx) A VxFx -+ VxGx VxFxjQ (3xFx -+ VxFx) A 3xFx -+ VxFx

A second way to obtain valid predicate formulas directly from tautologies is to substitute predicates for sentential variables uniformly and prefix the result with a universal quantifier as the next examples illustrate. PAQ-+P Fxj P, GxjQ Fx A Gx Vx(Fx A Gx -+ Fx)

-+

Fx

There are, however, valid predicate formulas other than those obtainable in this direct way from tautologies. Here is a list (there is an infinite number) of such valid predicate formulas: (1) (2) (3) (4) (5) (6) (7) (8) (9)

VxFx -+ 3xFx (VxFx -+ VxGx) -+ (VxFx -+ 3xGx) Vx(Fx -+ Gx) -+ (3xFx -+ 3xGx) Vx(Fx -+ Gx) -+ (VxFx -+ VxGx) Vx(Fx A Gx) -+ VxFx A VxGx 3x(Fx V Gx) -+ 3xFx V 3xGx VxFx V VxGx -+ Vx(Fx V Gx) VxFx A 3xGx -+ 3x(Fx A Gx) VxFx_ "",3x "'-' Fx (10) Vx,,-, Fx_ ,,-,3xFx (ll) ,,-,YxFx_ 3x "'-' Fx (12) "-,Vx",,, Fx_ 3xFx

Proving the Validity of Predicate Formulas

119

validity of these formulas depends on the meaning ofVx and 3x and erely a matter of how simple statements are. related by statement }S no t »1 . oonncctl'ves . Thus they cannot be obtamed from tautologIes. One way to establish the validity of these formulas is to use a .1 l'o nd ab8~t1'dum technique. For example, consider (1) and suppose t'elbUC .~ cc >YO

it is false: ,-...,(VxJ?x -+ 3xFx)

If this is true, then VxFx must be true and 3xFx must be false. If ,-...,3xF;r is true, then Vx "-' Fx. If VxFx is true, then Fa is true where ct is an individual in our universe of discourse. Similarly, if Vx "'-' Fx is t.rno, then ,-...,Fct is true. Thus we obtain a contradiction Fct A ,-..."Fct, so our original assumption must be false. To take another example, let us informally prove the validity of (6) by this RAA method, using successively numbered lines as follows: 1. ,-...,(3x(Fx V Gx) -+ 3xFx V 3xGx) 2. 3x(Fx V Gx) 3. ,-...,(3xFx V 3xGx) 4. Fct V Gct 5. ,-...,3xFx 6. VX"'-' Fx 7. ,-...,Fct 8. Fa A ",,-,Fa

In this informal proof, 2 and 3 are true if 1 is true. 4 follows from 2, our now taking 'ct' as the individual in V for which 2 is true. 5 comes from 3 by SO (DM, DN and S), 6 from 5 since (10) is valid; 7 from 6-if Vx "'-' Fx then for any individual, including 'ct', "'-'x is F. 8 comes from 7 and 4 by SO. The valid predicate formulas above have something in common. '1'hey contain only one-place predicates. For such formulas a truth-table decision procedure is possible. It has been established that a predicate formula with only one-place' predicates is valid iff it is valid in a domain of 2" individuals where n equals the number of kinds of one-place predicates in the formula. Thus formula (10) is valid iff it is valid in a domain of two individuals. Formula (2) is valid iff it is valid in a domain of four individuals. We may then establish the validity of (10) in this manner. First, write the equivalent to (10) in a V of two individuals as follows:

Second, use truth-table techniques to see if the resulting formula is a tautology. If it is, (10) is valid and if it is not, UO) is invalid.

120

Predicate Language

121

Such truth-table methods have their limitations in predicate logic. It has also been demonstrated that there is no truth-table decision procedure for all predicate formulas; in fact, there is no effective or mechanical decision procedure for all predicate formulas. But this fact does not prevent us from constructing a proof system similar to SO that can establish the validity of all valid predicate formulas. Such a system is introduced in the next chapter, and, as we will see, it makes use of the moves involved in the RAA proofs above, along with other moves.

A one-place predicate determines a class. For example, Fx dethe class of those things and only those things that have .ty F ~~l . If we let Fx: x is odd and V be the positive integers, then. . determines the class whose members are 1, 3, 5, 7, .... DyadIc redicates determine a special kind of class~what is called a class of P pairs. Fxy determines the class made up of x's and y's that stand . the relationship Fxy. Thus if Fxy: x loves y and V: living persons, In determines the class of x's and y's such that x loves y. The symbolization of dyadic predicates makes clear the order of the individuals or the direction of the dyadic predicates. Thus Fxy makes clear that x is F to y, and Fyx makes clear that y is F to x. Similar considerations hold for the other polyadic predicates. There are many occasions when we shall find it necessary to expose polyadic predicates in our symbolizing arguments in order to obtain the logical structure relevant to the validity of the argument. For example, mcdievallogicians were particularly troubled over a valid argument that defied validation in syllogistics. One version of the celebrated argument appeared at the beginning of this chapter and is

Exercises 1. In the following, establish the validity of the valid formulas by using one f

the three methods discussed. Be sure to use each method at least once. F~r those formulas that are invalid, demonstrate their invalidity.

(a) (b) (c) (d) (e) (f) (g) (h) (i)

VxFx V 3xGx ->- 3x(Fx V Gx) (3xFx ->- VxGx) ->- Vx(Fx ->- Gx) Vx(P ->- Gx) ->- (P ->- VxGx) Vx(Fx ->- Gx +-+ ,,-,Fx V Gx) (VxFx ->- VxGx) ->- (,,-,VxGx ->- "-'VxFx) Vx(Fx ->- Gx) ->- (3xFx ->- VxGx) (P V 3xFx ->- VxGx) ->- (P ->- VxGx) 1\ (3xFx Vx(Fx V (Fx V Gx) ->- Fx) "-'3xFx ->- Vx(Fx ->- Gx)

->-

VxGx)

2. Is the statement If all men are mortal, then some men are mortal. of the form VxFx -:+ 3x!x-a vaIidformula-or of the form Vx(Fx ->- Gx)--;. 3x(Fx 1\ Gx)-an InvalId formula? Can it be an instance of both forms? Is the statement analytic?

(1) All circles are figures. Therefore whoever draws a circle draws a figure.

If (1) were symbolically paraphrased in this manner:

vx(Cx ->- Fx) I- V x(Dx ->- Ix) where Ox: x is a circle, Fx: x is a figure, Dx: x draws a circle, and Ix:

x draws a figure, it is intuitively obvious that its conditional would not 5. 7

Polyadic predicates

So far most, though not all, of the n-place predicates we have considered are one-place or monadic predicates. As noted earlier, one-place predicates are predicate expressions with only one kind of free individual variable. But we may have two-place (dyadic) predicates, three-place (triadic) predicates, and so on. Such n-place predicates are called polyadic predicates. Examples of such predicates would be: Gxy: x is greater than Exy: x is equal to y Lxy: x loves y

Y)

+ 2y = 5 x loves y more than z} x +y > y +z

two-place predicates

exemplify a valid predicate formula. In fact what we have just done to (1) is an incorrect symbolization since it violates the principle for correct paraphrase (see Section 3.3). The trouble arises in leaving 'x draws a circle' and 'x draws a figure' as one-place predicates, rather than introducing a two-place predicate 'x draws y'. In many cases, for purposes of establishing validity, one can leave relations unexposed; indeed one can represent a statement as P, but not in this case. A polyadic or relational predicate needs to be introduced to display the structure of the conclusion relevant to the validity of the argument. How can (1) be correctly paraphrased~ If we select this interpretation:

Rxy: x Lxyz: Gxyz:

Pxyzw: x pays y to z for w '""/-,

.

three-place predICates four-place predicate

V: natural things Cx: x is a circle Fx: x is a figure Dxy: x draws y

122

Predicate Language

then (1) can be correctly symbolized. The first premise is as above Vx(Cx -+ Fx). One way to go about symbolizing the conclusion usin~ the two-place predicate Dxy is illustrated as follows: Whoever draws a circle draws a figure. 1. Vx(x draws a circle -+ x draws a figure) 2.

3y(Gy /\ Dxy)

3. Vx(3y(Gy /\ Dxy)

-+

3z(Fz /\ Dxz) 3z(Fx /\ Dxz»

The procedure is to first ask oneself if the statement is a universal or an existential statement. That is, does the statement have the form Vx( -+ ) or does it have the form 3x( /\ )? Clearly the statement 'Whoever draws ~ circle draws a figure' has the first form. To say whoever draws a CIrcle draws a figure is to say if anything is a drawer of a circle then it draws a figure. Having decided that our statement has the form Vx( -+ ), we write the statement, injecting this form, putting in the individual variable x, and leaving in the rest of the English. What we obtain is line one. The second step is to symbolize the remaining parts. In the antecedent within the parentheses we have x draws a circle

123

P"lvad.1c Predicates

Cy /\ Dxy. The rule to follow, however, is to change the individual variable as each new quantification is introduced when one is symbolizing the structure of a simple statement. In some cases no harm results from violating this rule, but in other cases it does (as with the inner expression), so a good policy is always to follow the rule. The final step 3 is to put the various pieces together to form the statement, maJdng sure the Vx has the scope of the entire formula since we wish the final x in Dxz to link up with Vx. Let us now use this procedure in symbolizing some more examples using the interpretation given at the right of each example:

Every student who passes knows something. Vx(x is a student who passes -+ x knows something) Vx(Sx /\ Px

-+

Sx /\ Px 3yKxy)

3yKxy

V: person Sx: x is a student Px: x passed Kxy: x knows y

Some student is liked by all professors. 3x(x is a student /\ x is liked by all professors) Sx Vy(Py 3x(Sx /\ Vy(Py -+ Lyx»

-+

Lyx)

(2) 3y(Gy /\ Dxy)

and the latter would be symbolized (3) Vy(Gy

-+

Dxy)

We want the former, (2). It is useful to note at this point that (2) does not say that x draws some particular circle, but that x draws some circle or other. (2) is true if x draws circle a or x draws circle b, etc. The same considerations apply to symbolizing the consequent found within the parentheses of 1. It should not be overlooked that in line 2 we put a y rather than an x in the first inner expression and a z rather than an x or y for the second inner expression. The outside x has the scope of the formula, so we needed y in the first expression to avoid ambiguity (and for other reasons). We need not have used z; we could have used y, for example, in the second inner expression since the 3y only has the scope ""/

..

V: person Sx: x is a student Px: x is a professor Lxy: x likes y

To say 'x draws a circle' is to say 'x draws some circle or other', not that x draws every circle. Given the above interpretation, the former is symbolized: There is no greatest prime number.

,.....,3x(x is a prime /\ x is greater than any other Px Vy(Py /\ ~Ixy -+ Gxy) ,.....,3x(Px /\ Vy(Py /\ ,.....,Ixy -+ Gxy» V: Px: Gxy: Ixy:

prime number) positive integers x is a prime x> y x = y

There is a mother and a father for every child. -+ x has a mother and father) 3y3z(Myx /\ Fzx) V: persons Vx(Gx -+ 3y3z(Myx /\ Fzx» Gx: x is a child Mxy: x is the mother of y Fxy: x is the father of y Vx(Gx

In an interpretation it makes no difference what individual variables we use. For example, the x and y in Fxy indicates only that we have a two-place predicate. This could be indicated by Fyz or, if we like, FCD®.

124

Predicate Language

To continue with some more examples of paraphrases of sentences using predicate symbols, consider: Politicians prefer wealth to power. Vx(Px ---+ x prefers wealth to power) (i.e., any wealth to any power) VyVz(Wy II Oz---+Exyz) V: persons . Vx(Px ---+ VyVz(Wy II Oz ---+ Exyz)) Ox: x is power Exyz: x prefers y to z Px: x is a politician Wx: x is wealth Sisters have a common parent. VxVy(x is the sister of y ---+ x and y have a common parent) Sxy 3z(Pzy II Pzx) VxVy(Sxy ---+ 3z(Pzy II Pzx)) V: persons Sxy: x is the sister of y Pxy: x is the parent of y

125

It will be observed that each of the following formulas is valid: (4) 3yVxFxy ---+ Vx3yFxy (5) 3xVyFxy ---+ Vy3xFxy

that the converse of each is not valid. Much is to be learned from this concerning the effect that the order of Vx and 3x can have on meanin? Let us try to bring out the sense of the antecedent and consequent.m each of the above conditionals and see exactly why (4) and (5) are valId, hereas the converse of each is not. We begin with (4). \V Suppose that our universe of discourse consisted of just three individuals: a, b, and c. And let Fxy: x loves y. In this case the general proposition below would mean the same as the accompanying string of disjuncts. 3yVxFxy

VxFxa

V

VxFxb

V

VxFxc

Consider, finally, the statement: Every gentleman prizes a blond. This statement can be understood in at least these two ways: (i) There is some blond who is prized by every gentleman. (ii) For every gentleman there can be found some blond whom he prizes. Using this interpretation: V: persons; Gx: x is a gentleman; Pxy: x prizes y; and Bx: x is a blond, (i) and (ii) are paraphrased as follows: (i) 3x(Bx II x is prized by every gentleman) Vy(Gy ---+ Pyx) 3x(Bx II Vy(Gy ---+ Pyx)) (ii) Vx(Gx ---+ x prizes some blond or other) 3y(By II Pxy) Vx(Gx ---+ 3y(By II Pxy)) The reader is encouraged to study carefully each of the above examples. The order of the quantifiers is unessential when the same quantifiers are used, since the formulas below are valid: VxVyPxy+--t VyVxPxy 3x3yPxy+--t 3y3xPxy

However, the order with Vx and 3x is essential, since a change in order means a change in sense.

To say 3yVxFxy is to say that some individual a or b or c is loved by every individual. If everyone loves a or b or c, then 3yVxFxy would be true. Thus 3yVxFxy reads "everyone loves the same person" or "some person is loved by everyone." With the consequent of (4) we obtain Vx3yFxy

3yFay II 3yFby II 3yFcy

which reads "everyone loves someone or other" or, simply "everyone loves." Thus if, say, Faa, Fbc, and Fca are true, Vx3yFxyistrue. Each person does not have to love one and the same individual for Vx3yFxy to be true, but they do if 3yVxFxy is to be true. We thus see why (4) is valid. However, if everyone loves someone or other, it does not follow that everyone loves the same person. Thus the converse of (4) is false. Turning to (5), the antecedent becomes 3xVyFxy

VyFay V VyFby V VyFcy

If a loves a and a loves b and a loves c, then 3xVyFxy would be true. In other words, if some one individual loves everyone, then 3xVyFxy is true. It thus reads: "someone loves everyone" or "everyone is loved by the same person." The consequent of (5) becomes Vy3xFxy

3xFxa II 3xFxb II 3xFxc

which is true if everyone is loved by someone or other. For example, if Fhc, Fbb, and Fac are true, then Vy3xFxy is true. Vy3xFxy thus reads

126

Predicate Language

"everyone is loved by some person or other" or, simply "everyone is loved." If everyone is loved by the same person, then 'everyone is loved' is true-(5) is valid-but the converse is false. As some of these examples bring out, when we have 'Vx3yF--

or

3x'VyF--

one way to go about reading them is to start with the first quantifier and if the x is in the first blank, read F in the active voice. But if x is in the second blank, read F in the passive voice. Thus 'Vx3y 3x'Vy 'Vx3y 3xVy

y causes x

y causes x x causes y x causes y

read, respectively, "everything is caused by some y or other," "some x is caused by everything," "everything causes some x or other," and "some x causes everything."

127

polyadic Predicates

To the interpretation above add a: Igloo, and symbolize the following. (a) If no one loves Igloo, then he does not love himself. (b) If anyone loves Igloo,. then it is Igloo. (c) If unless Igloo loves lumself, he loves no one, then Igloo loves someone only if he loves himself. (d) If everyone whom Igloo loves, loves Igloo, then if Igloo loves everyone, everyone loves Igloo. (e) If Igloo has a lover who has no lover, then Igloo is no one's lover. (f) If all the world loves a lover and no one loves Igloo, then Igloo isn't a lover.

4. A statement is analytic if it is an instance of a valid predicate formula. A predicate formula using a term like a-an individual constant-is valid iff it is true on all interpretations. We get an interpretation of a predicate formula using a term like a when we satisfy the conditions set down earlier (Section 5.4) and when we assign a name of an individual in the universe of discourse to each individual constant. Try to figure out which of the statements in exercise 3 are analytic.

1. Paraphrase into logical notation the following statements using the suggested interpretation. V: the set of human beings, Pxy: x influences y (a) There is somebody whom someone influences. (b) There is somebody whom everyone influences. (c) Everybody is influenced by someone or other. (d) Everybody is influenced by everyone. (e) Someone influences somebody. (f) Everyone influences somebody or other. (g) Someone influences everybody. (h) Everyone influences everybody.

5. Again, paraphrase with the interpretation given below. V: the set of positive integers, Gxy: xis greaterthany, Qxy: 2x y = 6, Px: x is prime (a) For every integer there is a greater integer. (b) There is an integer that is greater than every integer. (c) For every integer it is false that it is greater than itself. (d) For every integer x, y, z, if x < y and y < z, then x < z. (e) There is a prime number greater than O. (f) If 1 is less than any integer, then 0 is not an integer. (g) There is one particular integer that, if added to any integer multiplied by 2, gives 6. (11) If 2 is multiplied by any integer, it can be added to some integer or other to get 6. (i) For any integer it can be added to some integer or other multiplied by 2 to get 6.

2. Use the suggested interpretation and paraphrase each of the following into logical notation. V: the set of human beings, Ox: x is a coed, Px: x is a professor, Lxy: x loves y, Mxyz: x loves y more than z (a) Every coed is in love with some professor or other. (b) Some coeds are in love with every professor. (c) Some coeds only love professors. (d) Some coeds love themselves more than any professor. (e) Some coeds do not love any professors. (f) Every coed loves some professor more than she loves herself. (g) Some coeds love a professor only if the professor loves coeds more than himself. (h) There is a coed such that if any professor is loved by that coed then he " .loves her more than himself.

6. Establish the validity of those formulas below that are valid by using the RAA method discussed in Section 5.6. For those formulas that are invalid demonstrate their invalidity with abstract or concrete interpretations. Se~ answers for how to proceed. (a) 'Vx3yFxy -+ 3y'VxFxy (b) 3x(,-..,,(Fx -+ 'Vy(Gy -+ Fxy») (c) 3y'VxFxy -+ 'Vx3yFxy (d) 'Vy3xFxy -+ 3x'VyFxy (e) 3x'VyFxy -+ 'Vy3xFxy (f) 3x3yFxy -+ 'Vx3yFxy (g) 3x'VyFxy -+ ,-..,,'Vx'VyFxy (h) 3y3x'VzFxyz -+ 'Vz3y3xFxyz (i) 'Vx(Fx -+ Gx) -+ 'Vx(3y(Fy /\ Hxy) -+ 3y(Gy /\ Hxy»

Exercises

+

128

129

Answers

addition we need the following two kinds of logical constants:

1. (a) 3y3xPxy, (b) 3y'VxPxy, (c) 'Vy3xPxy, (d) 'Vy'VxPxy, (e)3x3ypxy (f) 'Vx3yPxy, (g) 3x'VyPxy, (h) 'Vx'VyPxy , 2. (d)

3x(Cx A 'Vy(Py

---+

Lxxy», (h) 3x(Cx A 'Vy(Py A Lxy ---+ Lyxy»

3. (d) 'Vx(Lax -+ Lxa) ---+ ('VxLax -+ 'V Lxa) (e) 3x(Lxa A 'Vy "-' Lyx) -+ 'Vx "-' Lax

6. (a) If V: {a, b}, (a) becomes

Fx, Fx

which is false under conditions indicated. (b) If V: {a, b}, (b) becomes the following nontautological formula: Faa) A (Gb

---+

Fab» V "-' (.Fb

-+

(Ga

---+

Fba) A (Gb

---+

Fbb))

(c) 1. ",,-,(3y'VxFxy ---+ 'Vy3yFxy)

2. 3. 4. 5. 6. 7. 8.

3y'VxFxy ,,-,'Vx3yFxy 3x'Vy,,-, Fxy from 3 'Vy"-' Fay from 4 'VxFxb from 2 ,,-,Fab from 5 Fab from 6 Note that 'a' on line 5 is taken as the something-the x ofline 4 such that 3x'Vy,,-, Fxy. We now need a different name to remove the 3y of line 2; thus b is put in at line 6.

5.8

Formation rules for the predicate language

Up to this time we have relied on the reader's having an intuitive understanding of what is and what is not a well-formed formula in the predicate language. It is now time to provide formation rules by which one can tell in all cases whether or not a formula made up of the terms of the predicate language is well-formed or not. The notion of a well-formed formula in the predicate language needs, in other words, to be made an effective notion. First let us set down the symbols we wish to include in the predicate language (omitting from now on the mention of parentheses). We want these kinds of variables: Sentential variables: P, Q, R, ... . Predicate variables: F, G, H, ... . Individual variables: x, y, z, ... .

-+, and~

example:

(Faa V Fab) A (Fba V Fbb) -+ (Faa A Fba) V (Fab A Fbb) FTTT TTFF FFTF TFF

-+

A, V,

We wish to count as well-formed formulas only those strings of these mbols that, on interpretation, have sentences expressing statements as :~stances. We should not overlook that this requirement does not restrict formulas with free individual variables from being wffs. For

4. Only (b) and (e) are not analytic.

,,-,(Fa -+ (Ga

Statement connectives: "-', Quantifiers: 'V and 3

-+

3xFx, 'VxFxy

may be included as wffs since we can extend the notion of in~erpretation sO that they yield statements. To provide an interpretation of a formula with a free variable, all that one needs to do is to replace free individual variables uniformly with a name for an individual in the universe of discourse. Thus if V: positive integers, Fx: x is odd, and Fxy: x = y, selecting I for x in the above formulas would give us: I is odd. If I is odd, then some integer is odd. All integers are equal to I.

It is worthwhile noting that, given this V and these predicates, depending on what numeral one substitutes for the free variable, the first formula will sometimes give true statements and sometimes false ones, the second formula will also give true statements, but the last formula will usually give false statements. The second formula, in fact, is a valid formula since no matter what interpretation one may provide, it always gives true statements. The decision needs to be made whether we care to admit formulas that involve vacuous quantification. Vacuou8 quantification occurs when one has a quantifier without a corresponding individual variable coming under its scope. Examples of vacuously quantified formulas would be: 'VxP, 'VxFy, 'Vx3yFxz, 'Vx3y3zFxy

No harm would come from allowing such formulas to be well-formed. However, such formulas contain quantifiers that do not work. So let us state the formation rules so that vacuous quantification is avoided: Also we wish to rule out formulas such as (I) 'Vx'v'xFx, 'Vx'VxFxx, 'v'x(Fx

-+

'VxFx)

130

131

Formation Rules for the Predicate Language

as wffs. We wish to rule out as well-formed, formulas such as \:fx\:fxFxx since one of the quantifiers is doing no work, the two occurrences of x after F being controlled by either quantifier. Eliminating each kind as well-formed formulas will simplify to some extent the statement of the rules to be introduced in the next chapter. If we said

All well-formed formulas of SO are well-formed formulas of iang ua ge . predicate language. In applying. these rules we continu~ to drop eses according to the conventIOns we have been followmg. paren th . . To take an example, the followmg IS a wff: \:fx(Fx A (",-,Gx

If A is a wff which contains an individual variable x, then \:fxA is a wff.

this would rule out vacuous quantification but would allow the (l)'s to be wffs. To avoid (1) we need to stipulate that all occurrences of x within the scope of the quantifier must be free in A. As with vacuous quantification, no harm results in allowing formulas like (1); however, as we have said, we wish our wffs to have straightforward statements as instances. To state the rules, it will be useful first to introduce the notion of an atomic formula. Let us mean by a te1'm an individual variable. An atomic formula is either (a) a sentential variable, or (b) a predicate variable followed by n number of terms We can now dispense with listing some capital letters as sentential variables and some as predicate variables. We can simply say that an atomic formula is a capital letter followed by n number of terms. Thus examples of atomic formulas would be: P, Fx, Fxy, Fxyz

In the first case, P, n = O. The definition for a well-formed formula of the predicate language can be set down in terms of the following rules: 1. All atomic formulas are wffs. 2. If A is a wff, then ",-,A is a wff. 3. If A and Bare wffs, then (A A B), (A V B), (A -+ B), and (A +-t B) are wffs. 4. If A is a wff that contains an individual variable v, then \:fvA and 3vA are wffs, provided all occurrences of v within the scope of the quantifier are free in A.

In these formation rules v in 4 is understood to be a variable for individual variables. We should note that these rules also generate all the wellformed formulas of the sentential language. These rules make clear that the predicate language is the result of making additions to the sentential

-+

\:fy3z "'-' Fyz))

because Fx, Gx, and Fyz ,-...;Gx and "'-'Fyz 3z,-...; Fyz \:fy3z,-...; Fyz C"",Gx -+ \:fy3z "'-' Fyz) (Fx A (",-,Gx -+ \:fy3z "'-' Fyz)) \:fx(Fx A (",-,Gx -+ \:fy3z I"-' Fyz))

are wffs by 1 are wffs by 2 is a wff by 4 is a wff by 4 is a wff by 3 is a wff by 3 is a wff by 4

Finally, if we have a formula that contains, say, Fx and Fxy, these are taken to be different predicate variables since Fx is a variable for one-place predicates, whereas Fxy is a variable for two-place predicates. With the introduction of the predicate language, we are not only in position to introduce systems that can generate all of t~e valid fo~~ulas in the predicate language, but we can; at last, prOVIde a prOVISIOnal clarification for some of the terms left undefined in the first chapter. 'fhe final clarification will be found in Section 7.3. First, jOl'mulas in logic are well-formed formulas of the predicate language. To say such a formula is valid is to say that no matter what interpretation we provide, the result is a true statement. Any instance of a valid formula in logic expresses a necessarily true statement of the kind called analytic. With respect to any argument, we can say it is valid if the premises logically imply the conclusion. The premises of an argument logically imply the conclusion if and only if the corresponding conditional is analytic. Exercises 1. For each of the following formulas state whether it is a wff. If it is a wff, give a demonstration of this. (a) \:fxFxy (b) (\:fyFy -+ \:fxGx) (e) \:fx(Fx -+ \:fxGx) (d) VxP (e) \:fy(Fy -+ \:fxGx) (f) (\:fz\:fxFz -+ Fa) (g) \:fz(Fxy -+ Gz) (h) (P -+ \:fxFx) (i) ",-,Fx

132 (j) (k) (1) (m) (n)

VxVxFx 3x(Fx --+ Vy3xFyx) 3x(Fx --+ Vz(Gz --+ 31lHxyu)) (Fa --+ VxFx) I- (VxFx --+ 3xFx) (0) (1 < 2 --+ 3xx < 2) (p) 3y(3xFx --+ Fy)

2. On interpretation, vacuously quantified formulas and double quantified formulas such as Vx(Fx --+ VxGx) do generate full-blooded assertions that are true or false. Provide interpretations for the following and indicate if ' possible, their truth-value. (a) (b) (c) (d) (e) (f) (g) (h)

VxP VxFy Vx3yFxz VxVxFx VxVxFxx Vx(Fx --+ VxFx) 3x(Fx --+ VxFx) Vx(Fx --+ 3xFx)

pc

3. As indicated in an earlier exercise, a formula has a model when it has an' interpretation that results in a true statement. Think up some arguments in support of the following true generalizations. (a) For any finite number n, there are formulas that do not have models in a domain of discourse containing less than n individuals. (b) If a formula has a model in some finite domain, then it has a model in an infinite domain. (c) If a formula contains only one-place predicates, then it has a model in some finite domain. 4. Show that for there to be a model for the set of formulas below, the domain of discourse must be an infinite domain. VxVy(Fxy --+ r-..lFyx) VxVyVz(Fxy --+ (Fyz Vy3xFxy

natural deduction system

--+

Fxz))

In the previous chapter the point was made that there is no effective decision procedure for the validity of all well-formed formulas of predicate logic. However, this does not preclude the construction of a system that is capable of generating all the valid predicate formulas as theorems. Such a system is called the predicate calculus. In this chapter one such system will be described. The system of this chapter will be an extension of the natural deduction system SO. As with se it will prove useful to introduce inference rules with reference to argument patterns. Later in this chapter the system will be explicitly set out and some of the theorems of the system will be proven. In other words, we will follow the same order of exposition that was used for se. The basic strategy in constructing proofs for argument patterns whose validity is a matter of how predicates are related by quantifiers, such as

Answers Vx(Fx --+ Gx), Vx(Gx --+ Hx) I- Vx(Fx

1. Only (a), (b), (e), (g), (h), (i), (1), and (p) are wffs.

--+

Hx)

will be this: First, we will drop the quantifiers from premise assumptions according to two elimination rules, one for the universal quantifier and one for the existential quantifier. Second, we apply the rules of system se to derive the conclusion minus the quantifier. 133

134

6. 1

Natural Deduction System PC

135

Finally, we add quantifiers accord_ ing to two introductory rules for each type of quantifier. These new rules for taking off and adding quantifiers constitute the primal' extension of SC that will give us tl1e y . macIunery to construct proofs for all valid predicate formulas. These rules together with the ten rules f SC will make up a natural deductio: system for predicate logic that will be called system PO after 'predicate calculus' .

Ohapter 4. It will be well at the outset to make clear how the apparatus SO can be applied. To do this we must first give the precise account of counts as a tautology. A formula will be spoken of as basic iff it is an atomic formula or if it is of the form V xA or 3xA. A formula P that is not basic is formed from formulas called basic components of P. The basic components of a formula must be related by statement connectives. A formula is a t(Putology iff it takes the value T for every assignment of truth-values to its basic components. Thus, for example, the following are tautologies:

Individual constants and tautologies

In o~'der to simplify to some extent the statement of the four quantificatIOn rules found in PC, it will prove useful to introduce individual constants into our bag of symbols. (Why the four new rules can be stated in a simpler form, with the consequence of increasing the number of terms and wffs in PC, will be indicated in exercise 6.4.2.) We will let ~, b, c, ... be individual constants. An individual constant is a term that IS taken as a name for an element in the universe of discourse. If pre~icate form~la contains an individual constant, an interpretation i: achIeved by umformly supplying a name for an individual in the universe of discourse. Thus an interpretation for

P

-+

P

V

Q, r--;VxFx

r--; r--; r--;

Fa, Fa /\ Gb, 3z(Fz /\ Gaz), Fa

-+

P, Fb

-+

3xFxy, VxFxa

At the beginning it was said that the basic strategy in constructing proofs for predicate argument patterns is first to remove quantifiers and then make tautological inferences using the apparatus of SC in

r--JVxFx, Fa

V

r--;Fa, Fa

-+

Fa

V

3xFx

We will be allowed in the new system PC to make use of SC since, as we will see later, PC is constructed by making additions to SC. At this time the apparatus of SC may be used either to make tautological transformations of the basic components of a formula or to infer a tautological consequence from a formula. As was indicated in Chapter 4, B is a tautological consequence of A iff A -+ B is a tautology, and A or a part of a formula containing A may be tautologically transformed into B if A t--t B is a tautology. And, as just indicated, A -+ B or A <--+ B are tautologies iff they take the value T for every assignment of truth-values to their basic components. The import of these remarks for what is at hand is that we may make inferences such as those below since the second formula is a tautological consequence of the first.

(1) Fa /\ Ga

might be V: posi~ive integers, Fx: x is odd, Gx: x is even, a: 1; or it might be V: natural obJects, Fx: x is solid, Gx: x is heavy, a: Earth. Individual const~nts differ from individual variables in the fact that they cannot be quantIfied, though they can come within the scope of quantifiers. In the formation rules given at the end of the last chapter, an atomic formula is a capital letter followed by n number of terms. We will now count as terms not only individual variables but also individual constants. To illustrate, we now count the formulas below as wffs:

-+

3x F x

Fa /\ Ga Fa

VxFx /\ Fa Fa

Fa VxFx

V

Exercises 1. Which of the following now are win (a) VxFxa (b) VxP (c) VxFm (d) Vx(Gx -+ VzHxz) (e) 3xFx -+ VzGxz (f) 3x(Gx /\ Ga)

2. Which of the following are valid formulas? Demonstrate the invalidity of the invalid formulas. (a) Fa -+ VxFx

(b) Vx Fx -+ Fa (c) Fa -+ 3xFx (d) 3xFx -+ Fa

136

Natural Deduction System 1'C

3. Which of the following are tautologies? (a) P /\ 'VxFx -+ P (b) ,-...,'Vx(Fx /\ ,-...,Fx) (c) 'V xFx /\ 3xGx -+ 3xGx (d) Fa /\ P -+ Fa (e) 'Vx(Fx /\ Gx -+ Fx) (f) 3xFx /\ 3xGx ~ ,-...,(3xFx -+ ,-...,3xGx) 4. In each pair, is the second formula a tautological consequence of the first? (a) 'VxFx . ,-...,,-...,AxFx (b) 'VxFx Fa (c) 'VxFx -+ 3xGx ,-...,3xGx -+ ,-...,'VxFx (d) 'Vx(Fx -+ Gx) 'VxFx -+ 'VxGx (e) '-""('VxFx -+ 3xFx) 'V xFx /\ ,-...,3xFx (f) 'VxFx 3xFx (g) Fa -+ 'V xFx ,-...,(Fa /\ '-""'VxFx) (h) 'Vx'VyFxy 'Vy'VxFxy (i) 'Vx,-..., (Fx V Gx) 'Vx(,-...,Fx /\ ,-...,Gx) (j) Fa /\ Fb /\ ... 'VxFx Answers 1. Only (a), (d), and (f) are wffs.

2. Only (b) and (c) are valid. To demonstrate the invalidity of (d), let V: human beings, Fx: x is handsome, a: Charles DeGaulle. 3. All but (b) and (e). 4. The second lines in (a), (c), (e), and (g) are tautological consequences.

6.2

Universal elimination and existential introduction

W ebegin with a rule of inference that allows us to remove universal quantifiers from formulas of the form 'V xA. The rule expresses the truth that if we are given, say 'VxFx, then F _ follows no matter what term we place in _. For if in a given domain every x is F, then each element

137

Universal Elimination and Existential Introduction

the domain is F. In other words, given the domain {a, b, c, ... } ('VxFx

-+

Fa) /\ ('VxFx -+ Fb) /\ ('VxFx

-+

Fc) /\ ...

This general truth can be expressedin.the form of a.rule .comm~nl! lmown a S Universal Specification or Umversal QuantificatIOn EhmI• . n To express this rule, let A be a well-formed formula of the natlO . language, v a variable, an d t an III . d"d t an t . Then redicate IVI uaI cons ~(tlv) is the formula that results when each occurren~e of v in A is IS the result of rep1ace d by an occurrence of t. In other words, A(tjv) " bstituting t for v in A. To illustrate, suppose A IS 'VxFxy and v IS y. is a, then A(tJv) is 'VxFxa, and if tis b, then A(t/v) is 'VxFxb.

;;t

The rule of universal elimination is DE 'IvA A (tjv)

Examples of the use of this rule would be 'VxFx

'VxFx

Fa

Fib

'Vx3yFxy 3yFay

'Vx(Fx -+ 3yFxy) Fa-+ 3yFay

'VxFxa Faa

In using this rule we must remember several things. First, t goes in for v, and not for any other variable. It only goes in for the v found in 'VvA. Thus the following inferences are erroneous: 'Vx3z(Fxz /\ Gxz) 3z(Faz /\ Gaa)

'VxFxb Faa

since a went in for the second z and z is not v(x) in the first example, and in the second example a went in for band b is not v(x). Second, t goes in for each occurrence of v in A. Thus the following are erroneous: 'Vx(Fx -+ Gx) Fa-+ Gx

'Vx(Fx -+ Gx) Fa-+ Gb

Finally, the inferences below are also erroneous: 'V xFx -+ 'V xGx Fa -+ 'VxGx

'VxFx -+ 'VxGx Fa-+ Ga

since to UE we must begin with a formula of the form 'VvA, where the quantifier has the scope of A.

Natural Deduction System Pc

138

To illustrate a simple use of DE, consider the proof below:

Universal Elimination and Existential Introduction

We now illustrate some uses of EI and UE in the next two proofs. VxFx f- 3xFx A 1. VxFx 1, DE 2. Fa 2, EI 3. 3xFx

Vx(Fx -+ Gx), Vx(Gx -+ Hx) f- Fa -+ Ha 1. Vx(Fx -+ Gx) A 2. Vx(Gx -+ Hx) A 3. Fa -+ Ga I, DE 4. Ga-+Ha 2, UE 5. Fa -+ Ha 3, 4, Trans

The basic strategy of constructing proofs for predicate argument patterns is first to remove the quantifiers. This can be done with the premises of this pattern now that UE is available. Once t's are inserted for v's, the result is atomic formulas related by -+. 5 is a tautological consequence of 3 and 4. The second rule of inference reflects the valid move of going from, say, Fa to 3xFx. If an a in the domain of discourse has F, then something in the domain has F. The rule that warrants such a valid inference is called Existential Quantifier Introduction or Existential Generalization. The rule is EI

A (t/v) 3vA

As before, A (tlv) is the formula that results when each occurrence of v in A is replaced by an occurrence of t. In determining the correct application of UE, one's attention goes from the bottom line to the top line, whereas the reverse is the case with EI. With EI, one must see if t goes in for each occurrence of v in A. Examples of EI would be: Fa 3xFx

Faa 3xFxx

Faa 3xFxa

Fa -+ VxFx 3y(Fy -+ VxFx)

Note the third pattern, which is valid and accords perfectly with EI. Examples of erroneous uses of the rule would be: Fa -+ VxFx 3x(Fx -+ VxFx)

Fab 3xFbx

Fab 3xFxx

In the first case 3x(Fx -+ VxFx) is not a wff; also each occurrence of v(x) in A(Fx -+ VxFx) is not replaced by an occurrence of t(a). In the second case b was changed to a and we did not merely have b going in for x in Fbx. In the third case neither a nor b go in for each occurrence of x in Fxx.

139

Vx(Fx -+ Gx) f- Fa 1. Vx(Fx -+ Gx) Fa 3. Fa-+ Ga 4. Ga 5. 3xGx 6. Fa -+ 3xGx

-+

IT

3xGx A A I, DE 2,3, MP 4, EI

RCP

Exercises 1. Construct proofs for the following, using DE and EI. (a) VxFx f- Fa (b) Vx(Fx A Gx) f- 3x(Fx A Gx) (c) VxFx f- VxFx (d) Faa f- 3xFxa (e) ,.....,3xFxa f- r-.;Fba (f) ,.....,Faa -+ Vx r-.; Fax f- Fab -+ Faa (g) VX(f".'Fxb -+ Fab), r-.;Fab f- r-.;3xFxb (h) Fa, Vx(Fx -+ Gx) f- f'.'VX r-.; Gx 2 If formulas such as V xV xFx and V xFa were admitted as well-formed, what . restrictions would be needed on the rules to prevent invalid inferences such as

Vx(Fx -+ 3xVyFxy) Fa -+ 3xVyFay

Fa-+ VxGaa 3x(Fx -+ VxGxx)

Answers I. (c) 1. VxFx A 2. r-.;V xFx A 3. (I) A (2) I, 2, Conj 4. "" ~VxFx RAA 5. VxFx 4, DN A (f) 1. r-.;Faa -+ 'Ix r-.; Fax A 2. ",(Fab -+ Faa) 2, Imp 3. ",(~Fab V Faa) 3, DN,DM 4. r-.; r-.;Fab A '"'-'Faa 4, S 5. r-.;Faa 1,5, MP 6. 'Ix"", Fax 6, DE 7. r-.;Fab 8. r-.; r-.;Fab 4, S 8,DN 9. Fab 9,7, Conj 10. Fab A r-.;Fab RAA II. r-.; ,,-,(Fab -+ Faa) 12, DN 12. Fab -+ Faa

141

Y::E:xistentlaJ Elimination and Universal Introduction

2. A (tjv) must be the formula that results when each j1'ee occurrence of v in A is replaced by an occurrence of t.

, A (tfv) represents our substituting an indivi~ual constant, t, in lace v occurs in A. Examples of the use of thIS rule are eacI1 P 3xFxx Faa

3xFx ~ 6.3 Existential elimination and universal introduction

Both UE and EI are valid without any provisos. However, the rules to remove existential quantifiers and to add universal quantifiers need certain restrictions to insure their validity. To introduce the rule that allows the removal of existential quantifiers under certain conditions, we first consider this semiexplicit proof: (1) Vx(Fx -+ Gx), 3xFx I- 3xGx l. Vx(Fx -+ Gx) A 2. 3xFx A 3. Fa from 2 4. Fa-+ Ga 1, DE 5. Ga 3,4, MP 6. 3xGx 5, EI

This derivation may be described as follows: Assume that whatever has F has G. Now assume that something has F, and let us assume that a is the thing that has F (hence line 3). If a has F and if, given 1, whatever has F has G, then a must have G and thus there is something that has G. Such reasoning is sound. Note that in these reflections we take a as being one of the things in V which has F given line 2-3xFx. The idea is that if F is true for some individual, then we can choose some individual, a, for which F is true. If an individual term is used to stand for some chosen individual, then we will say the term is limited. When we validly move from 3xFx to Fa, 'a' is limited. It should be noted that if an individual constant is limited it will not follow that the term is used to stand for some chosen individual, for we wish to use the expression 'limitation' to cover this use of individual terms and a second and third use to be discussed shortly. As will become clear later, to say a term is limited is to say that we cannot do with it some of the things we do with other individual constants; we cannot use it in existential elimination and in universal introduction, the fourth rule to be introduced shortly. The rule of existential elimination can be stated as follows: EE

"'>/ ••

3vA A (tjv)

where t is not limited

3yVxFxy VxFxa

3xFxb Fab

provided a is not limited

Erroneous uses of the rule are _3xFx

1'f

3x(Fx -+ Gx) Fa -+ Gx

. l'ImI'te d , a IS

Fa

Vy3xFxy VyFay

If we were allowed to use EE with an individual constant that is limited, then we could "prove" the invalid pattern below: 3xFx, 3xGx I- 3x(Fx l. 3xFx 2. 3xGx 3. Fa 4. Ga 5. Fa /\ Ga 6. 3x(Fx /\ Gx)

/\ Gx) A A 1, EE 2, EE (erroneous) 3, 4, Conj

5, EI

Since a is introduced on line 3 by EE, a becomes limited, and conseuently in using EE on line 2 we cannot use a. Another term must be q use d , but this would then prevent our obtaining line 5. The idea here is that if we let a be the individual that is F, given 3xFx, then we can validly move from 3xFx to Fa, but then we cannot move from 3xGx to Ga. We need to introduce another term, which itself becomes limited and could not then be used if another existential quantifier were eliminated. We resume following the restriction on EE in the next proof. 3x3yFxy I- 3y3xFxy l. 3x3yFxy A 2. 3yFay 1, 3. Fab 2, 4. 3xFxb 3, 5. 3y3xFxy 4,

EE EE EI EI

In using EE we introduce certain constants into the proof. It is important to avoid having these constants appear in conclusion lines. To illustrate, note that the following argument patterns are invalid but could easily be established if we allowed limited constants introduced by EE in conclusion lines. 3xFx I- Fa Vx(Fx -+ Gx), 3xFx I- Ga

Natural Deduction System

14"

As was pointed out earlier, in providing an interpretation for a formula with an individual constant we supply a name for one of the individuals in the domain of discourse. This being the case, a counterexample for 3xFx f- Fa-would be: V: positive integers, Fx: x is a prime number, a:4. The last rule permits the introduction of universal quantifiers under certain conditions. Again let us consider a semiexplicit proof:

flif:~;~ti~I Elimination and Universal Introduction

in A. For example, the following are correct if we suppose that a not limited: Fa

v;;FX , "IT~,n,,~~T<'1'

Vx(Fx --+ Gx), Vx(Gx 1. Vx(Fx --+ Gx) 2. Vx(Gx --+ Hx) 3. Fa --+ Ga 4. Ga --+ Ha 5. Fa --+ Ha 6. Vx(Fx --+ Hx)

--+

Hx) f- Vx(Fx

--+

Hx)

A A 1, DE 2, DE 3,4, Trans

from 5

When we used UE in the above proof to obtain 3 from 1 and 4 from 2, We used a. But we could have used any name for an individual in the domain of discourse. In effect, a here is an arbitrarily selected element of the domain. Suppose, for example, there are aI' a 2 , • • • an individuals in the universe of discourse. From (1) by UE we can obtain

which in this V means that Vx(Fx --+ Hx). In general, when u is introduced by UE and it is proven that a has some property F, then it validly follows that VxFx, since in this context we could have proven that each individual in V or any arbitrarily selected individual in V has }!', The last quantification rule allows us to go from A (tlv) to VvA and is commonly known as Universal Generalization or Universal Quantification Introduction. To secure the validity of such an inference the rule must be stated with two provisos: DI

A(t/v) VvA

where t is not limited and does not occur in VvA

Again, the idea of this rule is that if A is true for t and t is not limited, and it stands for an arbitrary element of the domain, then A is true for every element of the domain. Again, A (tlv) indicates that the individual constant goes in each place where the individual variable in question is j.,

3yFya Vx3yFyx'

Faa VxFxx

these are erroneous: Fx--+ Ga Vx(Fx --+ Gx) ,

3xFxa Vx3xFxx'

Faa VxFxa

the first, a(t) does not go in each place where x(v) is found in A, and in the second, (tlv) is violated and Vx3xFxx is not a wff. In the third attern, u(t) occurs in V IvA. As it will turn out, if a term appearing in a p is not limited, then at that point it stands for an arbitrary element. It has already been stated that t is limited when it is introduced by Consequently, ur cannot be applied to a term with an EE source. fro illustrate the danger of ignoring the EE source of a term when using consider the erroneous "proof" below: 1. 3xFx 2. Fa 3. VxFx

and from 2 by UE we could obtain

143

A 1, EE 2, DI (erroneous)

A somewhat more sophisticated violation of the proviso on in this next example. 1. 3xFx 2. Fa r~' r--.;Fa

Li:

(2) 1\ (3)

5. r--.;r--.;Fa 6. Fa 7. VxFx

ur is found

A 1, EE A 2,3, Conj

RAA 5,DN 6, DI (erroneous)

Introducing an individual constant by EE is not the only way in which a term can become limited; there are two and only two other ways. The second way a term can become limited is by appearing in an assumption line. But when the assumption is discharged, it should be stressed, the term is no longer limited unless it is limited in virtue of being introduced by EE (see the last proof). To illustrate, consider the example below. 1. Fa 2. VxFx

A 1, DI (erroneous)

Natural Deduction System

In this example, a is limited since it occurs in an undischarged assumption line. However, in the next two proofs a is not limited when VI is applied since the assumption line in which a makes its appearance has been discharged. Prove: ,.....,3xFx f- "Ix,....., Fx

. U

A

1. ,.....,3xFx Fa 3. 3xFx 4. (1) A (3) 5. ,.....,Fa 6. Vx,.....,Fx

A 2, Er 1, 3, Conj RAA 5, Dr

"_'"'",..",,. Elimination and Universal Introduction

a is not introduced by EE, nor does it appear in an undisassumption both b and a become limited when EE is applied. chtLrge d ' b is limited follows from the use of EE. That a becomes limited is erhtLPS surprising until we see that Faa would not follow from ~:dyFXY and 3yVxFxy does not follo~ ~rom VX3yFxy. To prev~nt su~h £: lces a term which occurs withm the scope of an eXIstentIal iULel'eI , . . . .fier and which is later removed by EE wIll become lImIted. A quant 1 . good rule to follow with resp~ct to applying EE, by the way, IS not to use a t which previously occurs m the sequence. The converse of the above is a valid argument pattern and can easily be proven.

Prove: Vy(Fy ->- 3xFx) I~. Fa L! 3xFx 3. Fa ->- 3xFx 4. Vy(Fy ->- 3xFx)

RCP 3, Dr

Note that in this proof "Ix cannot be used in VI on line 3 since ->- 3xFx) is not a wff. It is worthwhile to note that Fa f- VxFx

is not a valid pattern. Given this interpretation: V: positive integers, Fx: x is even, and a: 2, we obtain a counterexample. It should also be remembered that since a term is limited in an undischarged assumption line, this term cannot be used in EE unless the assumption has been discharged. For from 3vA we can obtain A (tjv) only if t is not limited. Failure to keep this in mind might result in the following unfortunate sequence: A

Fa 3xGx Ga Fa A Ga 5. 3x(Fx A Gx)

3yVxFxy f- Vx3yFxy 1. 3yVxFxy A 2. VxFxa 1, EE 3. Fba 2, DE 4. 3yFby 3, Er 5. Vx3yFxy 4, Dr

A 2, Er

Vx(Fx

1. 2. 3. 4.

A 2, EE (erroneous) 1,3, Conj 4, Er

The reader can easily demonstrate the invalidity of Fa, 3xGx f3x(Fx A Gx).

The third and final way in which a term can become limited is by appearing in a line that is introduced by EE. To illustrate:

Note that the b in line 4 is not limited. The proviso on both EE and VI is that t not be limited. Let us conclude this section with a summary of how an individual constant, t, can become limited. t can become limited iff either (1) t is introduced by EE, (2) t appears in an undischarged assumption line, or (3) t appears within the scope of an existential quantifier later removed by EE.

Exercises 1. Find and explain the errors in the following items. (a) 1. VxVyFxyx

2. 3. 4. 5. (b) 1. 2. 3. 4.

5. (e) 1.

1. 2. 3. 4. 5. -"

Yx3yFxy 3yFay Fab VxFxb 3yVxFxy

A 1, 2, 3, 4,

145

DE EE Dr (erroneous) Er

2. (d)r1.

lb 3. 4.

VyFxya Fxba VxFxba VyVxFxyy ,.....,VxFx ""Fa ,.....,Fa V Ga ""Fa vVxGx Vx,-...., Fx V VxGx Fab --+ VxGax Vx(Fxb ->- VxGxx) Fm VxFx Fm --+ VxFx VyFy ->- VxFx

A 1, DE 2, DE 3, Dr 4, Dr A

1, 2, 3, 4, A

DE Add Dr Dr

Dr A 1, Dr

RCP 3, Dr

146

(e) l. 3xGxa 2. Gaa 3. 3xGax 4. 3y3xGyx (f) l. YxFx 2. Fa 3. Fb 4. Fa A Fb 5. Yx(Fx A Fx) (g) l. YxYyFxy 2. Faa 3. 3xFxa 4. Fba 5. YxFbx 6. 3yYxFyx

'''.n.f",!!'les

A 1, EE 2, EI 3, EI A 1, VE 1, VE 2,3, Conj 4, VI A 1, VE (2) 2, EI 3,EE 4, VI 5, EI

2. Numerals are proper names for numbers and thus are singular terms. Rath than let an individual constant stand for a numeral, we will leave the numera~r as they are. Find and explain the errors in the following. s

(a)ul. 1 >> 0 0) 2. 3x(x 3. a> 0 4. Yx(x > 0) 5. (b > 0) 6. Yx(x> 0) 7. (1) - (6) (b) l. "",(a> 3) 2.4> 3 3. 3x(x > 3) 4. a> 3 5. P A "",P - Q 6. (1) A (4) - (7) 7. 0 > 1 8. "",(a> 3) _ 0 > 1 9. "",(0 > 1) 10. a > 3 II. Yx(x > 3) (c) l. 2> 1 2. 3x(x> 1) 3. a> 1 4. a > 1 - a > 0 5. a > 0 6. Yx(x> 0)

A 1, EI 2, EE 3, VI 4, VE 5, VI RCP A

A

2, EI 3, EE SC theorem 5, substitution 1, 4, 16, MP RCP A 8, 9, MT 10, VI A EI 2, EE A 3, 4, MP 5, VI 7. (1) - «4) - (6» RCP (2) 3. Vsing VG and VI, construct proofs for the following. (a) YxFx f- YyFy (b) P A YxFx f- Yx(P A Fx) (c) Yx(Fx _ Gx) f- YxFx _ YxGx (d) Yx(Fx - Gx) f- Yx "'" Gx _ Yx"" Fx (e) Yx(Fx ~ Gx) f- YxFx ~ YxGx ~I·,

in Using Quantification Rules

147

f) VxFx V VxGx f- Yx(Fx V Gx) VxFx A VxGx f- Yx(Fx A Gx)

~g)

Using EE and EI, construct proofs for the following patterns. (a) Fab - 3xFxb, Fab f- 3yFyb (b) 3x(Fx - Fa), ""Fa f- 3x "" Fx (c) r-I3xFxb --+ Fab, ""Fab f- 3y3xFyx (d) 3x{Fx V Fx) f- 3xFx (e) 3xFx f- 3x3y(Fx A Fy)

Answers (f) Only the use of VI is erroneous since a{t) does not go in for each occurrence of x(v). (g) Only the use of VI is erroneous since a becomes limited on line 4.

3. In Section 6.6 see T2 for (g). (d) l. Vx(Fx --+ Gx)

li: :::"G:Gx 4. 5. 6. 7.

Fa- Ga r-IFa 'Ix"", Fx 'Ix r-J Gx - Yx

1"'0.1

Fx

4. (e) 1. 3xFx

2. Fa 3. FaA Fa 4. 3x3y(Fx A Fy)

A A 2, VE 1, VE 3,4,MT 5, VI RCP A 1, EE 2, Idem 3, EI (2)

6.4 Strategies in using quantification rules

There are several strategies that should allow the construction of proofs for most predicate argument patterns. We review some of these strategies at this time. STRATEGY ONE. If the conclusion is of a conditional forni, A --+ B, then assume antecedents for an RCP proof, as was done in SC. This strategy was used on examples and in some of the exercises. To illustrate the strategy with an only slightly more complicated proof, consider:

Yx(Fx _ Gx) f- 3xFx l. Yx(Fx _ Gx) 2.3XFX 3. Fa 4. Fa _ Ga 5. Ga 6.3xGx 7. 3xFx - 3xGx

Q

3xGx A A 2, EE 1, VE 3,4, MP 5,EI RCP

Natural Deduction System Po

In general, any of the strategies used in constructing proofs in SO can be used in PC. For example, Dis can often be used when the inner structure of a quantified formula is of the form A V B. To illustrate, consider: 3x(P V Fx) I- P V 3xFx 1. 3x(P V Fx) A 2. P V Fa 2, EE .P A 4. P V 3xFx 3, Add 5. Fa A 6. 3xFx 5, EI 7. P V 3xFx 6, Add 8. P V 3xFx Dis

U

STRATEGY TWO. Sometimes the conclusion will have the form VxA or 3xA, with A having the form B ->- O. Often B with t going in for v can be taken as an additional assumption in an RCP proof. To illustrate: Vx(Fx --+ Gx) I- 3x(Fx ->- Gx) A 1. Vx(Fx ->- Gx) . Fa A 3. Fa ->- Ga 1, UE [} 4. Ga 2,3, MP 5. Fa --+ Ga RCP 6. 3x(Fx --+ Gx) 5, EI 3xFx --+ P I- Vx(Fx --+ P) 1. 3xFx ->- P A Fa A 3. 3xFx 2, EI 4. P 1,3, MP 5. Fa --+ P RCP 6. Vx(Fx --+ P) 5, UI

. [1

Note that with Fa (line 2) discharged in both proofs, a is no longer limited. STRATEGY THREE. RAA strategies can be employed in interesting ways. An interesting example appears below. ,.....,Vx,....., Fx I- 3xFx 1. ""Vx",-, Fx A ,2. ,....,,3xFx A

One possibility is to introduce an assumption Fa that will lead to a contradiction so that we can get f'..IFa with Fa discharged. If Pa has

• ' Using Quantification Rules strategieS III

149

then a is not limited in ",-,Pa, so we can use UI. discharged, F can con]' oin this to line 1 and use a final sed Dr on "'-' a, we . . 2 Let us continue wIth thIS strategy. u . 1 fl' to get the dema 0 me .

' G:

Fa 4 3xFx

~

(2) /\ (4) 6. ",-,Fa 7. Vx"'-' Fx 8. (7) /\ (1) 9. "'-' ,-..;3xFx 10. 3xFx

A 2, EI 2,4, Conj

RAA 6, UI 7,1, Conj

RAA 9,DN

. hen the conclusion is of the form VxA or 3xA we. may find sometImes w . . RAA strategy. The next proof Illustrates . usef u1 to deny A m usmg an Jt strategy. A 3xGx I- 3x(Fx A Gx) VxFx /\ 3xGx A 3xGx 1, S Ga 2, EE 4' VxFx 1, S Fa 4, UE ro-'(Fa /\ Ga) A 7 ",-,Fa V r-JGa 6, DM 8' ",-,Ga 5,7, DS 9: Ga /\ r-JGa 3, 8, Conj 10. r-J "",,(Fa /\ Ga) RAA 11. Fa A Ga 10, DN 12. 3x(Fx A Gx) 11, EI

VxFx 1. 2. 3

. U 5:

Exercises . . UG UI EG and EI construct proofs for the followmg: (A 11- B 1. Usmg , " 'd b th 'ays) means the derivation is purporte to go 0 " . (a) 3xFx V 3xGx I- 3x(Fx V Gx) (b) Vx""" Fx I- ,-.....,3xFx (0) 3x r-J Fx 11- ,--,VxFx (d) Vx(Fx V Gx) I- VxFx V 3xGx (e) VxFx V 3xGx I- 3x(Fx V Gx) (f) ......,3x,-....; Fx 11- VxFx (g) Vx3y(Fxy --+ Gxy) I- 3xVyFxy --+ 3x3yFxy (h) Vx(Fx ->- P) 11- 3xFx ->- P (i)vy(Fy -+ Gy) I- Vx(3y(Fx A Fxy) ->- 3y(Gy A Fxy» . the four rules then the 2. If individual variables were taken as terms m tl stat~d' inferenoes below would be justified by the rules as presen Y . Vx3yFxy 3yVxFxy 3yFyy VxFxx

150

Natural Deduction System

Pc

and yet these inferences are invalid (demonstrate this). The trouble is that when t goes in, it becomes accidentally bound. Why can't this happen with individual constants? What proviso must be added to which rules to prevent the rules from being invalid if individual variables are allowed as instances of t?

· U 5. P

6. 3xFx--+ P -no 3xFx --+ P Fa 3. 3xFx 4. P 5. Fa --+ P 6. Vx(Fx --+ P)

. [i

A A 2,EE 1, VE 3,4,MP ROP

A A 2, EI 1,3,MP ROP 5, VI

2. Proviso added to VE and EE: t must be free for t/v, t does not become bound by a quantifier in A.

6.5

have been working with, formulas such as Fx--+ 3xFx Fx --+ 3xFx~ Fa --+ 3xFx

and they are valid formulas. However, though ell-formed woU Jd b e w ' . . ~e could derive, say, Fx --+ Fx, we could de1'lve neIther of the above only individual constants are instances of t in the four rules. We 'ndividual variables as values of t, but then the rules would could a11ow I the adjustment discussed in exercise 6.4.2. We wish to keep the as simple as possible. So the decision is. made to keep the r~les as are. Not a great deal is lost in not havmg such theorems smce as biconditional suggests, whenever we have a valid forn: ula with a individual variable this is equivalent to a formula WIth the free individual variables uniformly replaced by an individual constant (and versa). Nevertheless the PC with the previously discussed for-

Answers 1. (h) 1-1. Vx(Fx --+ P) 3XFX 3. Fa 4. Fa --+ P

151

System PC

V.

t is free for v iff ,vhen

lUation rules could not be complete. . . To provide for the possibility of PC's completeness we w~ll SImply not count formulas with free individual variables as wffs. Agam no loss . results since the equivalence described just now obtains. The rules of inference are those for SO plus the followmg, where, again, A(tfv) is a formula that results when each occurrence of v in A is replaced by an occurrence of t:

System PC

We turn now to the construction of the system that will have valid predicate formulas as theorems. This system, PO, will be in part made up of the sentential calculus, specifically system SO. Thus the system will have theorems that are tautologies and theorems that are valid predicate formulas. These are the only kinds of theorems the system will have. In addition, all tautologies will be theorems of the system and all valid predicate wff of the system will be theorems. Since PC will be built on SC, we count among the symbols of the system all those of SC plus 1. Predicate variables: F, G, H, ... . 2. Individual variables: x, y, z, ... . 3. Individual constants: a, b, c, ... .

4. Quantifiers: V, 3

It will be advisable to alter slightly the formation rules in setting up PC. We wish PC to be complete. This means we wish all valid wellformed formulas to be theorems of PC. Now given the formation rules

VE

VvA A (t/v)

EE

3vA A (t/v)

where t is not limited

EI

A (t/v) 3vA

VI

A (t/v) VvA

where t is not limited and does not occur in V vA

We should note that the metavariables in the ten rules taken from SO are now variables for any well-formed formula in PC. This fact allows us to make tautological transformations and tautological inferences on basic components of formulas. The definition for the biconditional is also carried over from SO. A proof in PC has the same description as a proof in SC, with the rider that a theorem line cannot contain a limited term introduced by EE (see Section 6.3). We recall that a theorem is obtained when all assulUptions have been discharged. Exercises 1. What changes need to be made to the primitive basis of PO if V is taken as the only primitive quantifier?

Natural Deduction System p()

2. List some other valid formulas containing predicate expressions that ar r and not theorems of PC. What is the counterpart of each where indiev:da ld constants are used? I uaJ 3. Set down the formation rules for PC.

,,-..,3xFx I- "Ix """ Fx r--.JVx""" Fx I- 3xFx

~~. Ji'x !2, TI (T4) r;: :~:]i': "Ix""" Fx (1) 1\ (3) 5. "" ,......,3xFx

2. For example VxFx ->- Fx and VxFx ->- Fa

Theorems of PC

In this section we will prove a few theorems, introduce the two d . rules of PO, and list some major theorems. e1'lved T1

T2

Y,x(Fx 1\ Gx) ->- VxFx 1\ VxGx 1. Vx(Fx 1\ Gx) A 2. Fa 1\ Ga 1, VE 3. Fa 2, S 4. VxFx 3, VI 5. Ga 2, S 6. VxGx 5, VI 7. VxFx 1\ VxGx 4,6, Conj 8. T1 RCP \f.xFx 1\ VxGx ->- Vx(Fx 1\ Gx) 1. VxFx 1\ VxGx A 2. 'VxFx 1, S 3. VxGx 1, S 4. Fa 2, VE 5. Ga 3, VE 6. Fa 1\ Ga 4,5, Conj 7. Vx(Fx 1\ Gx) 6, VI 8. T2 RCP

The first derived rule will be a carry-over of Theorem Introduction (TI) from SO: Now, however, we understand theorems to include not only tautologIes .but all theorems of PO or any instances of them. The next two proofs Illustrate the use of TI. T3

Vx(Fx 1\ Gx)~ VxFx 1\ VxGx 1. T1 TI

2. T2 3. T1 1\ T2 4. T3 ""f.,

T4 T5

Li

Answers

6.6

153

TI 1, 2, ConJ. df

6. 3xFx 7. T5

1, 3, Conj RAA 5, DN RCP

Note that TI can be eliminated from this proof and the steps of the proof for T4 can be inserted. (T5 can also be proven using a different strategy ,vithout T1 or T4-see Section 6.4.) Such eliminability is a necessary feature for a derived rule such as TI. Also note that we have again explicitly collapsed introducing the theorem (with or without substitution) and using MP into one step, and we have justified this with TI (T4). The Rule of Replacement (R) from SC may be applied to welIformed formulas in PO. As R is a derived rule in SO, so it is a derived rule in PC. For example, having proven T3 we can now distribute quantifiers within quantified conjunctions and justify this by appealing to R, or we can omit using R or T3 (with or without substitution) and MP and go through the laborious process of recreating an instance of the proofs of T1, T2, and T3. In proofs it is sometimes convenient and time saving to use R. For example, having proved

we could make the following move: 1. 3x(Fx ->- Gx) 2. """"Ix"",, (Fx ->- Gx)

1, R (T6)

In this case Fx ->- Gx is a substitution for Fx in T6. The move from 1 to 2 above could also be made by TI and MP. However, if 1 appeared as part of a longer formula, R would have to be used. Often when TI or R are used, substitutions are made in formulas. The rule that makes the notion of substitution effective in PC is a difficult rule to state. Since.the student intuitively substitutes correctly when he uses R or TI with substitution, and since both Rand TI are luxury rules, things we can get along without, we will not set down the rule for substitution. Also in constructing proofs for formulas, normally the only occasion we have for using TI or R is to change existential quantifiers into universal

Natural Deduction System

nUf,lanIBSS, Consistency, and Completeness of PC

quantifiers (and -:ice versa) in order to drop the quantifier. In other words, TI or R WIll normally be used with T6 plus the theorems below.

Yx,...."", Fx ~ """""3xFx

T9

,...."",3x,...."", Fx ~ YxFx

TI-T9 a~ong with the theorems below make up the major theorems the predICate calculus. of TIO TIl TI2 TI3 TI4

TI5 TI6 TI7 TI8 TI9 T20 T21 T22 T23 T24 T25

T26 T27 T28 T29 T30 T31 T32 T33 T34 T35

T36 T37 T38

Prove the following theorems. (a) Vy(YxFx -- Fy) (b) 3y(3xFx -- Fy) (c) Vy(Fy -- 3xFx) (d) 3y{Fy -- YxFx)

3x,...."",Fx~,...."",YxFx

T7 T8

Establish the analyticity of those statements that are analytic in exercise 5.7.3.

Answers 1. f"'oooJ3y(Fy -- YxFx)

A

YxYyFxy~ YyYxFxy

2. 3. 4. 5.

I, R (T8) 2, UE 3, Imp

Fa--* 3xFx

6.

YxFx--* Fa YxFx --* 3xFx YxFx~YyFy

3xFx~3yFy

7. 8. 9. 10. II.

3x3yFxy~

3y3xFxy 3xYyFxy --* Yy3xFxy Yx(Fx --* Gx) --* (YxFx --* YxGx) Yx(Fx~ Gx) --* (3xFx~ 3xGx) Yx(Fx~ Gx) --* (YxFx ~ YxGx) Yx(Fx --* Gx) --* (3xFx --* 3xGx) (YxFx --* YxGx) --* 3x(Fx --* Gx) Yx3y(Fxy --* Gxy) --* (3xYyFxy --* 3x3yGxy) YxFx /\ 3xGx --* 3x(Fx /\ Gx) Yx(Fx V Gx) --* YxFx V 3xGx YxFx V YxGx --* Yx(Fx V Gx) YxFx V 3xGx --* 3x(Fx V Gx) 3x(Fx --* Gx)~ YxFx --* 3xGx (3xFx --* YxGx) --* Yx(Fx --* Gx) 3x(Fx /\ Gx) --* 3xFx /\ 3xGx 3x(Fx V Gx)~ 3xFx V 3xGx Yx(P /\ Fx)~ P /\ YxFx Yx(P V Fx)~ P V YxFx 3x(P /\ Fx)~ P /\ 3xFx 3x(P V Fx)~ P V 3xFx 3x(P --* Fx)~ P --* 3xFx Yx(Fx --* P)~ 3xFx --* P 3x(Fx --* P)~ YxFx --* P

Exercises l. Using .the derived rules as needed, construct proofs for the unproven theorems fol~owmg the numbered order. It is permissible to pass up any theorem for

whI~h the corres~onding argument pattern has been proven in the previous sectIOns or prevIOus exercises.

155

6.7

Yy,....., (Fy -- YxFx) f"'oooJ(Fa -» YxFx) ,.....,(,.....,Fa V YxFx) f"'oooJ ,.....,Fa /\ ,.....,YxFx f"'oooJ,-."JFa Fa YxFx YxFx /\ ,.....,YxFx ,.......,,.....,3y(Fy-- YxFx) Td

4,DM 5, S 6,DN 7, UI 5,8, S, Conj

RAA 10, DN

Soundness, consistency, and completeness of PC

If one of the uses of PC is to establish predicate formulas as valid, then PC must be consistent; otherwise it could be used to establish any predicate formula. Also the formulas it establishes as theorems must be valid if PC is to have this function. In other words, PC must be sound. Finally, we wish all valid predicate formulas to be theorems of PC; that is, we wish PC to be complete. If PC is complete, then we can say a predicate formula is valid only if it is a theorem of PC. To prove the soundness of PC one must first prove that each of the four quantification rules is valid. The next step is to show that in a proof each line is a valid consequence of its assumptions. If A is a valid consequence of no assumptions, A is valid. Since theorems in PC are assumption free lines, all the theorems would be valid. If one proves PC sound, then it follows that it is negation consistent since no theorem can have the form A /\ ,.....,A. However, one can prove PC negation consistent without relying on a proof of the soundness of PC. Let us outline one such proof. To each formula A we assign a formula A', called an associated sentential formula (asf). We obtain the asf of a quantified formula by simply deleting all the quantifiers, individual variables, and individual

156

Natural Deduction System PC

constants, leaving everything else, including all the "-' signs, and substituting sentential variables for the predicate variables. Thus, the asfs of VxFx

--?

3xFx,

"-'Vx3y(Fx V Gy)+-+ VxFx V 3yGy

are, respectively, P--? P,

,.....,(P V Q)+-+ (P V Q)

We note that. all the theorems proven or to be proven in this chapter have tautologiCal asfs. All the formulas of SO are already in transformed form, and all theorems of SO are tautologies. If it can be shown that all theorems have tautological asfs, then A /\ ,,-,A cannot be a theorem and PO is negation consistent. All the theorems of PO would have tautological asfs if the rules will only allow theorems that have tautological asfs. If we consider each predicate rule, we see each one has the asf form A' A'

Any asf derived by such rules must be a tautological consequence of the asf assumptions. If an asf line derived from previous asf lines by the rules of PO is a tautological consequence of its asf assumptions then since theorems are assumption free lines, each theorem must have ~ tautological asf. If PO is consistent, this does not imply that all the theorems of PO are valid-that PO is sound. To show this we need but note that the asf form of the consistency proof would work even if the restrictions were removed from EE and Ur. PO with unrestricted quantifications rules can be proven consisten t. However if the restrictions were removed , invalid formulas such as those below would be theorems. Fa --? VxFx 3xFx --? V xFx 3xFx /\ 3xGx --? 3x(Fx /\ Gx) Vx3yFxy --? 3yVxFxy

The asf of each of these formulas is a tautology, we should note. Oonsequently, eliminating provisos on the quantification rules does not affect the consistency of the system, but it does affect the soundness.

):Intllldne.;~, Consistency, and Completeness of PC

157

PO is not only consistent but is sound and complete. We will to the question of the soundness of PO in Part II. The proof for completeness of PO or of any predicate calculus is now generally 'zed as the demarcation point between beginning and advanced rccog nI . and will not be attempted in this book. One final note. Earlier it was pointed out that there is no mechanical .dCCISl . 'on procedure for determining whether any predicate formula is or not. This means that we could not program a computing achine such that it would always give the answer 'yes' if the formula is m and the answer 'no' if the formula is invalid. However, we could use a system like PO to program a machine that would give a 'yes' answer if the formula is valid. We could program a machine that would construct a proof for all well-formed predicate formulas that are valid. However, with respect to some formulas the machine, no matter how it is programmed, would never give a 'yes' or a 'no' answer. In effect, the machine would be inadequate with respect to 'no'. Some formulas that are invalid would be greeted with no answer at all by the machine. Logic, in a word, can never be fully mechanized. This important truth was first noted (in 1936) by the American logician Alonzo Ohurch, when he proved that there is no mechanical decision procedure for predicate logic. Human intuition-providing counterexamples-will thus always be necessary for recognizing invalidity ..

Exercises I. Give the asf of the first ten theorems of PO and demonstrate that they are

tautologies. 2. Would proving PO complete imply that PO is sound? Why? 3. Discuss how one might go about proving that each of the quantification rules is valid. 4. Establish the validity of each of the following arguments that are valid using any of the theorems or derived rules of PO. If the argument is invalid, establish the invalidity of its argument pattern by providing a counterexample. (a) No student who is a member of YSA is dishonest. Some students are dishonest. Therefore, not all students are members of YSA. (b) Some students are athletes. Some students like all athletes. So some students are not disliked by all students and some athletes are liked by some students. (c) Students are animals. So students' tales are animals' tales.

Natural Deduction System

(d) Converse of (c). (e) Some students are radicals. Some faculty members do not like any radicals. Therefore some students are not liked by all faculty members. (f) Anyone who can please any student can please anyone any student can So if Homer cannot please Judy and Judy can be pleased by some student' then Homer cannot please all students. ' (g) If anyone pleases anyone, then someone dislikes both of them. No on dislikes anyone unless he knows them. Judy pleases Homer. Therefore Judy is disliked by someone who knows her and Homer. e 5. In the exercises for 4.8 the tree method for sentential logic was described. Th~s ~ethod ca~ be expanded so as to 0 b~ain a techniqu? for establishing the valIdIty of predIcate formulas. To do thIS we need to mtroduce two rules. The first relates to dropping universal quantifiers when we have formulas of the form 'v'vA and can be stated as follows: UE Given an open path with a sentence of the form 'v'vA for each individual constant, t, that appears anywhere in the path write a sentence of the form A (tfv). If no t appears in th~ path, choose a term. The rule for dropping existential quantifiers is: EE Given an open path with a sentence of the form 3vA, write a sentence of the form A (tfv) where t is an individual constant that does not previously appear in the path. When a formula has either the form ,,-,'v'vA or the form ,,-,3vA, we remove the denial sign by employing ,,-,'v'v A +-+ 3 v "-' A "-'3vA +-~'iv "-' A as if they were rules. We also remove double denials whenever they may occur. The EE rule must be used before the UE rule for the method to work adequately in all cases. To illustrate, examine the following: 'v'x(Fx --+ Gx), 3xFx f- 3xGx 1. 'v'x(Fx --+ Gx) 2. 3xFx 3. "-'3xGx 4. 'v' x "-' Gx 5. Fa 6. Fa --+ Ga 7. ,,-,Ga

~

8. ,,-,Fa Ga x x In using the tree method to establish an argument pattern we continue to enter the premises-lines 1 and 2-and the denial of the conclusion-line 3. Line 4 comes from 3 by using one of the denial equivalences. 5 comes from 2 by EE. 6 comes from 1 by UE. Since only 'a' appears in the path we write Fa --+ Ga. If 'b' had also occurred, then we would write Fb --+ Gb under Fa --+ Ga. 7 comes from 4 by using UE, and 8 comes from 6. An argument is valid iff each path closes. The x's indicate that both paths close, establishing the validity of the argument pattern. ~I·.

Consistency, and Completeness of PC

159

To take another example: 'v'xFx --+ P f- 3x(Fx --+ P) 'v'xFx --+ P ,.....,3x(Fx --+ P)

~

,...,'v'xFx 3x ,....., Fx ,.....,Fa 'v'x,....., (Fx --+ P) ,.....,(Fa-+ P) Fa

P

'v'x "-' (Fx --+ P) ,.....,(Fa --+ P) Fa ,.....,P

x

,.....,P cXh path does close in this finished tree, the argument is valid. Note . . . Smce ea n go on forever-and thus are mvalId-for example, the tree

some t re es ca . for Vy3xFxy (EE introduces a new name, thus UE demands we put I?another 3xFxt where t is the new name, and so on). Thus the tree method If programmed on a computer would have the result that for some sets ~f J! ulas the machine would give no answer. The method, however, IS form lete' thus if the set of entered formuI as . . t th e mach'me IS ,mconsIsten, ::rd gi;e a 'yes' answer. Remember.: ~ is inc.onsistent iff all its ~aths ?lo~e on a tree. P f- Q is valid iff P A ""Q IS mconsIstent, and P --+ Q IS valId Iff ,...,(P -+ Q) is inconsistent. (a) Use the tree method to establish the validity of some argument patterns in this chapter. (b) Use the tree method to establish the validity of some theorems in PC. (c) When a finished tree has an open path, this provides us with a counterexample. Read off some counterexamples from trees for invalid arguments. . (d) Why does this method work? (e) Use the tree method to establish the validity of the valid arguments m exercise 6.7.4.

Answers 2. No. A negation inconsistent logic system is necessarily complete. PC also can be complete but have invalid formulas as theorems.

161

Let us suppose someone said:

predicate logic with identity

7. 1

(1) The figure in the upper left hand corner of the diagram is the figure to the left of a triangle with a circle below it. (2) The figure in the upper left hand corner of the diagram is the figure in the lower right hand corner. To express that a loves b or that a caus~s b, we can Use the two-place predICates Lxy: x loves y, and Oxy: x causes y. To express that is identical with b we could Use Ix ~ ~ is i~entical with y, or the sign f~l: Iden~Ity, =. x = Y is a two-place predIcate constant just like Lxy and Oxy. The identity relation has some rather special properties that will be discussed in this chapter. This relation is also useful in expressing how many things there are that meet a certain description. This will be looked into. In addition, in this chapter, the system of natural deduction for predicate logic PC will be expanded so that the~rern~ containing = can be proven. But first we consider the interpretation for =.

Identity

~he notion of identity, which is of particular interest in logic, can be mtroduced by considering the following diagram.

OD

00 ""/

..

160

(1) identity would be expressed, whereas in (2) identity of type would he expressed. It is the use of 'is' in (1) that is our interest. In (1) we

have two expressions, called definite descriptions, that can be said to name the same thing, but in (2) we have two definite descriptions that name two different things that are exactly alike. To say line segment AB is congruent with line segment DO would be an example of this second notion of 'identical' or 'is'. These two uses of 'is' could be displayed by using the names given to the squares as follows: a is a (identity) a is b (identity of type)

If a happened to have two names-a! and a 2-then identity would be expressed by saying

Identity thus is expressed in saying Lewis Carroll is Charles Dodgson. where we have two names for the same person. The symbol = will be used for the identity predicate. We may interpret = as follows: a = b is true iff a and b are the same. What needs to be kept in mind is that the identity sign is used to express that two singular terms are names for one and the same thing. It is to be recalled from an earlier discussion (Section 5.1) that a singular term is any expression that names or is used to pick out an individual whether or not that individual exists in the universe of discourse. Proper names, demonstrative expressions, definite descriptions-expressions that take the form 'the so-and-so'-and abstract nouns are all used to pick out individuals and are singular terms. Exercises 1. Distinguish between the 'is' of one-place predication, the 'is' of identity, and

the 'is' of type identity in the following: (a) Beauty is truth. (b) The worst president in U.S. history is yet to be born.

163 (c) Arizona is a state. (d) Aristotle is a philosopher who taught Alexander. (e) 2 + 2 is 4. (f) These exercises are easy. (g) This car is the car they made back in 1930. (h) This is the dress you wore at the party. (i) The author of The Sound and the Fury is Faulkner. (j) The only worthwhile thing in life is love.

A relation may be such that whenever it holds between one individual and another in some domain it does not hold between the second and the first. For example, 'is the uncle of', 'is taller than', and '>' are likethis. A relation that has this property is said to be asymmetrical. Thus a relation R is asymmetrical in V iff VxVy(Rxy -+ ~Ryx)

2. Is identity the same as equivalence? Answers 1. (a) predication, (b) predication, (c) predication, (d) identity (e) ident"t

(f) predication, (g) can be understood to be type identity, (h) 'can be un~ y~ stood to be type identity, (i) identity, and (j) identity. er

2. No. Equivalence is a relation between statements, whereas identity is a relation between names.

7.2

Some properties of two-place predicates

There are several important properties of relations. A relation may be such that in some given domain of individuals, some V, whenever it holds between one individual x and another individual y, it also holds between y and x. For example, the relation 'next to' has this property in the domain of natural objects. If, for example, object a is next to object b, then b is next to a. A relation that has this property is said to be a symmetrical relation. In general, a relation R is symmetriwl in some V iff: VxVy(Rxy

-+

Ryx)

Other examples of symmetrical relations would be 'conjoined to' (in domain of woodworking) and 'sibling of' (in domain of historical persons). In the latter domain we may note that 'brother of' is not symmetrical. If x is the brother of y, then it does not follow that y is the brother of x-y may be a sister. However, if we selected as our domain male persons, then 'brother of' would be symmetrical. Similarly, in some people's ideal world 'loves' would be symmetrical, though as almost all discussions of symmetry point out, love, tragically enough, is not symmetrical in our world. In any domain, however, identity is symmetrical. In other words, in any universe of discourse VxVy(x

is true.

=

y-+y

=

x)

A wodd in which no one loved an individual in return would give us a domain in which 'loves' would be asymmetrical. In mathematics the relation'S;;' (being less than or equal to) is such that if x and yare distinct objects and x S;; y, then ,,-,(y S;; x). We may speak of such a relation as being. antisymmetric. R is antisymmetric is some V iff VxVy(x =F y II Rxy

-+

f".IRyx)

It is interesting to note that identity is antisymmetric since VxVy(x =F y II x = Y -+ y =F x). A relation can thus be symmetrical and antisymmetrical. All relations are either symmetrical, asymmetrical, or neither. If a relation has this last property, it is said to be nonsymmetrical. A relation R is nonsymmetrical in some V iff: 3x3y(Rxy II Ryx) II 3x3y(Rxy II ,,-,Ryx)

An example of a nonsymmetrical relation in propositional logic would be 'implies'. There are some propositions such that if x implies y then y implies x, but there are others such that x implies y but y does not imply x. 'Brother of' is also nonsymmetrical in our world, as is 'loves'. A relation may be such that whenever it holds between x and y and between y and z it holds between x and z. Such a relation is said to be transitive. That is, a relation R is transitive in some V iff: VxVyVz(Rxy II Ryz -+ Rxz)

'Implies', 'parallel to', '>', 'stronger than', and 'subset' are all transitive. Identity is transitive in any universe of discourse. On the other hand, the relation 'is the mother of' is not transitive. In fact, 'is the mother of' is such that in any domain if Mxy and Myz, then ~Mxz. If Mxy and Myz, then x is the grandmother of z. The relation 'is the mother of' is said to be intransitive. A relation R is intmnsitive in some V iff VxVyVz(Rxy II Ryz -+ ~Rxz)

165

operties of Two-Place Predicates Some P r

There may be some relations in a given V that are neither transitive nor intransitive. They are said to be non transitive. A relation R is nontran8itive in a given V iff 3x3y3z(Rxy

A

Ryz

A

Rxz)

A

3x3y3z(Rxy

A

Ryz

A

'v'xRxx Other examples of reflexive relations are' :::;;', 'is the same age as', and probably 'loves' in the domain of natural persons. A relation R is irreflexive in a domain iff 'v'x "-' Rxx

Examples of such relations are 'is the mother of', '<', and 'runs faster than'. Finally, a relation may be neither reflexive nor irreflexive. Such a relation is said to be nonreflexive. A relation R in a given domain is nonreflexive iff 3xRxx A 3x "-' Rxx Examples of nonreflexive relations would be 'loathes' and 'is the square of' (1 is the square of 1 but 2 is not the square of 2). A relation may have various combinations of these properties. For example, a symmetrical relation such as 'is equal to' may be transitive, but the symmetrical relation 'lives next door to' is intransitive in the natural world, given linear construction procedures, and the symmetrical relation 'is one foot from' is nontransitive. However, some properties entail the presence of other properties. For example, all intransitive relations are irreflexive. This can be formally demonstrated as follows. We wish to prove that if Fxy is intransitive, then it is irreflexive. In other words, we wish to prove:

j.,

'v'x'v'y'v'z(Fxy A Fyz -+ '"'-'Fxz) Faa A Faa ---+ r--JFaa Faa ---+ '"'-'Faa '"'-'Faa V '"'-'Faa '"'-'Faa 6. 'v'x r--J Fxx

1. 2. 3. 4. 5.

"-'Rxz)

'Loves' in our world can again be used to illustrate this property, along with 'two inches from' and 'defeats'. Every relation is either transitive, intransitive, or nontransitive. A relation R may be such that for every x in a domain, Rxx. Such relations are said to be reflexive. Identity is reflexive in any domain i.e., 'v'x(x = x) is true in any universe of discourse. We may thus sa; that a relation R in a given domain is reflexive iff

'v'x'v'y'v'z(Fxy A Fyz -+ r-...JFxz) f- 'v'x"-' Fxx

may do this as follows: A 1, DE (3) 2, Idem 3, Imp 4, Idem 5, DI

e relational properties of identity, the following are true To sum up th ny universe of discourse:

, In a

'v'x'v'y'v'z(x = yAy = z ---+ x = z} 'v'x'v'y(x = y ---+ y = x} 'v'x(X = x} 'v'x'v'y(x =f. y A x = Y -+ Y =f. x} Identity is transitive, symmetrical, reflexive, and, oddly enough, ant~. 1 Each of these formulas will be a theorem when PC IS symmet rIca. . . 'n the next section to include IdentIty. d d expan e 1

Exercises . d . 1 'fy the following relations according to the relational I. Supposmg a omam, c aSSI properties they do and do not have. (a) Rxy: x < y + 1 (b) Rxy: x is the mother of y (0) Rxy: x :::;; y (d) Rxy: x =f. Y (e) Rxy: x is a female br~ther of y (f) Rxy: x is good and y IS good 2. Give examples of relations whioh are, in some domain, (a) transitive and no~symm~trical (b) symmetrical and Irreflex.lve (0) transitive and nonreflexlve (d) transitive and intransitive 3. Prove the following, if true: . . . (a) Every irreflexive and transitive relatIOn IS nonsym~etrI.c. . (b) Every relation that is both .symme~rical and tranSItIve IS also refleXIve. (0) Every asymmetric relation IS reflexlv.~. and irreflexive is also asym(d) Every relation that is b oth t ranSl t Ive metrical. (e) No relation can be intransitive and reflexive. . (f) Every asymmetrical relation is irreflexive. (g) No relation can be transitive, reflexive, and asymmetrIC.

Predicate Logic with Identity

. d'IVI'd u al variables or constants is a well-formed by two In f familiarity these two definitions are then For purposes 0 , " ' bl be the same or different mdIvIdual vana es or (let t an d t 1

Answers 1. (a) transitive, asymmetrical, and irreflexive. (f) nonreflexive, symmetrical, transitive. 2. (a) x follows from y. (b) x is different from y. (c) (f) in 1.

t

3. (d) 'v'x'v'y'v'z(Fxy 1\ Fyz --+ Fxz) 1\ 'v'x,-.., Fxx I- 'v'x'v'y(Fxy --+ ,-..,Fyx) 1. 'v'x'v'y'v'z(Fxy 1\ Fyz --+ Fxz) A

" also write ---,(t = t I ) as t -=1= t I , . may t d ,'ve all valid well-formed formulas uSIllg =, It IS enable us 0 en , d"d To I l the first rule let t be any m IVI uaI a.: ' nt to add two ru es. n iCle

2. 'v'x,-.., Fxx

A

SUll

3. 4. 5. 6. 7.

I, DE (3) 2, DE

constant, then

3, 4, MT 5, DM, Imp, DN 7, UI (2)

IDENTIT

Fab 1\ Fba --+ Faa ,.....,Faa ,.....,(Fab 1\ Fba) Fab --+ ,-..,Fba 'v'x'v'y(Fxy --+ ",-,Fyx)

Predicate logic with identity: System PCI

The logician's intuition, to a large extent, seems to control whether = is classed as a logical constant along with 'v' and 3 and statement connectives. It must be so classed to obtain predicate logic with identity, and it is convenient in mathematics and the sciences and in establishing the validity of arguments to use a predicate logic with identity. With identity introduced into logical formulas, we need to add to what constitutes an interpretation of a formula. We get an interpretation for a formula in predicate logic when we follow the previously given instruction, and, in addition, we assign to = the identity predicate where a = b is true iff a is the same individual as b. A formula is valid iff it is true for all interpretations. From allowing = into formulas and from this expanded definition of an interpretation, it follows that formulas such as 'v'x(x = x) 'v'x'v'y'v'z(x

=

y 1\ Y

= z --+ x = z)

are valid formulas. The system of predicate logic, PC, may be expanded so as to include such formulas as theorems. The usual method of doing this is first to introduce Ixy into the list of primitive terms. Ixy is introduced as a predicate constant and interpreted to mean identity. The rules for a well-formed formula are then adjusted so as to allow formulas using Ixy. This adjustment is merely a matter of indicating that I immediately ",?,

= tl =df Ittl = t I ) =df r-Jlttl

,-....;(t

(d) any relation in which in the domain r-v3x3yFxy.

7.3

167

l:I.,,,H~,ate Logic with Identity: System PCI

Y INTRODUCTION (II): . f . t d ce into a proof at any time any Illstance 0 t We may III ro u

=

t.

. b d the validity of 'v'x(x = x). The second rule is This rule t~e t~:~h ~:at if a = b, then if a has property F, then b has based on F Let At be a wff containing the term t and let At l . be a wff prope:t! . h t t The rule may be stated as follows, lettmg t and e conta~lll~g !h s::e ~~ different constants and letting Atl be At with tl tl agaI~ e. e more occurrences but not necessarily all occurrences: replaCIng t m one or ' IDENTITY ELIMINATION (IE): If I- t = tl and I- At, then I- AtI · The system resulting from these ~odifications is usually called the predicate calculus with identity (abbre~Iated POI). Let us noW prove some theorems m POI. T1

'v'x'v'y(x

= Y --+ Y = x)

::~

U 2: 3. b 4. a

=

a

= b --+ b =

a

5. T1

A II 1,2, IE RCP DI (2)

In this proof we follow the familiar strat~gy ~f ass~min~ the ante~de~t within the quantified formula. IE appbes smce III thIS case t - tl IS

a

=

b, At is a T2

= a,

and Atl is b

. U

= a.

'v'x'v'y'v'z(x = Y 1\ Y = z -->- x = z) 1. a=bl\b=c A a = b 1, S 3. b = c 1, S 4. a = c 2, 3, IE 5. (1) 6. T2

-->-

(4)

RCP DI (3)

T3

· U Fa

2. 3. 4. 5.

T4 T5 T6

rreUlCate Logic with

---+

169

3x(x = a II Fx)

Fa a= a a = a II Fa 3x(x = a II Fx) T3

A II 1,2, Conj 3, EI RCP

VxFx ---+ 3x(x = a II Fx) a = b ---+ (Fa ---+ Fb) Fa II ",-,Fb ---+ "'-'(a = b) . Fa II ",-,Fb A 2. a = b A 3. ",-,Fb I, S 4. "'-'Fa 2,3, IE [ 1,4, S, Conj 6. "'-'(a = b) RAA 7. T6 RCP

~ FaA~Fa

. VxVy(x

() Fa II ,....,.,Fb (f)

T7

T9 TlO

(a = b Va = c) II Fa ---+ Fb V Fe VxVy(Fx II x = Y ---+ Fy) rI. Fa II a = b A L1: Fb I, IE 3. Fa II a = b ---+ Fb RCP 4. T8 3, DI (2) VxVy(x

=

y

---+

VxVy(~(x = y)

x

=

-+

x II Y

=

Y ~ (Fx

~ 3x3y(x

*

>

LewIs . Carro11 was Charles Dodgson. Carroll wrote Alice in Wonderland. So Dodgson wrote Alice.

A d validation is forthcoming using a simple a pp!icatiodn o~:! once :.he n a . correc , t ly sym bolized using =. Here IS the eSll e proo. argument IS Prove: c = d, Ac ~ Ad I. c=d A 2. Ac A 3. Ad 1,2, IE

x)

7. '"'-'(a = b) 8. '"'-'(b = a)

~'"'-'(a=b)

=

Y II Fy)) y ---+ Fy))

natural world Lewis Carroll Charles Dodgson x wrote Alice in Wonderland

Chicago is crowded. Any crowd~d city has no slums. So this city is not ChICago. Prove: Ga, Vx(Gx ~Sx), '"'-'Sb ~ '"'-'(b = a) I. Ga A 2. Vx(Gx~Sx) A 3. ,...,Sb A 4. Ga~Sa 2, DE 5. Sa 1,4, MP

1. Prove the unproven theorems in this section.

=

V: c: d: Ax:

. I y pI'oven identity theorems are used in the next validation 'l'wo prevIOUS of an argument.

6. Sa II '"'-'Sb

2. Prove the following theorems. (a) VxVyVz(x =1= y II Y = z ---+ x =1= z)

y)

y)

· easI'1y see n . Often the nce of PC with identity to argumen t s IS . to an argument can convemently be re1eva d .c: ula that correspon s . 101m . _ A d when = is employed III such a way, by employIllg -. n t' .;l"flV~'<"''-'~ is made POSSI'ble b Y USIll. . g PCI For example , this argumen IS trans pa rently valid:

Exercises

-"I·,

Fy))

~(y = x))

The definitions for some of the working notions used in this book may now be unconditionally stated. Formulas in logic are well-formed formulas of the predicate calculus with identity. A statement is analytic if and only if it is an instance of a theorem of PCI. Form accounts for analyticity, and a form of a statement is obtained by obtaining a formula in logic exemplified by the statement. Often analytic statements are spoken of as logically true. Finally, the premises of an argument logically imply the conclusion if and only if the corresponding conditional of the argument is analytic.

(b) Vx(Fx ---+ 3y(x (c) Vx(Fx ---+ Vy(x

f--+

PrOD f.s for arguments

One could construct a proof for T6 without RAA by using T5.

T8

=

(!) VxVy(Fx II x = Y ~ Fy II x =

its slums. This city has V: natural world a: Chicago Gx: x is crowded Sx: x is a slum b: this city

T6 6,5,3, Conj, MP 7, T10

. the III . t 1'0 d u~ t'IOn of T6. on1'f line'ustifica6. TIO, TI and substitution are used III d MP are used to obtain line 8 from hne 7. To SIm~ 1 y J . , not explicitly indIcated. tion,' an these relatively ObVIOUS moves ale

UE

Predicate Logic with Identity

!lVIl~"oU"'''''!l

In this next somewhat longer validation, another theorem is used: No one but Ivan and Joe had the plans. Someone who had the plans took the briefcase. So either Ivan or Joe took the briefcase. Prove: Vx(Px -+ x = a V x = b), 3x(Px II Bx) f- Ba V Bb 1. Vx(Px -+ x = a V x = b) A V: natural world 2. 3x(Px II Bx) A Px: x is the holder 3. Pe II Be 2, EE of the plans 4. Pe -+ e = a Ve = b 1, DE a: Ivan 5. Pe 3, S b: Joe 6. e = a V e = b 4,5, MP Bx: x is a briefcase 7. Be 3, S taker 8. Ba V Bb 6, 7, T7

Using the Identity Sign

17l

. ne person who is identical with everyone. Therefore everyone d' . (f) There IS 0 . . . everyone is not Ivme. !'t.. 'f they are not one and the same then eIther one is dIvme or (g) For any two po 1 IClhans.I . rupt or the ot er IS corrupt . Therefore either any politician, if he IS cor . corru t or Anderson is corrupt. is not Ande~s~~, IS' f he i~u"t Anderson then he is honest. Therefore for (b) any For any poll~tt~c~an 1 'f they are not one and the same, then either one is two po 1 !Clans, 1 honest. . h t For any politician, if he his nott Anderson, then he (1') Anderson IS ones. . h ones., t So happily , all politicians are ones. also IS

Answers 3x(x =f- 10 II Ex) f- 3xEx II (ElO ---+ 3x3y(x =f- y II Ex II Ey» =f- 10 II Ex) A E -.L 10 II Ea 1, 2, aT A E 3 ElO 2,3, Conj a =f- 10 II Ea II ElO 4, EI (2) 3x3y(x =f- y II Ex II Ey) RCP 6. (3) ---+ (5) 2, S E 7. a 7 EI 8 3xEx ' . 9: (8) II (6) 8, 6, Con] f 3 Vx(x - y) f- VxDx V Vx I " . ' Dx » /xVy(x -; y ---+ Ox V Oy) f- Vx(x =f- a ---+ Ox) V Oa (f) Vx(x =f- a ---+ Hx) f- VxVy(x =f- y ---+ Hx V Hy) (i) Ha, Vx(x =f- a ---+ Hx) f- VxHx (e) 1. 3x(x

We have omitted the explicit introduction of a substitution instanoe of T7, and the use of MP. Indirect proof using IE can be used to construct a validation for the argument that ends this section. The agent who found the bomb was in the hotel. Now if anyone Was in the hotel he was in the city. If anyone was in New York, then he was not in town. In fact, Bond was in New York. Therefore Bond was not the agent who found the bomb. Prove: Ha, Vx(Hx -+ Ox), Vx(Nx -+ "-'Ox), Nb f- "-'(b = a) 1. Ha A V: natural world 2. Vx(Hx -+ Ox) A Hx: x is a hotel 3. Vx(Nx -+ ,,-,Ox) A Ox: x is in the city 4. Nb A Nx: x is in New York 5. b = a A a: the agent who found the 6. Nb -+ ,,-,Ob 3, DI bomb 7. ,,-,Ob 6,4, MP b: Bond 8. ,,-,Oa 4, 5, IE 9. Ha-+Oa 2, DI 10. ,,-,Ha 9, 8, MT II. Ha II ,,-,Ha 10, 1, Conj 12. "-'(b = a) RAA

Exercises I. Establish the validity of the following arguments. (a) There is an integer equal to 5 and odd. Therefore 5 is odd. (b) 2 is even. Therefore if any integer is 2, then it is even. (c) Jones is Smith. Therefore Jones hit the car iff Smith hit the car. (d) Only Jones and Smith are unemployed. Jones and Smith are both sleeping. So everyone unemployed is sleeping. (e) There is an integer that does not equal 10 and is even. Therefore there is an integer that is even and if 10 is even then there are at least two integers that are even.

r:: Li

~

7.5 Symbolizing using the identity sign

The introduction of the sign for identity ~llow~ us. to expr~ss a ~etw array of statement forms. Specifically, the IdentIty SIgn can e use 0 express statement forms such as: (i) At least n things are F. (ii) At most, n things are F. (iii) Exactly n things are F. Let us consider these in this order. (i) At least n things are F. To say at least one thing is F, all we need to do is to use this form: 3xFx

173 Symbolizing Using the Identity Sign

Predicate Logic with Identity

but if we wish to say that at least two things are F, then (I) 3xFx 1\ 3yFy

or

d d to show that we are speaking of all numbers other than y is nee e G' To obtain our statement we merely negate (5). x when we say xy.

X

3x3y(Fx 1\ Fy)

(ii) At most, n things are F. will not do. For (I) is true if only a in V has F, and 'At least two things are F' is not true if only a has F. The inadequacy of (I) is shown by remembering that the following are valid: 3xFx 1\ 3xFx~ 3xFx 3x3y(Fx 1\ Fy)~ 3x3x(Fx 1\

Fx)~

3xFx 1\ 3xFx

. 's F is to say something that is true if no at most one tlllng 1 . . F To say thO . F Thus to say at most one t,hmg IS F 'f only one mg IS . things are or 1 t I t two things are F. In other words, at most one is to deny that a eas . . F can be expressed by thIllg IS (6) r-./3x3y(Fx 1\ Fy 1\ x =1= y)

We can adequately indicate that at least two things are F by using the identity sign in this way:

(6) is equivalent to

(2) 3x(Fx 1\ 3y(Fy 1\ x =1= y))

(7) VxVy(Fx 1\ Fy

Note that since x =1= y appears at the end, 3x must have the scope of the entire formula and 3y must have the scope of Fy 1\ x =1= y. (2) is read: "there is an x such that x has F and there is a y such that y has F and x is not y". (2) is equivalent to:

-----l-

X

=

y)

. h t mary way in which 'at most one thing is F' is paraand (7) IS t e cus o"F 11' d all y if x and yare both F, x is identical ora x a n , . ase d (7) reads: pl:trl ,; To express 'at most two things are F', we WrIte WIly.

VxVyVz(Fx 1\ Fy 1\ Fz

-+

x

= y

Vx

=

z Vz

= y)

(3) 3x3y(Fx 1\ Fy 1\ x =1= y)

(iii) Exactly n things are F. In turn, 'at least three things are F' is expressed by: To say exactly one thing is F is to say (4) 3x3y3z(Fx 1\ Fy 1\ Fz 1\ x =1= y 1\ x =1= z 1\ Y =1= z)

Note that it is not enough to have x =1= y and x =1= z, for z may be the same thing as y. If we wish to say that something is F and has a property, but that everything else that is F lacks this property, then x =1= y is indispensable. Suppose we wish to paraphrase It is false that there is a number greater than any other number.

'There is a number' would be

at least one thing is F and at most one thing is F Thus 'exactly one thing is F' can be phrased: (8) 3xFx 1\ VxVy(Fx 1\ Fy

x

=

y)

. I b t we can use a shorter formula (8) adequately expresses what we WIS 1, u which is equivalent to it, namely: 3x(Fx 1\ Vy(Fy ~ Y

3xNx

Now we wish to say that this number is greater than any other number, so if we let Gxy be the two-place predicate 'x is greater than y', we may write

-+

=

x))

. h IS . rea d : "Tllere I'S something that has F, and anything which has the property F is that thing." To paraphrase w 1nc

There are exactly two honest politicians in 'Vashington. (5) 3x(Nx 1\ Vy(Ny 1\ x =1= y "").,

-----l-

Gxy))

175 Descriptions

(c) F 0 r

with V: persons and Hx: x is an honest politician, we have: 3x3y(Hx /\ Hy /\ Vz(Hz

-+

z = y

V

z

any two numb ers x a

nd y if x is y then the successor of x is the ,

successor of Y x it is false that x is less than O. ber (d) For any numb "f 's less than or equal to 0, then x is zero. (e) For any num er x, 1 x 1

= x) /\ x =F y)

We conclude this section by giving some further examples of paraphrasing into predicate symbolism employing the identity sign.

ShoW that 3x(Fx /\ Vy(Fy

<J

.,.

-+

. ) y = x)

and

3xVy(Fy<:-'t Y = x) h . that there are at least three ways to . lent to (8) thus s owmg are eqUlva ' F' . expresS 'there is exactly one

There is one and only one building in Yuma and it is older than any building in Phoenix. 3x(Yx /\ Vy(Yy -+ x = y) /\ Vz(Pz -+ Oxz)) where V: objects; Yx: x is a building in Yuma; Px: x is a building in Phoenix; and Oxy: x is older than y.

Answers 1.

Everyone admires himself more than he admires anyone else. Everyxadmires himself and Vy if Y =F x then x admiresxmorethan y. Vx(Axx /\ Vy(y =F x -+ Mxxy»)

~;~

3x3 (Px /\ Py /\ x =F y /\ Bxy). ';'C(P; --+ 3y3z(y =F z /\ Cy /\ Cz /\ Byx /\ Bzx)).

2. (a) Vx(x (e) Vx(x

=

<

0 V 3y(x = yl»~ 0 V x = 0 -+ x - 0).

where V: persons; Axy: x admires y; and M xyz: x admires y more than z. Everyone wants to make himself happy; only somB, not everyone, want to do this for others. VxMxx, and some people want to make others happy, and not all want to do this. VxMxx /\ 3y3x(x =F y /\ Mxy) /\ ,...../v'x3y(x =F y /\ Mxy)

1.6 Definite descriptions

. f h '1'he subject expressIOn 0 eac

(1) The yak in Nebraska is fat.

The largest mammal at Berkeley plays football. . hn Kennedy was an anarchIst. The man wh 0 shot Jo The author of Waverley was Scottish.

where V: persons; Mxy: x wants to make y happy. Exercises

,.

hen called upon to paraphrase into symbols singular bJ'ect terms that are definite descriptions, we have proposItIOns WI su d d used the form Fa. For example (1) may be ren ere t

1. Symbolize the following propositions using the indicated interpretation. V: set of buildings in the U.S.; Ax: x is of architectural merit; Px: x is in Phoenix; t: Tribune Tower; Bxy: x looks better than y; Cx: x is in Chicago; x = y: x = y (a) There is only one building of architectural merit in Phoenix. (b) At least one building in Phoenix is better looking than the Tribune Tower. (c) There are at least two buildings in Phoenix. (d) There is one building in Phoenix that is better looking than any building in the U.S. (e) There are no two buildings in Phoenix such that one is better looking than the other. (f) For every building in Phoenix there are at least two buildings in Chicago that are better looking.

2. Symbolize the following. V: positive integers; x = y: x = y; x': x' is the successor of x; 0: 0; x x is less than y (a) For any x, either x is zero or x is the successor of some y. (b) For any x, x plus 1 is the successor of x.

f the following is a definite description. 0

Tn the. ~as '':th

(2) Fa

Fx: x is fat; a: the yak in Nebraska.

Let us use the upside-down iota 1XYX

1

and rea d

1

" as "the.

Yx: x is a yak in Nebraska.

. a yak in Nebraska." (2) may thus be will read: "the x such t h at x IS further paraphrased as: F·/xYx

<

y:

A puzzle or paradox suggests itself when one reflects on nega:~ng statements of the form F1xGx. At first sight it seems that the nega IOn

.L/U

177

Definite Descriptions

of, say, (I) The yak in Nebraska is fat.

is

If (4) is taken to be a proper paraphrase of (I), then our paradox is resolved. (3) becomes (5) 3x(Yx /\ 'v'y(Yy

(3) The yak in Nebraska is not fat. Now if this is so, then (I) V (3) is an instance of the tautology p V and thus (I) V (3) must be true. But there are no yaks in Neb k ,,-,p . N b k ras a not e~en m eras a zoos. This being the case, both (I) and 3 are' smce, apparently, both (I) and (3) imply that there is a yak' (N) b false If (I) d m e raska an (3) are both false, then (I) V (3) is false. This then is . puzzle: (I) V (3) seems to be both true (since it is an instance of pour and false (since there are no yaks in Nebraska). V "-'P) The c~stomary logical analyses of such statements as {l) and (3 tIns paradox, and others, by maintaining that (3) is not th~ negatIOn of (I). ~ clue to the analysis of (I) comes from notin that. any of the followmg are true. g If

resol~e

(i) There are no yaks in Nebraska. ~~~) There is more than one yak in Nebraska.

(m) The yak isn't fat. then (I) would be said to be false in a familiar sense of '.cal ' W d' 'd i. se. e may IVI e statements into those for which there is a complex I'n th . e UllIverse f d' o Iscourse corresponding to what is asserted in the st t t th £. . a emen , and ose or wInch there IS no complex-the former can be called t rue d the latter false. And it is in this sense of 'false' that we ma sa IaI~ false if any of (i) to (iii) are true. y y ( ) IS '" If (i), (ii), and (ii~) im~ly (I) is false, then the denial of (i), (ii), and (m) o.ught to appear m (I) s logical analysis to bring out the validity of such mferences. As we saw in the last section, this expression 'There is exactly one yak in Nebraska', may be symbolized: '

3x(Yx /\ 'v'y(Yy---+y

=

x))

~h~ property of being fat can be conjoined within the scope of 3x thus gIvmg us ' (4) 3x(Yx /\ 'v'y(Yy---+y = x) /\ Fx)

:vh~cl~ reads: "Some individual is a yak in Nebraska and only one mdlvldual is a yak in Nebraska and that individual is fat."

---+

Y = x) /\ r-->Fx)

Now (4) and (5) are contraries-they both cannot be true-not contradictories; thus (5) is not the negation of (4). The negation of (I) IS simply obtained as follows: (6) r-->3x(Yx /\ 'v'y(Yy

---+

y = x) /\ Fx)

'(I) or (6)' is thus an instance of P V r-->P, not '(I) V (3)'. (4) and (5) both imply 3x Y x, but (6) does not. In fact, if r-->3x Y x, then (6) can be seen to be true. In proving the validity of the arguments that follow, we will occasionally paraphrase propositions of the form F1xGx as 3x(Gx /\ 'v'y(Gy --+ x = y) /\ Fx). Some logicians have questioned such a procedure on the grounds that any statement of the form Fa is neither true nor false if there is no a. Some, for example, have argued that if one utters statements of this form and there is no a, for example, if one utters (I) and there is no yak in Nebraska, then the utterance (I) is neither true nor false. A necessary condition for the utterance being either true or false, it is argued, is that the substitution for 1XGX have a bearer. There are serious difficulties with this account, but there is no doubt that often, though not always, in the nonformal use of language, ifthe description or singular term fails to designate anything existing, the statement as a whole would not be said to be either true or false. The explanation for this is quite simple. Typically we would understand It is false that the yak in Nebraska is fat.

as implying that the yak is lean, so we would hesitate in saying this if we knew that there are no yaks in Nebraska. In other words, the negation of the form Fa is commonly understood to imply a is not F and this implies 3x r--> Fx (by EI). So if there is no a, to avoid possible misunderstanding, one does not very well want to negate Fa. However, to say Fa is neither true nor false if there is no Ct means, it seems, that neither Fa nor a is not F is true. It does not mean that neither Fa nor ""Fa is true. If this is so, Fa is neither true nor false if there is no a-Fa and a is not F are both false-and yet r-->Fct would be true. Specifically, r-->Fa is true when there are no a's. Paraphrasing definite descriptions in the manner above will enable

179 Definite Descriptions

us to establish the validity of certain arguments. 'ro illustrate: The philosopher who taught Alexander was a Greek. Aristotle was the philosopher who taught Alexander. Therefore Aristotle was a Greek. Prove: 3x(Px /\ Vy(Py -+ x = y) /\ Gx), Pa f- Ga 1. 3x(Px /\ Vy(Py -+ x = y)/\ Gx) 2. Pa 3. Pb /\ Vy(Py -+ b = y) /\ Gb 4. Vy(Py-+b = y) 5. Pa-+b = a 6. b = a 7. Gb 8. Ga

A A 1, EE

V: natural world Px: x is a philosopher who taught Alexander Gx: x is a Greek a: Aristotle

3,8 4, UE 2, 5, MP

3,8

6,7, IE

Exercises

1. Using the indicated interpretation of exercise 7.5.1, symbolize the following. (a) The one and only building in Phoenix is worse looking than a building in Chicago. (b) Only the building of architectural merit in Phoenix is better looking than the worst looking building in Chicago. (c) There are no buildings in Phoenix. (d) If there are any buildings in Phoenix, then the best looking building looks worse than the worst looking building in Chicago. 2. Validate the following arguments. (a) The philosopher who taught Alexander was a Greek. Therefore some philosopher was a Greek. (b) There is exactly one unscrupulous politician. Harry Fogg is clearly an unscrupulous politician. Therefore Fred Zilch is not an unscrupulous politician. (c) The teacher who wrote that book writes only logic texts. Therefore that book is a logic book. (d) The teacher of logic at the university is very smart. 80 it is false that some teachers of logic at the university are not smart. (e) For any x there is exactly one y such that y = x. (Prove as a theorem.) (f) The only brother of Mary is the father of Harry. Therefore Mary is Harry's aunt. (g) The highest mountain in Washington is higher than any mountain in California. Mount Rainier is in Washington and Mount Shasta is in California. Mount Rainier is the highest mountain in Washington. Therefore Mount Rainier is higher than Mount Shasta. (h) Every integer added to 0 equals itself. For any two integers x and y, x y = y -+ x. Therefore there is exactly one integer such that for any integer x, x y = x. (Remember numerals are terms.) (i) Every boy can jump higher than any boy who is smaller. Therefore if there is a tallest boy then there is the highest jumping boy.

+

-, ..

+

that we can paraphrase F1xGx as 3x(Gx /\ Vy(Gy -+ x = y) ./\ Fx) 'fhe theory 11' d . ly one among a number of such theOrIes for "" t d Russe s an IS on . ., h n1 is Ber ran. definite descriptions. The above obJectIOn IS not teo y one. paraphrasmlg ld one take all of the following to be true: e, cou f 1 Flor examp (a) F1XGx is always either true or. a se. (b) The God of Islam does not eXIst. (c) Malcolm X worshipped the God of Islam. hold to Russell's account? and yet A ShoW that G "'. , 3x(Fx /\ Vy(Fy -+ y = x) /\ Xi <J

is equivalent to . 3x(Gx /\ Vy(Fy+-t Y = x») . , F . G' . 'that there are at least two ways of expressmg the IS a or thus showmg . , 'the one and only F IS a G . . . If . I'd f rmula 3x(x = a) can be derived from It (do thIS). 5. Vx(x ~ x) IS advapl. 0 I'd then'Q is valid. But is 3x(x = a) valid? How can P implIes Q an IS va 1 , this puzzle be resolved? . . t I gic with identity applies to mathematIc reasomng, ' l! 'fo see how pre d ICa e 0 d th u. k me derivations in a theory calle group eory, let uS ma e so h The axioms (abbreviated A) of the t eoryare Al VxVyVz(x (y z) = (x y) z) A2 Vx(x 0 = x) A3 Vx(x x = 0) f 't' d . r r'etation that we will select is as follows: V: set 0 POSI Ive ~n T~le I~te p 0 d _ have their usual arithmetical interpretatIOn. nega~~e m!~~e~~antifi~:~on and identity rules are used, 0 is a term (a~ . en t) Wh n _ is used with a term the result is a term. When IS f . mstance o e . d t used between two terms the result IS treate as a erm. The formula Tl VxVyVz(x z = y z-+x = y) . . . . b he three axioms of the theory. It is called a theorem, is logICally ImplIed ylt (bb . ted Tl) We establish that Tl is logically a reVIa ' specifically, theorem implied by the axioms as follows. 1. VxVyVz(x (y z) = (x y) z) !~ 2. Vx(x + 0 = x) A3 3. Vx(x x = 0) A 4, a c= b c 5. (a c) c = (a c) c II 6. (a c) c = (b c) c 5,4, IE 7. a (c 8) = (a c) c 1, DE 8. a (c c) = (b c) c 6,7, IE 9, b (c c) = (b c) c 1, UE 10. a (c c) = b (c 8) 8, 9, IE 11 0 3, UE . c c= 10 11 IE 12. a 0= b 0 21m' 13. a 0= a 2' UE 14. b 0= b 12, 13, 14, IE 15. a = b _ RCP 16. a c = b + c -+ a - b 16 Dr (3) 17. T1 '

+ + +

+

+

+ +

+

+

+

+ +

+ + + + + + + + + + + + + + + ++ + + + +

+ + + + + +

+ +

+ + + + + +

180

Predicate Logic with Identity

It is the practice in mathematics to shorten such proofs by not expli 'tl ' d own. t h e premIses, . . Y settmg t h e aXIOms, but entering them as needed withClDE already applIed. These proofs can be shortened even further if we go from ,

M~

+ + + +

+ + + +

n. (a c) c = (b c) c directly to a (c c) = b (c c) with our justification being merely 'from n by AI'. In effect we collapse the two uses of DE and two uses of IE into the justification 'n, AI'. . We may make use of th~orem one in another proof since proven theorems logICally follow from the aXIOms. We will use the above short cuts in the next proof. T2 Vx(x 0= 0 x) 1. 0 (a a) = (0 a) a Al 2. 0 0 = (0 a) a 1, A3 3. 0 = (0 a) a 2, A2 4. a a = (0 a) a 3, A3 5. a = 0 + a 4, TI 6. a 0= 0 a 5, A2 7. T2 6, UI Derive the remaining theorems below: T3 VxVyVz(x y = x z-+y = z) T4 Vy(Vx(x y = x-+y = 0» T5 Vx(x x = x x) T6 VxVy(x y = 0 -+ y = x) T7 Vx(x = x)

+ + + + + + + + + + + + + + + + + + + + + +

7. The tree method discussed in previous exercises can be used to establish the :alidity o~ formulas. ~nd argument patterns using the identity sign. All that IS needed IS the addItIOn of two rules corresponding to II and IE. In fact no restatement of these rules is needed; they can be used with the tree method as they are. IE will continue to be used to replace equals with equals. II will be used to close paths when a formula of the form t -# t appears. Use the tree method to establish some of the argument patterns and theorems found in this chapter.

8. System PC is extended in this chapter so as to obtain the lowe?' predicate calculus with identity. As the reader might suspect, if there is a lower predicate calculus there must be a higher predicate calculus. Let us outline how lower predicate logic differs from higher order predicate logic. We have already had occasion to note that at times we wish to express statements about all properties of a certain kind or some properties. For example, the rule of IE introduced in the earlier section can be expressed as follows: For any individual x and y, if x = y, then for any property, if x has it, then y has it, and vice versa. This can be symbolized by quantifying properties as well as individual variables as follows, where the universe of discourse is everything. VxVy(x = y -+ V F(Fx+--+ Fy»

181

Definite Descriptions

Similarly, we can obtain these results, supposing the universe of discourse again to be everything. 3F3xFx: "Some property belongs to some individual." VFVxFx: "Every property belongs to every individual." 3FVxFx: "There is a property which belongs to every individual.' , 3xV F Fx: "There is an individual which has every property." V F3xFx: "Every property has at least one instance." VxVy3F Fxy: "Any two things bear some relation to each other." The defining feature of the lower predicate calculus ~s that in its well-fo:med ~ rmulas all predicate variables are free. The defimng feature of all hIgher o~der predicate formulas is that one also has bo.und predicates. . (a) Selecting the domain of discourse as everythmg, paraphrase the followmg into logical notation. (1) It is false tha~ there are t~o.things t~at .have the sar.ne properties. (2) Every irreflexIve and tranSItIve relatIOn IS asymmetrIc. (3) Honesty is a virtuous prop~rty. (4) Nothing is related to everythmg. (b) Construct an argument of some sort to show that the valid formulas below are valid and the invalid formulas are invalid. (1) VxV FFx+--+ V FVxFx (2) V F3xFx -+ 3x3F Fx (3) 3x3y3FFxy -+ 3FVx3yFxy (4) 3 FVxFx -+ Vx3F Fx (c) What differences do you find between these two formulas: V FVxVy((Fx+--+ Fy) -+ x = y) VxVy(V F (Fx+--+ Fy) -+ x = y)

Answers 2. (h) Vx(x 0 = x), VxVy(x y = y 1. Vx(x 0 = x) 2. 'Ix Vy(x y = y x) . 'Ix (x a = x) 4. a 0= a 5. 0 a= 0 6. 0 a = a 0 7. 0 = a 0 8. 0 = a 9. Vx(x a = x) -+ 0 = a 10. Vy(Vx(x y = x) -+ 0 = y)

+

~

+ + +

+ + +

+ +

+

+

+

+

+ x) f- Vy(Vx(x + y =

x)

-+

0

= y)

A A A 1, DE 3, DE 2, DE 5,6, IE 4,7, IE Rep 9, DI

5. We must remind ourselves that in giving an interpretation of a formula we give a nonempty domain and assign to each individual constant of the formula an individual in the domain. 8. (b) Only (2) and (3) are not valid.

II LOGICAL AXIOMATIC SYSTEMS

-,.

formal aXiom systems

Up to this point, we have two methods of demonstrating the validity of well-formed formulas in predicate logic, the natural deduction system pcr and the tree or tableau method (introduced in the exercises). A very different approach proceeds by assuming axioms and some rules of inference and deducing theorems-the theorems being valid predicate formulas and all predicate formulas being theorems. This is a logical axiom system for predicate logic. Such a system will be described in this part of the text. There are at least two reasons for introducing an axiom system for predicate logic at this point. First, it is useful to be acquainted with the axiomatic technique. Second, the important metalogical questions concerning consistency, soundness, and completeness can be more easily answered with respect to an axiom system than with a natural deduction system. The procedure we will follow in Part II is first to describe the notion of a formal axiom system and discuss again the properties of consistency, soundness, and completeness. This will be followed by an axiom system for sentential logic and one for predicate logic. For each system the questions concerning consistency, soundness, and completeness will be raised. We will stop short of a proof for the completeness of predicate logic.

185

187

186 8. 1

The development of geometry

The development of the axiomatic method has made a profound impact on human thought. To many it seems the ideal way to organize some field of knowledge. The axiomatic method was first applied to geometr ~nd the hist~ry of. the development of geometry provides many insighr~ mto the aXIOmatIC method. For our purposes, the most interesting feature is how the development of geometry made people conscious of the conditions for a formal axiom system. Let us very briefly review some aspects of this development. Most of us will recall that geometry is taught as a deductive discipline. It is not presented as, say, a collection of contingent statements that are adopted because they agree with observations. Rather a few statements, axioms, are given without proof (e.g., 'a straight line can be drawn from any point to any other point', and 'things that are equal to the same things are also equal to one another') and from these, other statements, theorems, are deduced by using some logical rules. This procedure, begun by Euclid around 300 B.C. in The Elements, is called the axiomatic method. The ancient geometers saw two advantages in organizing geometry in an axiomatic way. First, all the geometry truths that had been collected for hundreds of years by the Egyptians and others, could be systematically organized. Second, if the axioms could be so selected that they were necessarily true, then, if the deductions were valid, the truth of the theorems would be guaranteed, no matter how complex they might be and no matter whether they were self-evident. The Greeks regarded the axioms in Euclid's system as self-evident, with the exception of the fifth postulate. The fifth postulate found in Euclid's system is equivalent to the assumption that through a point outside a given line, only one parallel to the line can be drawn. It has become known as the parallel postulate. The ancient geometers made attempts to derive the parallel postulate from the other axioms. All of these attempts failed, since, as it was later proven, the parallel postulate cannot be derived from the other axioms. The effort to derive the postulate continued through the middle ages, and in the eighteenth century an Italian mathematician, Gerilama Sacchieri, assumed the negation of the parallel postulate and tried to deduce a contradiction. If he had succeeded in this reductio ad absurdum strategy, then he could have concluded that the parallel postulate is deducible from the other axioms. Though Sacchieri succeeded in deriving many counterintuitive and odd-sounding theorems, he was unable to '">1·,

derive a contradiction. Sacchieri had, in fact, developed a nonEuclidean geometry. Non-Euclidean geometries were intentionally developed in the nineteenth century. In hyperbolic geometry, for example, through a point not on a given line, always more than one line can be drawn parallel to the given line; and the sum of the angles of a triangle is always less than two right angles. Out of the development of these geometries emerged the pattern of how an axiom system should be constructed and developed. Since familiar spatial diagrams and intuition could not be used to clarify the axioms of the systems, and neither could they be used as an assistance to the derivation of theorems, a way had to be devised of developing the systems independent of intuition. A proof of a theorem must be a matter of what logically follows from what. We must pay nO attention to the meaning of the primitive terms of the system, but think only about the logical form of the statement. But for this to be possible, geometry had to be constructed as a formal axiom system. We turn next to the notion of a formal axiom system, along with a more precise account of the notion of theory, axiom, and theorem. Exercises 1. What would be the consequence if each axiom of Euclidean geometry could be derived from any other axiom? 2. At what points did intuition break down in the development of geometry? 3. Pascal is said to have desired a system for superhuman minds in which all terms would be defined and all statements demonstrated. Is such a system possible? 4. Must an axiom in a system have the feature of being intellectually selfevident? Answers 1. Only one axiom would be needed. 3. No. Thus in a system some statements-axioms-must be supposed and some terms-primitive terms-undefined. 4. The logical status of an axiom is that of being an assumption from which some set of statements are logically derived. Logically speaking, they need not even be true. 8.2

Formal axiomatic theories

A theory can be identified with two sets of statements. First, there is the set of statements concerning the subject matter of the theory. In

188

189

geometry, all those statements about lines, points, and so forth, would make up this first set. In nuclear physics, all those statements about subatomic particles would ma~e up this first set. In sentential logic, aU well-formed formulas would make up this first set. The second set is the subset of the first set that is regarded as acceptable. All the state_ ments regarded as true in geometry or nuclear physics fall into this second set. All valid well-formed formulas of sentential logic fall into this second set. A theory is an axiomatic theot·y when some acceptable statements are taken as axioms, as sta~ements acceptable without proof, and the remaining acceptable statements are theorems, or are the hoped for theorems. A theorem is a statement deducible from the axioms. A theorem T is deducible from a non-logical axiom set S if and only if S ~ T is analytic, that is, if S ~ T .is an instance of a valid logical formula. A theory is ajormal system when it has the following features: 1. The notion of a statement of the system is effective, i.e., there is a

mechanical procedure by which in a finite number of steps one can tell whether some marks are or are not statements or formulas of the system. If the notion of a statement of the system is effective, then a digital computer can be programmed to check whether a string of marks of the system is or is not well-formed. 2. The notion of.a.proof is effective, i.e., there is an effective procedure for deCIding whether a sequence of statements is or is not a proof. The last step in a proof is a theorem; thus a system is a formal system only if checking a proof of a theorem can be reduced to a mechanical procedure. If a system is formal, if one can check mechanically whether a string of marks is well-formed or not, and if one can check mechanically whether a sequence of lines is a proof of a theorem, then theorems can be proven in a system independent of what meaning one gives to the marks of the system. If a computer can check whether a sequence is a proof, then a proof does not depend on the meaning of the marks but relies only on the form and the rules for the manipulation of forms. We can usefully distinguish the syntax from the semantics of a theory. With logical formal theories the rules for a wff and proof are parts of the syntax of the theory. Syntactical questions in logi~ are questions independent of any interpretation given to the symbols of the system. Semantics in logic consists in giving interpretations to the symbols, e.g. basic truth-tables for statement connectives, and in specifying when formulas are true in a given interpretation. Formulas

are true in every admissible interpretation are, of course, the valid formulas. In this book our primary interest is in formal theories for sentential . nd predicate logic with identity. In Part I the formal _theory. for . . IogiC a these logic systems is a natural deductIOn system. In thIS part we WIll be interested in formal axiom systems for logical sys~ems. . The difference between the natural deductIOn system for lOgIC presented in Part I and those presented in this part is not in terms of being formal. All are formal ~heories. Ra~her the difference re~ts in the fact that the natural deductIOn systems III Part I have no aXIOms but only rules. In an axiom system one starts with a, set of well-formed formulas that are accepted without proof. Using rules, one then derives In a natural deduction system for fr om these axioms the theorems. , logic there are no axioms, but there are more rules. For a logical theory, as the rules increase the axioms decrease (and vice versa) and when one has the maximum number of rules (no axioms), then one has a natural deduction system. In a natural deduction system for logic, we should note, there must be some rules that allow us to begin proofs by intro,ducing assumptions, rules like Rep and RAA. In axiom systems we can start proofs with axioms. To illustrate some of the remarks in this section, let us set down as simple a formal axiom system as wl} can imagine. l The primitive symbols of the system are these five: 1, 2, 3, =, and~. A well-formed string of these symbols may be defined as follows: 1. preceded by 1, 2, or 3 and followed by 1, 2, or 3 is a wff. 2. If A and Bare wffs then (A ~ B) is a wff.

Our only rule of inference is MP and the theory has exactly two axioms Al A2

1= 2

~

(1 = 2 ~ 2 = 3)

1= 2

A proof in the theory is a finite sequence of formulas such that each formula is either an axiom or is inferred from previous lines by the rule of inference. A theorem is the last line in such a sequence. This is a formal axiom system. First the notion of a wff is effective. For example, Al and A2 are wffs, but

= 1, 1 2 = 10, 2 = = 1, 2 = 2 = 3, 4 = 2 + 2 1 The idea for this and for many of the definitions in this section is taken from Angelo Margaris, First 01'der Mathematical Logic (Waltham, Mass.: Blaisdell Publishing Co., 1967).

191

VB

190

SYMBOLS:

are not wffs. The definition for a wff settles in all cases whether a string of symbols is ~ wff or not. The notion of an axiom is effective since .A is an axiom iff it is either Al or A2. Finally, the notion of proof is effective. The system also has exactly four theorems, AI, A2, 1 = 2 ~ 2 = 3 and 2 = 3. And this can be determined even though no meaning has been specified for 1, 2, 3, =, and ~. Formal proofs for two of the four theorems would be Proof 1: 1 = 2 ~ (1 = 2 ~ 2 = 3) Proof 2: 1 = 2 ~ (1 = 2 --* 2 = 3) 1=2 I=2~2=3

Exercises 1. Why is a proof with an infinite number of steps not permitted in a formal axiomatic system? 2. In the natural sciences, what controls whether a well-formed formula is accepta ble?

3. Why is a formal system not affected when you drain its terms of meaning, that is, obviate any interpretations for the primitive terms?

4. Give the proofs for the remaining theorems of the formal system set down at the end of this section. 5. Many of Euclid's proofs relied on features of diagrams. Was Euclid's system a formal system?

6. Discuss how one would go about making the tree method into a formal system. 7. Why are questions of soundness and completeness questions relating syntax and semantics? Answers 1. It would not be possible to know when one had a theorem.

3. A machine can be programmed to register 'yes' if a sequence is a proof and 'no' if it is not a proof.

1. a, b, and m 2. A and I FORMATION:

1 A or I preceded by a, b, or m and followed by a, b, or m is a wff. 2: If A and Bare wffs, then ",-,A, (A A B), and (A --* B) are wffs. AXIOMS:

AI: A2: A3: A4:

aAa ala mAa A bAm --* bAa mAa A mlb --* bla

RULES OF INFERENCE:

.

Rule 1: If A is a well-formed formula, then we may umformly substitute a, b, or m for a, b, or m. Rule 2: Modus Ponens. If I- A --* B and I- A, then I- B.

This axiom system will make use of sentential logic by ~sing.tautol. . the derivations and by using statement connectIves m conogles m ' . I structing wffs as indicated in th~ ~or~ation :ules. How the tauto~oglCa formulas may enter into proofs IS mdlCated m the last rule of CS. Rule 3: Tautology introduction rule. Any tautological wff may be entered in a proof. DEFI,NITIONS:

1. bEa 2. bOa

=

=

df

.......,bla

df ",-,bAa

A proof in CS is a sequence of one or more wffs, each of w~ic~ either is one of the axioms, an instance of a tautology, a theorem, or IS mferred from preceding wffs in the sequence by modus ponens or Rule l. theorem is the last line of a derivation. In writing out a proof ;ve ~Ill take the liberty of numbering each line and indicating to the rIght ItS

!"-

justification. 8.3

System CS

It will prove useful to present a second, somewhat more complicated

TI bEa --* ",-,bla l. ",-,bla --* ",-,bla 2. bEa --* ",-,bla

Identity, ",-,bla/P, R3 df

formal axiom system. In order to make absolutely clear that this is a formal system we will first lay down the primitive basis of the system, which will be called CS, and derive a few theorems without considering what interpretation is intended for the primitive symbols.

T2 ",-,bla l. ",-,bla 2. T2

Identity, ",-,bla/ P df

--* --*

bEa ",-,bla

'Identity' is the name for the tautology P -+ P If a tautol . £ . . ogy IS needed or whICh we do not have a familiar name" we will list it Qt th b . . . e eglnnln of the proof, as we do in the next proof. g T3

bla -+ ,,-,bEa -+ "-'Q) -+ (Q -+ ",-,P) (bEa -+ "-'bla) -+ (bla -+ ,,-,bEa) bEa -+ ,,-,bla T3 ,,-,bEa -+ bla bOa -+ ,,-,bAa ",-,bAa -+ bOa bAa -+ ",-,bOa ,,-,bOa -+ bAa bla -+ alb (bAb A bla -+ alb) -+ (bAb -+ (bla -+ alb))

(i) (P

1. 2. 3. T4 T5 T6 T7 T8 T9 1.

2. rnAa A rnlb -+ bla 3. bAb A bla -+ alb 4. bAb -+ (bla -+ alb) 5. aAa 6. bAb 7. T9 TlO

Tll

Exportation, bAb/P, bla/Q, alb/R A4 b/rn, bfa, alb (RI) 3,1, MP Al b/a 6,4, MP

bAa -+ bla mEa A bIrn -+ bOa

(i) (P A Q -+ R)

-+

2. (1) -+ (3)

3. ,,-,bla A rnlb -+ ,,-,rnAa 4. bEa A rnlb -+ rnOa 5. Tll

TI2 rnAa A blrn -+ bla TI3 rnAa A bArn -+ bla 1. rnAa A bArn -+ bAa 2. bAa-+ bla 3. TI3

symbols. . .. . . OS is an axwmatlzatwn of SyllOgIStICS. The system presented here, with certain modifications, is taken from the Polish logician Jan ..:t:;ukasiewicz' work, Aristotle's Syllogistic. A and I are syllogistic quantifiers, and a, b, and m are general name variables. A general name is an expression that determines a class. Examples of general names would be: man, five-cent hamburgers, unicorns, and class. Al reads "All a is a." Instances of Al would be: 'All men are men', 'all roaring mice are roaring mice', and 'all positive integers are positive integers'. A2 tells us that there is something which is a-a is not an empty term. It reads: "Some a is a." A3 is the valid formula corresponding to the valid syllogism called Barbara and A4 is the valid formula corresponding to the valid syllogism called Datasi. The system is strong enough to derive the conditionals corresponding to the 24 argument sequences thought to be valid in syllogistics and, in addition, all the laws of immediate inference (conditionals with simple components as antecedents and consequences), omitting inferences involving negative terms. The formation rules can be seen to be designed to give us categorical propositions (see Section 5.2) and categorical propositions linked by statement connectives.

Exercises

("-'R A Q -+ ,,-,P)

1. rnAa A rnlb -+ bla

TI4

(i) bEa/P, bla/Q TI 1,2, MP

interpretation of OS, i.e., the intended interpretation for the primitive

1. Construct proofs for the unproven theorems. A4 (i) rnAa/P, rnlb/Q,

bla/R 1,2, MP df rn/b, b/rn

A3 TlO

1, 2, Trans, substituting, andMP

mEa A bArn -+ bEa

. This exercise should have convinced the reader that one can have an aXIOm system without an interpretation for the primitive symbols of the sy~tem: In OS, theorems are derived without supplying any interpretatIOns for a, b, m, A, and 1. Let us now supply the principal

2. Give some instances of the axioms and theorems.

3. Provide a second interpretation for CS. 4. What is the logical status of a definition in a formal system, for example, the definitions in CS? 5. State some theorems of SC other than those given above. 6. Distinguish between giving an interpretation for a system and giving an interpretation for a formula.

Answers 2. A3: If all babies are illogical and all philosophers are babies, then all philosophers are illogical. 3. Let a, b, and rn stand for classes, and let xAy be x c y (reads: "class x is included in class y"), and let xly be x n y i= A (reads: "the intersection of x and y is not an empty set"). 4. Logically speaking, a definition in a formal system introduces no new content; it is a decision or a convention useful for purposes of abbreviation and familiarity.

.

.

-""orma! Axiom Syste

6, To gIve an mterpretation for a s stem IS . . of the system; to give an iute y t t. to mterpret the primitive symbols rpre a IOn of a formula is to interpret the variables of the formula.

8.4

Meta/ogica/ properties of CS

There are many reasons for develo in . respect to syllogistics one such p ? a formal aXIOm system. With , reason IS system t' II the logical truths in syllogistics from a fe ,..a lCa y ~o deduce all of rules. Another reason is to sh ' w pnnCIples (axIOms) and a few , . ow preCIsely the relat' b gIStlCS and sentential logic A thO d . IOn etween syIIoh . 11' reason IS to . th wether the system has certain m t I . I ~aIse e question of are: consistency soundness co e at oglCa propertIes. These properties .' ,mp l e eness, and independ W now;o a dIscussion of these properties and as. ence. e turn system, whether an axiom system 0 is negation consistent when t h ' l' a natural deduction system e aXIOms and/or l' I d ' wff and its negation to be deduced as. . u es 0 not allow both a cannot both be derived "T . IS c~nslStent, then, if A and ",A . 0 say a system IS d . theorems are acceptable. To say r I sou~ IS to say that all its . 't' a ~orma system IS compl t . th pnmi Ive sense of 'complete" t e e m e most formed formulas of the th IS °h say that all the acceptable well'd eory are t eorems of the s t A . ys em. system is sal to be independent when n 0 aXIOm or rule can be d . d f th o er axioms or rules of th t enve rom the . d e sys em. as can easily b m ependent and consistent Th I e proven to be problematic matter. . e comp eteness and soundness of as is a How does one prove that any axiom s . . Obviously one cannot prove that an a . ystem IS ~egatIO~ consistent? fact that the theorems alread d d dXIOm system IS conSIStent by the e it is possible that undeducelth uce do no~ contradict each other, for A . eorems may mtroduce t d' n mconsistent system cannot be i . . a con ra lCtion. the axioms and theorem . t g ven an mterpretatIOn such that all s are rue (a model) Th' . prove the consistency of as T thO . IS provIdes one way to . . 0 IS end let us I th mterpretation for the prim't' b ' supp y e following lIve sym ols of as. Let xAy be x can be put in t xly be x is included in ya one- o-one correspondence with y a be the set of positive integers m be positive integers divisible by 2 b be the set of even positive integers

On this interpretation each of the fou . statement. To prove SO consist ..1' aXIOms of as becomes a true ent, It IS now sufficient to show that if

Metalogical Properties of CS

195

rules are applied to true lines, the result must be true lines. For if the rules are valid, then if the axioms are true, all the theorems must be true. In other words, given this interpretation for as, which results in each axiom being a true mathematical statement, as is consistent if the rules are valid. It is easy to prove that the rules of as are valid. A second way to prove consistency of the axioms of a system is to interpret the system in terms of another system. That is, if one can show that the axioms of system one can be interpreted so as to be theorems of system two, that system two has the same inference rules as system one, then system one must be consistent if system two is consistent. This second method was needed to establish the consistency of nonEuclidean geometry since the axioms were initially regarded as plainly false of any familiar domain of things. Interestingly, each axiom of elliptical geometry can be interpreted so as to become a theorem of Euclidean geometry ('plane' is made to signify the surface of a Euclidean sphere, 'point' a point on this surface, 'straight line' an arc of a great circle on this surface, and so on). On this interpretation, the replacement for the parallel postulate reads: "Through a point on the surface of a sphere, no arc of a great circle can be drawn parallel to a given arc of a great circle." A numerical interpretation of both Euclidean and nonEuclidean geometry can also be provided. Thus if number theory is consistent, so are these geometries. A third way to prove consistency of an axiom system is to find a property such that (i) the axioms have the property, (ii) the rules pass on the property, and (iii) two formulas of the form A and ,....,.,A cannot both have the property. If each axiom has the property and the rules pass it on, then all the theses of the system must have the property. If A and ,....,.,A cannot both have the property, then the system must be simply consistent. The property in question in such a proof may be an interpreted or uninterpreted property. We should note that a variation of this method was used to prove SO and PO consistent. 'True' is of course such a property; thus to use a model to prove consistency is an example of the third way in which the property is an interpreted property. Let us use this method again to sketch a second proof of the negation consistency of as. The formulas for categorical propositions in the axioms of as can each be rewritten by following this procedure: (i) Replace the small letters with sentential variables. (ii) Drop the A and put in a ~. (iii) Drop the I and put in a ~.

196

197

Metalogical Properties of US

Following this procedure, Al to A4 become P+--+ P P+--+ P (Q+--+ P) II (R+--+ Q) (Q+--+ P) II (Q+--+ R)

--+ --+

(R+--+ P) (R+--+ P)

As we can see ~y following this procedure, each axiom becomes a tauto.logy. We wIl.l call formulas obtained by following this procedure as~oClated sententIal formulas (asf's). It can thus be said that all the aXIOms possess the property of having valid asf's. It can next be seen that the rules pass on this property. If A has a valid asf and one of ru~es is applied, .then the formula that results has a valid asf. Now if ~~: aXIOms have thIS feature and the rules pass it on, all the theses of OS must have t~is feature. Finally, it cannot be the case that both A and ,......,A have thIs feature, so OS must be consistent. To prove an axiom system sound-to prove that all theorems are acceptable-it is sufficient to prove that each axiom is acceptable und the pri~cipal interpretation of the system, and that the rules of inferen:: are valId. The proof that the rules of OS are valid presents no real problem. However whether each of the axioms is acceptable is problematic. Under the principal interpretation of OS the axioms are: Al A2 A3 A4

All a is a. Some a is a. All m is a II all b is m --+ all b is a. All m is a II some m is b --+ some b is a.

where a and b are general name variables. A well-formed formula in OS is acceptable or valid if on every interpretation for its variables it is a true statement. Let us SUppose such an interpretation occurs for a formula in OS when (i) a non empty domain of discourse is indicated and (ii) . general names in the domain are assigned to the general ~ame varIables. If we SUppose a formula in OS is valid if true under all such interpretations, then AI, A3, and A4 are all valid. However, A2 presents us with problems, for we can give an interpretation for A2 that res.ults in a false statement. For example, let V: natural world, a: Ulllcorns. If, however, we modify the definition for an interpretation of a OS formula by adding that the domain must be such that there is at least one thing that has a, b, and m (remember rule one allows us to substitute small letters), then A2 is valid. But now 'valid' has a different sense in OS than it does in predicate logic. Another powerful motive in formalizing a theory is to answer questions concerning the completeness of the theory. An axiom system """/-,

is compIete , in what is sometimes called the relative sense of completeh h en all the acceptable formulas of the theory are theorems of t e nesS, .c 1·Ize d , . mW s stem. Supposing the field of knowle d ge h as b een .lorma axlOmay Yay we s a system is complete if it is sufficiently powerful to generate all its acceptable well-formed formulas. The completeness of ~e~ t 0 f . . . problems beyond the scope of thIS book. OS may OS raIses . .be SaId . to . be in another sense in that all the truths of syllOgIStlCS, Ignormg compIet e . . . · terms , are theses of the system. When EuclId aXIOmatIzed nega t lve he tried to select his axioms so that all the truths of geometry t geome ry, d· db could be derived from his axioms-that is, not only those Iscovere y the Egyptians and Greeks, but all possible truths of geometry. . The axiom systems and natural deduction systems for sententIal logic and predicate logic are complete in the way Eu.clid hoped his geometry was. Not all formal systems, howeve~, are relatIvely c~mplete. For example, a complete, consistent set of aXIOms cannot be gIven for number theory (deductive system for the acceptable stateme~ts found in the arithmetic of positive integers). No matter what aXIOms are an se1ec t ed , Kurt Godel in 1931 showed how one could construct . acceptable sentence of number theory that would be true Iff ~ot rovable in that system. Godel's theorem showed that mathematIcal iruth cannot be identified with derivability from any particular set of axioms. There is a second sense of completeness, called absolute completenesS. A system is absolutely complete if and only if a nonderivable wff produces an inconsistency if added to the axioms of the system. Few systems are absolutely complete. In logic, only the s~ntenti~l calcul~s (and fragments of it) is absolutely complete. There ~s no dIfficulty III showing that OS is not absolutely complete. OS IS not absolute~y complete since there are underivable formulas that do not result III contradictions if added to the axioms, e.g., (I) alb --+ (r-.>aAb (2) bla --+ bAa

--+

bAa)

The asf's of (I) and (2) are tautologies, so if (I) and (2) were added to the axioms of OS, what would result would not be an inconsistent system. But how does one show that (I) and (2) are not derivable in OS? Again, mere failure to find a proof is not enough to demonstrate that (I) and (2) are not derivable. This brings us to the question of the independence of a wff of a system from other wffs of the system, and thus to the question of the independence of an axiom system. In using the axiomatic method, the aim is often not only to establish a system in which all the basic features are made explicit, but to establish

199 J.l10

a simple or nonredundant system. An unnecessary axiom or rule would violate this latter demand. When all the axioms are necessary, that is when no axiom can be derived from the other axioms, the axioms ar~ said to be independent. Since rules can be derived from the axioms and rules of a system, we can also speak of the rules being independent. Independence is not an essential property of an axiom system; in fact in many cases a number of dependent axioms make proofs of theorem~ easier. Many system builders, nevertheless, value elegance and conceptual simplicity and thus independence is sought after. ConSistency is essential; soundness is needed; completeness is most desirable, though not always obtainable; independence is obtainable, aesthetically pleasing, and can reduce one's labors in metalogic. The question of independence has played an important role in the history of thought. As mentioned earlier, the parallel postulate was regarded by the Greeks as not being as self-evident an axiom as: The whole is greater than its parts. Attempts were made to prove the postulate dependent. All of these attempts failed since the parallel postulate is, as has been proved, independent. The efforts to prove its supposed dependence led, however, to the development of non-Euclidean geometry. Since such systems have been made use of in the theory of relativity, and since these theories demanded clarity concerning the requirements for a purely formal system, this question of independence of an axiom has had a significant role in the history ofthought. It should be noted that if the parallel postulate had been removed from the Euclidean axioms, as many wanted, some of the theorems of Euclidean geometry that are regarded as expressing true geometric principles would be underivable. This follows from the fact that the parallel postUlate is independent in Euclid's system. An axiom can be proven independent of other axioms if all the axioms can be given an interpretation that results in the latter axioms' having a property that the first lacks, a property that must be passed on by the inference rules. The usual procedure to show that an axiom is independent in this way is to provide an interpretation that will make it false and the rest of the axioms true. The proof that Al is independent may proceed along these lines. Let A be 1\ I be--+ a be F (false) Al becomes on this interpretation: AI:

F

1\

F

The other three axioms become: A2: A3: A4:

F --+ F (m 1\ F) 1\ (b 1\ m)--+b 1\ F (m 1\ F) 1\ (m --+ b) --+ (b --+ F)

. kl be confirmed that under this interpretation Al has the It can qlUC! 1 ' s the other three axioms have the truth-value T. th value 1./ , W lereac . t,I'U d . d dent Obviously another or different Illter1 's thus prove III epen . A I.. ded to prove the independence of A2, A3, and A4. pretatlOn IS nee Exercises Prove the independence of A2, A3, and A4. 1. h t Al plus the other three axioms is a consistent set. What does 2. Show tar-.; this prove? . .. Ab cannot be derived from the axioms and that Its addItIOn 3 Showthat aIb --+ a .' . to tIe I axIO . ms would not result in an IllconsIstent system. u ose a system is consistent if and only if there is at least one well-forme~ 4. ~or~ula of the theory that is not a theorem. Show that the system below IS inconsistent: . (1) If P and Q are wffs, then P --+ Q IS a theorem. (2) Rule of inference: MP. 5. Raise and try to answer the three metalogical questions with respect to the system at the end of Section 8.2. 6. How do you prove that a rule of a system is not independent?

Answers 4. P --+ Q is a theorem. (P --+ Q) --+ P is a theorem. Thus. by MP, P is a theorem, and if P is a theorem, every wff of the system IS a theorem, and hence the system is inconsistent. 6. One way: show that what can be done with the rule can be done without the rule. (Recall exercise 4.7.3)

8.5 Axiom systems and logic

The advantages of a formally developed axiom system ~or any theory are many indeed. These would be some of the most l~portant advantages: (1) There are no hidden assumptions that may gIVe .us trouble since everything is explicitly set out. (2) The basis from whICh .all the theorems follow is perspicuous and thus the system can be ratlOn~lly evaluated. (3) Deductions are made rigorously, relying only on logICal

201

2UU

form, and no appeal is made to intuition. Intuition in this context would be the process of making inferences in an informal m tt . hi' a er WIt out re ymg on formal proofs. But intuition is a relative thin ' . .... g, and I ymg on mtUItIOn IS not always a safe procedure for avoiding er re . 1'01', and m ~ formally developed axiom system, it need not be relied on. (4) An u~mterpreted system opens the possibility of providing an interpretation dIfferent from the one the system builder intended. (5) Finall tl . t t . f . y, 1e lmpor an questIOns 0 consIStency, soundness, and completeness can be asked and answered. Most sciences have axiom systems as their ideal though ' . . ' axlOmatlzat~on has been ~chieved in a relatively few scientific fields such as mechamcs, mathematIcs, set theory, and logic. In these fields, the initial development was, of course, nonaxiomatic and intuitional . A n axlO' matization of a field, normally, can only be attempted when the fundament~l no~ions, pro~erties,. and principles are believed known. The first step IS to Judge whICh notIOns are basic, which are to be the primitive terms, and which acceptable statements or formulas are basic th ' ese provl'd'mg th'" e lmtlaI set of axioms. axiomatizations, . . In most ' .one usually presupposes certain theo rIes, a~IOmatlzed theories previously constructed. Thus the empirical SCIences presuppose mathematics. Customarily, in the axiomatization of an empirical science, mathematical terms are not counted as primitives or defined terms, nor are mathematical truths axioms or theorems of the system. Nevertheless these terms are found in the theses of the system and mathematical principles are assumptions in derivations. Formal mathematical theories are constructed by making additions to the predicate calculus with identity. Only in axiomatized logic is no system presupposed. In a science such as physics, some laws are deduced from other laws for example, Kepler's laws are derivable from Newton's laws. Also som~ terms are defined in terms of others. Obviously these deductions and definitions must stop somewhere. All statements cannot be derived in a system, nor can all terms be defined. This is one motive for arranging the true statements in physics and in an axiom system. Another motive is providing a few basic well-corroborated truths or principles that will be axioms and from which other statements in the field can be deduced. The degree of confirmation obtained by the axioms will thus be passed on, to some degree, to the theorems. Since an infinite number of theorems can be derived, new laws can be derived and experiments performed to corroborate them. Furthermore, since physical sciences develop partly in terms of what nature, so to speak, tells the scientist, these new experiments will reveal new aspects of nature that may prompt new -'/

..

ents For example the failure to verify a consequence from a deveIomp ' , . . entails the necessity either to modIfy the aXIOms (or any other theory . I nonmathematical principles use d'm t h e d enva . t'IOns ) or t 0 non1ogwa, . late new physical entities and forces to explam the apparent postLt'u t between theory and phenomena-th' b e h aVIOr . b' . ell' emg mconulC . 'ated into the theory through new aXIOms. corpol . ' In passing, it is worth noting that many pllliosophe~s of SCIence argue · the most advanced scientific theories three mam features can be t m tha . I D" th . t' . (bs lllg uI'shed'. (1) the formal axiomatIC calcu us. rammg e meanmg . of the nonlogical, nonmathematical t~rms in the axioms of ~ phYSICal system by replacing them with arbItrary marks would dIsplay the abstract calculus of an advanced scientific t~eory. H~re we cou~d spe.ak tIl e implicit meaning of such marks, meamng by thIS the relatIOnshIps of . d to them in the axioms Terms such as ,mo IecuI' e an d 'k'me t'IC aSSlg ne " ' . . ., molecules' in physics can be saId to be ImplICItly defined energy of . by the . s And to a large extent their meaning is exhausted m these aXIOm. implicit definitions. (2) A set of rules assigning empirica.l contexts to ~he abstract axiomatic calculus. These rules or laws permIt the deductIO~ from the calculus of empirical laws, laws that can be tested by expenment. Since the nonlogical, nonmathematical rules are premises in deriving testable statements, the disconfirmation of such derived statements does not imply that the axioms must necessarily be altered or added to. (3) An interpretation of the axioms that provides a model for the abstract axiomatic calculus. These models usually serve to aid in the understanding of the calculus. Since the primitive terms in an advanced theory in physics are not given explicit definitions, models are needed heuristically to picture the theory and thus aid in the development of the theory. It is a mistake, of course, to think one must have a model, and it is a mistake to assume that models necessarily have counterparts in nature. Axiomatization in mathematics brings to mathematics all of the desirable features mentioned earlier. A further feature, and a timely one, is the application of axiom systems to computers. Computers have a fixed set of directives programmed into them. These directives correspond to the explicit rules and axioms of a formalized system. Godel's proof, by the way, is thought by most to have as one of its consequences the impossibility of constructing a supercomputer capable of solving every problem in mathematics. All deductions in science and mathematics, whether formal (theorems derived in an axiom system) or informal, are moves in which some statement logically follows from other statements. A conclusion or step in a proof logically follows from premises or previous lines only if the

202

Formal Axiom ~_y~",,",,;_

conditional corresponding to the inference is an instance of a formula in predicate logic with identity. For mathematics to be fully axiomatized, it must begin by buildin on a formal system for predicate logic with identity, just as a natul'a~ science to be fully axiomatized must build on a formal axiom system for mathematics, including set theory. This being the case, there needs to be some systematic way to generate all valid formulas in predicate logic. In a word, there needs to be a formal system for predicate log'IC. The natural deduction system POI given in Part I is a formal system for predicate logic. Our interest in Part II is first to give a formal axiom system for predicate logic and, second, to raise metalogical questions.

Exercises 1. Wh~, if a theo~y is presented as a formal system, must the presuppositions

reqUlred for thIS purpose be understood in their full significance and under. stood as true? 2. Why are the terms 'primitive' or 'undefinable' and 'axiom' or 'undemon_ strable' to be understood only in a relative sense? 3. We could define a system or calculus in terms of theorems or in terms of a particular organization. How are 'sentential calculus' and 'predicate calculus' defined in this book? 4. Can disagreement arise over this question: Is P a theorem of formal, nonlogical theory T?

Answers 2. A term is undefinable and a statement undemonstrable only within a system constructed in a particular way.

aXioms for the sentential calculus

Any system that is consistent and that has all tautologies and only tautologies as theorems-as 'tautology' has been defined in this book-may be called a classical sentential calculus. (See exercises 9.8.7 for a nonclassical sentential calculus.) The natural deduction system SO found in Chapter 4 is one such sentential calculus. At this time we wish to study an axiomatic system for the classical sentential calculus. To do this in a formal way we need to list primitive symbols, give rules for a wellformed formula, and give an effective criterion for what is to count as a proof. In addition, we need to indicate some tautologies that will be axioms of the system and some valid rules of inference that will allow us to make derivations of theorems. The axioms selected will be the well-known set found in Russell and Whitehead's Principia Mathematica. Later in this chapter the system will be proven consistent, sound, and complete. We will refer to the system as PM, after Principia M athematica. There are, it should be noted, other possible sets of axioms which will allow a consistent, sound, and complete axiom system for sentential logic (see Exercises 9.8.4). The axioms for PM will be finite in number; there are, in fact, only five axioms in PM. In Chapter 10 an axiom system that corresponds to PM and that has an infinite 203

204

Axioms for the Sentential

>rellimirmry Discussion of PM

number of axioms, will be How this is possible will be clear at the beginning of Chapter I 9.1

Preliminary discussion of PM

L~ttle preliminary. discu~sion of. PM is needed. A well-formed formula WIll be defined . as It was III SC WIth one change. The logical conne ct'rves ~ and V wIll be taken as primitive, undefined connectives and th others will be introduced by definition. For example --+ will be intro~ duced as follows:

This definition allows us to write the first lines of each pair below in favor of the second. ~(P

/\ Q)

~~(P

--+

R

/\ Q) V R

~,--.,PVQ

~P--+Q

However, this definition will not allow us to do the following transformations: ~(P

/\ Q) --+ R (P /\ Q) V R

PVQ ~P--+Q

since the same shape is not preserved. /\ and <-+ will be :introduced by the following definitions: A /\ B =df ~(~A V ~B) A<-+ B =df (A --+ B) /\ (B --+ A)

There are five axioms in PM. In Russell and Whitehead's system one of the axioms, namely, P V (Q V R) --+ Q V (P V R)

can be derived later from the other four axioms of PM. Thus this axiom is not an independent axiom. However, we will include this dependent axiom in the axiom set for PM. The axioms will be expressed using -+ for purposes of familiarity. Beside the axioms, there will be two rules, modus ponens and a rule of substitution. The rule of substitution will allow us to uniformly substitute any well-formed sentential formula for sentential variables. For example, we can substitute ~P/P and "'QIQ f·,

205

the above formula, obtaining

A proof in PM is a finite sequence of wffs such that each of the wfIs the sequence either is an axiom or is inferred from a preceding formula formulas in the sequence by means of one of the rules of inference. We regard a definitional transformation of A as just another way of .<~'t"""l" A. A theorem is the last line in a proof. We will continue to follow our conventions for eliminating paren-

The turnstile, 1-, is again used to indicate that a formula is a thesis (theorem or axiom) of the system when we have I- A.

1. Why is every axiom of PM a theorem? ~.

Are the notions of well-formed formula, axiom, and proof in PM effective? Can a theorem appear twice in a proof sequence?

4. Are Russell and Whitehead to be censured for having an axiom that can be derived from their other axioms? Can a definitional transformation be regarded as an inference on a par with substitution or MP? 6. Which of the following are not correct definitional transformations? (a) f"o/P --+ Q PVQ (b) "'(f"o/P --+ Q) f"o/(,"" r-JP V Q) (c) ",(P /\ Q) f"o/f"o/(~P V ,.....,Q) (d) P V '""Q ",(,""P /\ Q) (e) ",(P<-+ Q) ",(P -+ Q) /\ ,.....,(Q -+ P) (f) ",p -+ ~P P V ,""P (g) f"o/(P V '"" ",-,Q) f"o/P /\ '""Q (h) "-' ,,-,(~P V "-' ,,-,(P /\ ,.....,Q)

~Q)

1. Every axiom is a theorem because the sequence consisting of a single stepthe axiom itself-is a proof, and the last formula in any proof is a theorem.

206

4. If ease of proof construction is valued above other considerations, they should be applauded. 5. Yes, but it will make the discussion to follow slightly more complicated d nothing much would be gained by it. ' an 6. (a), (d), (e), (f), and (g).

9.2

2. No. It doesn't appear in the list of symbols, primitive or defined. 3. Yes, but earlier we agreed not to repeat each time we had formation rules that parentheses are terms of the theory.

9.3

System PM

PRIMITIVE SYMBOLS:

l. Sentential variables: P, Q, R, ...

2. Statement connectives: """,

V

FORMATION RULES:

207

Development of PM

Axioms for the Sentential Calculus

Development of PM

\Ve now proceed to prove some theorems of PM. The reader is urged, as usual, to work carefully through the proofs. It is normally only in this way that one understands the system and acquires the ability {;o construct proofs. The first theorem we will prove is:

l. A variable is a wif. 2. If A is a wif, then """A is a wif. 3. If A and Bare ,,,ffs, then (A V B) is a wif.

Here is the formal proof of T 1 :

AXIOMS:

l.PVP-+P 2. P-+Q V P 3. P V Q -+Q V P 4. (P -+ Q) -+ (R V P -+ R V Q) 5. P V (Q V R) -+ Q V (P V R)

P-+QVP P-+r-...>Q V P P-+(Q-+P)

RULES OF INFERENCE:

l. Rule of detachment (MP): If I- A and I- A -+ B, then I- B. 2. Rule of substitution: Let M be a single sentential variable and N be any wif and let AM be a formula in which M occurs in A. Let AN be the result of replacing every occurrence of M in AM with N. Then if I- AM then I- AN. DEFINITIONS:

l. A -+ B =df """A V B 2. A A B =df """("",,A V """B) 3. A+--+ B =df (A -+ B) A (B

-+

A)

No justification for the steps is given since the definition of a proof in PM makes no provision for such an analysis. Each step can be effectively checked to see if it is an axiom or is inferred from earlier lines by modus ponens or by substitution. Checking the above sequence, we confirm that the first line is an axiom, Axiom 2. The second line comes from the first by using the rule of substitution and making this substitution: "'-JQ/Q. The last line comes from the second by the definition for -+. Since this discussion is intended for human beings rather than computers, a justification will be supplied, and it will take the form with which, by this time, the reader is familiar. The proof and analysis of the proof of T 1 will be

Exercises l. Set down system PM using as the only logical constant the Sheffer stroke

(see Section 3.6).

l. P-+QvP 2. P-+ """Q V P

A2 From 1 by thc rule of inference 2, making this substitution:

3. P

From 2 by df for-+

2. Is I- a symbol of PM? 3. Are ( and) symbols of PM? -"'/,

"'-'Q/Q -+

(Q

->-

P)

209

n~"Alor)ment of PM

T2 1. 2. 3.

T7

(P -+ Q) -+ ((R -+ P) -+ (R -+ Q» (P -+ Q) -+ (R v P -+ R v Q) (P -+ Q) -+ (,......,R V P -+,.....,R V Q) T2

A4 ,-..,RjR df

The justification for T2 has been shortened. Since ,-..,R/ R on line 2 indicates a substitution, we did not indicate Rule 2. Since 2 comes from the preceding line, we did not indicate 1. Only when we use a rule, such as MP, that involves more than one line need we indicate lines in the justification. We omitted the name of the definition on line 3, and to reduce writing we indicated the theorem by number on line 3 in accord_ ance with the practice we have been following. T3

(P

-+

Q)

-+

((Q

-+

R)

-+

(P

-+

R»

(Transitivity of 1mplication)

T4 P -+ P (Identity) 1. PvP-+P 2. P-+Q V P 3. P-+ P V P 4. (P -+ Q) -+ ((R 5. (1)

-+

((3)

-+

-+

P)

-+

Al A2 (R

-+

PjQ T2 P V PjP, PjQ, PjR 1,3,5, MP (2)

Q»

(6»

6. T4

r"-'

1. P

,.....,P -+ P

-+ ,......,

T6

----P

2. 3. 4. 5. 6. 7. 8. 9.

,.....,P -+ "" r-->,....,P ",-,PjP (P -+ Q) ->- (R v P -+ R v Q) A4 (2) ->- (5) ,....,PjP, ,...., "" ,-..,PjQ, PjR P V ........,P -+ P V,.....,,-.., ,.....,p 2, 4, MP Pv.......,P T5 P V ,-..."·....-',....,,P 5,6, MP P V Q -+ Q V P A3 P V "'-' r--..I "",p -+........, "'-' ,......,p V P "" "'" ""PjQ 10. ""-' r-...J """p V P 7, 9, MP

11. T7 T8

df

(P ->- Q) -+ (.......,Q -+ ,.....,P) 1. (Q -+ ,......, ,....,Q) -+ ((P -+ Q) -+ (P -+,....." ""Q» 2. Q ->-........,----Q 3. (P -+ Q) -+ (P -+ "" ,-..,Q) 4. "",-,p V ",-,,......,Q-+........,,.....,Q V """p 5. (P -+........, ,-..,Q) -+ (........,Q -+ ,-..,P) 6. (3)

-+

((5)

-+

(7»

7, T8

T2, QjP, ,-.., ""QjQ, PjR T6, QjP 1,2, MP A3, ""PjP, "" ,.....,QjQ

df T3, P-+QjP, P-+r-..o,-...., QjQ,,.....,Q -+ ~PjR 3 t 5, 6, MP (2)

The last proof is shortened by entering needed axioms and theorems with the desired substitutions.

Line 4 is not an axiom nor does it come from the previous lines in the sequence by a rule or definition. T2 at the justification place on line 4 means: "At this point insert the proof for T2." Thus we do not have a formal proof but a proof outline. To lessen our proof burdens we will frequently resort to such outlines. (We are in effect operating with the derived rule theorem introduction of se.) Using line numbers on line 5 saves time. It also helps to make clear how T2, substitution, and MP can do the work of a rule corresponding to Trans, namely: If f- A ->- B and f- a -+ A, then f- 0-+ B. T5

T6

P v,-.."p 1.P-+P 2. ,-..,Pv P 3. P V Q -+ Q V P 4 . .......,p V P ->- P v""P 5. T5 P-+r"-''''-'P 1. P v........,P 2. "",P v........,,.....,P 3. T6

T4 df A3 ""PjP, PjQ

2,4, MP

T5 ,-..,PjP df

T9 C. .·. -'P -+ """Q) -+ (P -+ Q) TI0 P V (Q V R) ->- (P V Q) V R 1. QvR-+RvQ 2. (Q V R -+ R V Q) -+ 3, 4, 5. 6. 7. 8. 9.

(P V (Q V R) -+ P V (R V Q» P V (Q V R) -+ P V (R V Q) P V (R V Q) -+ R V (P V Q) (3) -+ ((4) -+ (6» P V (Q V R) -+ R V (P V Q) R V (P V Q) -+ (P V Q) V R (6) -+ ((7) ----;. (9» TlO

A3 A4 1,2, MP A5 T3 3, 4, 5, MP (2) A3 T3 6,7,8, MP (2)

To shorten the above proof we have omitted indicating the substitutions made for the axioms and T3. (P V Q) V R -+ P V (Q V R) P -+ (Q -+ P A Q) (Conjunction) 1. (""P V ........,Q) V r_;(r-"P V r-...;Q) T5

TIl

T12

2. (1)

-+

(3)

3 . .......,p V (,-...,Q V ,-.....,(o--'p V,-.....,Q»

4. T12

TIl

1, 2, MP df (3)

211

Axioms for the Sentential

We may derive biconditional theorems using T12 and ha proven A ->- Band B ->- A. This procedure is illustrated in the proof· of the next theorem.

to shoW is that the rule is dispensable. This is shown as follows: For DR1

LA T13 1. 2. 3. 4.

P+--+,....., I"-'p

P

->- ,....",

,-,..,P

r-..J,,-,p-+ P (1) -+ ((2) ->- (4)) (1) A (2) 5. Tl3

T6 T7 T12

2. B 3. P

-+

4. A

-+

(Q -+ P A Q) (B -+ A A B)

5. A A B

Given as an assumption in DRl Given as an
1,2,4, lVIP (2)

1, 2, 3, MP (2)

df

'j'

t tl 's shows is that whenever we move from A, B to A

A

B, by

\Vw 1 t11 DR.1 we can convert tIns . ' . move lllto a proo f 0 fA A Bb Y USlllg 11ppeasu~stitution, and modus ponens twice. It will prove helpful to

Exercises

delllonstrate a second derived rule. 1. Prove the unproven theorems. It may be found convenient to prove some

relatively simpler theorems prior to proving one of the listed theorems. reader is encouraged to do this.

DR2

If f- A -+ B - ((Q -+ B) -+ (P -+ B)) T3 2: A -+ B -+ ((B -+ C) -+ (A -+ C)) AlP, BIQ, CIB 3 A-+B A (B -+ C) -+ (A ->- C) 1,2 MP 5. B-+C A

4:

9.4

Derived rules

6. A

The proof of theorems is made considerably easier by using derived rules (DRs). A derived rule is merely a substitute for what can be carried out by using the axioms and primitive rules. The use of derived ~'ules thus does not increase the class of provable formulas. They in fact produce more proof outlines. The justification for derived rules is similar to the justification for definitions. A definition is a convenient way to write in shorter form a longer expression. Similarly, a derived rule allows us to do in fewer steps exactly what we could do using the primitive rules and axioms of the system. This means that whenever a derived rule is used in a proof, we could give the unabridged proof if we wished. The proof of a derived rule is thus, in effect, to set down all the machinery needed to make unabridged proofs. If one wished to dispense with any use of a derived rule in a particular proof and write it out in full, all that would be needed is the machinery employed in the proof of the rule. 'DRn' in a proof is to be read: "Followtheproof ofDRn to convert this lineA into a proof of A." Let us take as our first derived rule Conjunction: DR1

If f- A and f- B, then f- A A B.

(Conjunction)

To demonstrate a derived rule, or to show a rule is a derived rule, all we

->-

C

4, 5, lYIP

rf3 is derived from the primitive basis of PM, so the .proof here is .a p~oof tl t what is done with DR2 can be carried out by USlllg T3, substItutIOn, 1a mocl1lS ponens twice . The next l)roof makes use of this derived rule. nnc . 1 T14

P A Q ->- P

1. ,.....,P -+ ,.....,Q V ,....,P 2. '"'-'Q V ",-,P -+ ,....,P V ,.....,Q 3. ,...."P -+,.....,p V ,.....,Q 4. (3)

-+

(5)

5. ,.....,(,.....,P V ,.....,Q) ->- o--',......,p 6. ,-..,; I"-'P ->- P 7. 'P V I"-'Q) -,.. P S. T14

r_+__

TI5 TI6 TI7 TIS TI9 T20 T2I

A2 A3

1,2, DR2 TS 3,4, MP T7 5,6, DR2

df

P A Q-+Q (P -+ (Q -+ B)) -+ (Q -+ (P ->- B)) PvQ+--+,.....,P-+Q (P+--+ Q) -+ (r-..P<.-> r-..Q) (P -+ Q) -+ (P vB ->- Q V B) (P -+ (Q -+ B)) -+ (P /\ Q ->- B) (P -,.. Q) A (P ->- R) -+ (P --+ Q /\ R)

In some of these proofs we find, for example, the need of commuting disjuncts. For example, the proof of T19 would be quite simple if we

213

212

°

one may conclude that all the positive integers plus have the erty F in the first case and that all the positive integers have F in

could go from A4 (P

---+

Q)

---+

(R V P

---+

R V Q)

(P

---+

Q)

---+

(P V R

---+

Q V R)

to

Even if we obtained the derived rule: If I- A V B, then I- B V A, this would not allow the switching of Rand P and Rand Q in A4. It would allow us to go from, say, (P II Q) V R

to R V (P II Q)

but it would not allow us to alter the inner structure of a wff. A rule that would allow us to alter the inner structure of a wff by replacing B with when B +-+ 0, would be useful in lightening our burden. We need a replacement rule for PM. One more additional theorem is needed to prove this desired derived rule, namely

°

T22

(P+-+ Q) II (R+-+ S) ---+ (P V R+-+

Q V S)

To prove the needed derived rule we must first prove what is called the equivalence theorem. To state the theorem let M and N be formulas. Let AM be a formula in which M occurs. Let N be the result of replacing one or more occurrences of M in AM by occurrences of N , then

prop the second case. .., Proofs using mathematIcal mductlOns often can be used when the tement to be proven involves the natural numbers. Theorems in logic sa t can involve numbers in a variety of ways. For example, a theorem may be about 1, 2, 3, ... , n variables in a formula or it may be about n number of logical connectives. Or a theorem may be about n number of mptions or about n number of steps in a proof. Frequently, aSS U ma,thematical induction may be used to establish such statements. 1'he proof of the equivalence theorem is by induction on the number n of statement connectives in AM. We suppose that only the primitive connectives are employed. Basis: If n = 0, then AM is M. Then AN is N, and (M +-+ N) ---+ (AM +-+ AN) by T4. Induction step: Assume the theorem holds for every formula with n or fewer connectives and consider AM with n + 1 connectives, where n is greater than 0. The + 1 connective must be either ""-' or v. Thus the two cases below exhaust the possibilities for AM. CASE 1: AN is ,......,BM. By our assumption (the theorem holds for every formula with n or fewer connectives): OvI +-+ N)

---+

(BM +-+ BN)

where BN is the result of replacing the occurrences of }JII in B1V] by occurrences of N. Then

THE EQUIVALENCE THEOREM:

(JY1 +-+ N)

---+

by TIS

(AM +-+ AN)

We will make use of (strong) mathematical induction in the proof of this theorem and some later theorems. Mathematical induction is often used to prove theorems in number theory. For example, if one shows that Basis Induction step

(i) 0 has property F, and (ii) for any positive integer n if every integer equal to or less than n has F, then n + 1 has F,

or if one shows Basis Induction step

(i) 1 has the property F, and (ii) for any positive integer n, if every integer equal to or less than n has F, then n + 1 has F,

CASE 2: AM is BM V OM. Let BN and ON be defined as in case 1. By our assumption:

(JY1 +-+ N) (JY1 +-+ N)

---+ ---+

(BM +-+ BN) (CM +-+ CN)

'['hen (M +-+ N) ---+ (BM V OM +-+ BN V ON) by T21 and T22 }1'rom the equivalence theorem we may prove the desired replacement theorem. Let M, N, AM, and AN be as in the equivalence theorem. 'l'HE REPLACEMENT THEOREM

(R):

If I- 111 +-+ Nand f- AM, then I- AN

214

Axioms for the Sentential Calculus

Proof: 1. JvI +-t N 2. 3. 4. 5.

AM (111 +-t N) --+ (AM +-t AN) AM+-t AN AN

A A Equivalence theorem 1,3, MP 2, 4, df, T14, MP

Exercises 1. Without using replacement, prove the unproven theorems through T22. If

necessary, you may prove other theorems in order to prove the numbered theorems. Prove some of the 55 theorems found in Chapter 4 that have not been proven so far, using replacement if appropriate. However, do not Use a biconditional theorem with replacement unless it has been proven. 2. Select some of the proofs' outlines using derived rules, and construct a

formal proof not using any derived rules. 3. A derived rule corresponding to RCP can be proven for PM. It is usually called the deduction theorem and may be stated as folluws: If /'-" P f- Q, then /'-, f- P --+ Q

This reads: "if there is a deduction of Q from /'-, (a set of formulas which may be empty) and P, then there is a deduction of P --+ Q from /'-,." A ded'llction here is a finite sequence of formulas such that for each formula at least one of the following holds: (1) It is an axiom or a substitutional instance of an axiom. (2) It is an assumption. (3) It is inferred from the preceding formulas by MP. Use the deduction theorem in constructing a proof for a few of the 55 theorems found in Chapter 4.

Consistency and Soundness of PM

215

vVo have already had occasion to review why consistency in the first, the negation sense, is an essential property for a sentential calculus (Section 4.8). If PM is negation inconsistent, then any wff is a theorem. Consistency in the other two senses also prevents a system from being useless. In any formalization of a field only some of the possible wffs-those regarded as acceptable-are desired theorems. If a system were absolutely inconsistent, then every wff would be a theorem and thus the system would be trivialized. And if a system were inconsistent in Post's sense, then given the rule of substitution, it would also be the case that every wff would be a theorem. There are good reasons why consistency in the second sense is called absolute consistency. First, a formal theory might not have a negation sign as one of its terms. If so, it could not be negation inconsistent and thus it would be vacuously negation consistent. In turn, a formal theory with no sentential variable as a term would be vacuously consistent in Post's sense. However, a formal theory cannot be vacuously absolutely consistent. Second, a system that is vacuously consistent in the negation and Post's sense could be a totally useless system if it were absolutely inconsistent, since a formal theory with every well-formed formula a theorem is useless. However, if a system is absolutely consistent, it is saved from such a fate. Third, if a formal theory is absolutely consistent, then it must be negation consistent, if P /\ ,,-,P --+ Q is a theorem, and consistent in Post's sense. This can be proven by proving: (1) If a system is negation inconsistent, then it is not absolutely

consistent. (2) If a system has a theorem that consists of a sentential variable then it is not absolutely consistent. 9.5

Consistency and soundness of PM

There are three senses in which an aXIOm system can be said to bo consisten t: (1) Consistency with respect to negation, or simple consistency: A

system is consistent in this sense if no well-formed formula and its negation are both theorems of the system. (2) Absolute consistency: A system is absolutely consistent if not every wff of the system is a theorem. (3) Post's sense of consistency:2 A system is consistent in this sense if there is no theorem in the system consisting of a sentential variable alone. 2

The American logician E. L. Post introduced this sense in 1921. -,.

If a system is negation inconsistent, then any wff is a theorem if p /\ ,-..,p --+ Q is a theorem. Thus (1) is true. If P is a theorem, then any wff is a theorem. Then (2) is true. We shall now proceed to prove three consistency theorems. We wish to prove that PM is negation consistent, absolutely consistent, and consistent in the sense of Post. To prove PM negation consistent, we first will prove that every theorem of PM is a tautology. For if every theorem is tautological, then A /\ """A cannot both be theorems. As previously indicated, a system is sound when all its theorems are acceptable. Since tautologies are the acceptable formulas of PM, to prove that every theorem of PM is a tautology is to prove the soundness of PM. To show that every theorem in PM is tautologous, we interpret P, Q, R, ... as sentential variables having only two values true or false,

216

Axioms for the Sentential Calculus

and we interpret '"'-' and V as statement connectives defined by the standard truth-tables. On this interpretation, every axiom of PM is a tautology and the rules of inference applied to tautologies give a tautology.

217

Independence of the Axioms of PM

Theorem 1 establishes the soundness of PM. Thus: Theorem 2: PM is negation consistent.

Lemma 13: Every axiom of PM is a tautology. Theorem 3: PM is absolutely consistent. Proof: Standard truth-tables show that each axiom takes the value T for every assignment of truth-values to its components. Lemma 2: M Od11S ponens applied to two tautologies gives a tautology.

Proof: Select any axiom, say AI. ,.....,AI is not a tautology by Lemma 1. Thus ,......,Al is not a theorem of PM by Theorem 1.

Proof: Suppose A -+ B and A take the value T for every assignment to the components A and B. Then B cannot have the value F since if A-+B

TT occurs, B must take the value T. Lemma 3: The rule of substitution applied to a tautology gives a tautology. These lemmas establish: Theorem 1: Every theorem of PM is a tautology. Proof: If every axiom is a tautology and the rules applied to tautologies can only give tautologies, then every theorem is a tautology since a theorem is the last line of a sequence of which each line either is an axiom or results from applying a rule. Using mathematical induction we can precisely set out this argument: Proof: Let Av ... , An be a proof. We prove by induction on n that every step in a proof is a tautology. Basis: If n = 1, then Al is an axiom. By Lemma 1 every axiom is a tautology. Induction step: Suppose that for every proof with n steps, every step is a tautology and consider a proof At> . . . ,An' A)/+l with n + 1 steps. By the induction hypothesis At> . .. ,A)/ are all tautologies. An+! is an axiom or is inferred by modus ponens or substitution. If A)/+l is an axiom, then An+l is a tautology by Lemma 1. If An+l is inferred from Ai and Aj by modus ponens, then An+! is a tautology by Lemma 2, and if A"+l is inferred from A.I by substitution'then A 1 is a n+ tautology by Lemma 3. 3 A lemma is a preliminary or auxiliary theorem demonstrated or accepted for use in_-l·,a demonstration of a main theorem.

Theorem 4: PM is consistent in the sense of Post. Proof: Let A be any wff consisting of a single sentential variable. Then A is not a tautology, and by Theorem 1 is not a theorem of PM.

Exercises 1. Show that if P-+Q

were added to PM, the resulting system, PM', would be inconsistent in all three senses of inconsistency. 2. Show that a sentential calculus is negation consistent iff it is absolutely consistent. 3. In what sense is a system without

,-...J

consistent?

4. Show that if a wff A were added to the axioms of PM and the resulting set alone were inconsistent, then ""A would be a theorem of PM.

9.6

Independence of the axioms of PM

An axiom A of a theory T is independent iff A is not provable in T from the other axioms of T. A dependent axiom does no harm; it is merely not needed. Earlier it was pointed out that A5 of PM is not an independent axiom. A5 can thus be proven as a theorem using the other four axioms. To prove the independence of the first four axioms of PM, we cannot rely on our failure to derive each axiom from the remaining ones, for such failure may be due to lack of ingenuity or luck. We must use another method. Up to this time, to give an interpretation for a formula in the sentential language has meant to assign truth-values to its components. vVe can also speak of giving an interpretation for the primitive symbols

219

Independence of the Axioms of PM

218

Given the above interpretation, the value of the axioms can now be determined in a way analogous to determining the truth-value of a formula on the standard truth-tables. First we employ the definitions rewrite the axioms in terms of "'-' and V as follows:

of PM. In fact, the following is the pTincipal interpretation oj P)J!I: (1) We understand P, Q, R, ... as having the values true or false. (2) We give a standard truth-table interpretation for the two primitive constants "-' and V. :Under this. interpretation. e~~h axiom is tautologous. However, other mterpretatlOns of the . prImItIVe symbols are possible . Now sa Y tllat . under such an . mterpretatlOn we find that axioms A2 ' A3 , n a d A4 possess a Cel'tam feature. And let us also suppose that the primitive rules pass on this feature. This being the case, all the theorems derived from A2, A3, and A4 must have this feature. But suppose Al lacks the feature. This would mean that Al is not derivable from A2, A3, and A4 in this system and thus is an independent axiom in the system . Tl·. - lIS then, is the method which will be used to prove the independence of th~ first four axioms of PM. W·e cannot, of course, use the principal interpretation of PM. We need a different interpretation. Such an interpretation will look strange, but it can be as strange as we like. The independence of Al from the other three will be proved by means of an arithmetical interpretation. The variables will take the values 0 1 and 2, and the signs "-' and V are interpreted by these tables: " P

,...,.,p

P

Q

PvQ

0

I

0

I

0 2

I

0 0 0

0 0 0 0

2

2 0 I

2 0 1

2

I I I

I

2 0 2 0

2 2 2

-ciT 2

0 2

V

0

I

2

0

0 0 0

0 I

0 2 0

1

2

2

These tables provide an interpretation for ,-...., and V in the same manner as the standard truth-tables do this for these connectives.

",-,(P V P) V P <"""--'P V (Q V P) ..-.-;(P V Q) V (Q V P) r---'("'-'P V Q) V (~(R V P) V (R V Q))

We next determine the value of Al as follows: "'-'

I

0 1

(P

V

P)

V

P

0 1 2

0 1 0

0

0 0 2

0

I

2

I

2

Under the main connective of Al we find 0, 0 and 2. A2's value IS determined by this table: P

V

Q

V

P)

I

0

0 2

1

0 0 0 0 0 0 0 0 0

0 0 0

0 0 0 0

0 1 2 0 1 2 0

I

2 0

0

I

2 I

2 0

0

I 2

2

which can be more conveniently written as:

I

Al A2 A3 A4

I I I

2 2

2

I

2 0 2 0

I 2

As we can see, under the main connective we have all O's. A2's value is thus O. The reader, as an exercise, can confirm that A3 and A4 are also O-formulas. This property of having a O-value is passed on by the two rules just as having a T-value is passed on. Thus all the formulas derived fi'om A2, A3, and A4 have the same value. Consequently, Al is independent of the other three axioms. In the exercises immediately to follow, values and matrices to define ~ and V are to be devised that prove the independence of A2, A3, and A4, and thus complete the proof -of the theorem below. Theorem 5: The first four axioms of PM are independent.

220

Axioms for the Sentential

Completeness of PM

Exercises

every formula in 6 is a theorem-where 6 is the class of acceptable Let us continue to call this completeness relative completeness. Since, under the principal interpretations, we wish all tautological formulas to be theses, PM is complete in the relative sense if and only if:

1. Prove the independence of the remaining axioms.

2. Show that the following is not a thesis of PM: (P -+ Q) A Q -+ P 3. Dependent axioms do no harm' they are merely superfluous S . orne t'Imes .. ' h owe vel', a deCISIOn whether an axiom is independent can have' , . Important . consequences. How does the brief account of the history of geometr" Chapter 8 illustrate this point? y gIven In

4. Why ca~not the principal interpretation be used to prove the independen of an aXIOm of PM? ce 5. Derive A5 of PM from the other four axioms . Use no theorems 0 l' d enved . ruI es t h at h ave been proven by using A5.

6. Try to provide an interpretation for the primitive symbols of PM that results in the axioms being (a) true statements and (b) names.

Answers 1. We can demonstrate the independence of A2 from Al and A3 and A4 by . the following interpretation: usmg

v

o

1

2

3

o

o

o o

o

3

3

3 3

o

3

o

1 2 3

2 1

1 2

o

3

o

o o

o

6. (~l Let P, Q, R, ... denote 1, 2, 3, ... respectively, V be vA. A

9.7

221

3 3 3

+, and

r-...J

A be

Completeness of PM

In most general terms an axiomatic system is complete if it is sufficiently powerful to make possible the derivation of all the acceptable well-formed formulas. The point of PM is to provide a system in which all tautologies are theorems. The point of having PM is to have a consistent system in which we set down a few valid formulas and a few rules, and from which all valid formulas can be generated. Let us now consider the question of the completeness of PM. There are several definitions for completeness. To sayan axiom system is complete, as just indicated, can be taken to mean that, given the principal interpretation, the system is sufficiently powerful to generate all its acceptable wffs. A theory is complete relative to D. if and only if -,.

Theorem 6: All tautologous wffs of PM are theorems of PM.

If a system is relatively complete, then the system is sufficient to generate the set of all its acceptable wffs. This suggests that we might distinguish a further sense of complete and say that a system is complete iff an inconsistency will result if it is made more powerful. In other words, if any underivable wff is added to the axioms, an inconsistent system will result. This is called absolute completeness. Absolute completeness is not a relative notion since a system is consistent or inconsistent independent of an interpretation. PM is absolutely complete if and only if an inconsistency arises when there is added to the axioms a formula not provable. Since PM can be inconsistent with respect to negation, absolutely or in the sense of Post, there are three corresponding senses of absolute completeness for PM. There are several ways to construct a proof of relative completeness. One way rests on our ability to construct truth-tables for well-formed formulas in PM; another rests on our ability to transform any wellformed formula in PM into a certain kind of form. We will use this second method. To prove PM complete in the second way we must first show that every well-formed formula can be transformed into conjunctive normal form (CNF). Every wff of PM can be put into CNF by using certain combinations of the following: PM definitions, rule of replacement, double negation law, distribution laws, commutation laws, and association laws. All of this is available in PM. We are able to establish the lemma below: Lemma 1: If A is a wff in PM, then there is a wff in PM, A I , in CNF such that r A +-+ A'. However, we will ignore the proof of this lemma. A second lemma is needed, namely: Lemma 2: Every tautologous wff of PM in CNF is a theorem of PM. This is the_fundamental lemma to proving the relative completeness of PM.

223

i
222 All tautologous ONF formulas will contain in every conjunct mutually contradictory disjuncts. If a formula in ONF is tau each conjunct must be valid, and this can be possible only if in conjunct we have a variable appearing as one disjunct along with negation as another disjunct. If we can show that PM allows the derivation of ccny t formula in ONF, Lemma 2 will be established. The following LVLLeUbl(Y indicate that any formula in ONF that is a tautology is provable in

(1) We can derive P V ,......,P, and by substitution we can replace by any wff. (2) By using A2, P -+ Q V P, or the DR: If I- A, then I- A V we can add any formula to P V,......,P. (3) By repeated use of (1) and (2) and conjunction, we can any tautologous ONF. We then, if necessary, can Use O·HOIII''''c>., ment, association, and commutation to insure a perfect match, Let us consider an example to illustrate these steps: Derive: (,......,P V P) /\ ((R V Q) V,......,Q) 1. P V,......,p theorem 2. Q V "-'Q V ,......,Q) V R 4. (P V ,......,P) /\ ((Q V ,......,Q) V R) 5. (,......,p V P) /\ ((R V Q) V,......,Q)

3. (Q

Q/P

addition 1, 3, Oonj Comm, Assoc, replacement

Theorem 6 follows Lemmas 1 and 2. If for any wff I- A +--> A '(Lemma and if all tautologous A' are derivable (Lemma 2), then all wffs are derivable in PM, for I- A follows from I- A +--> A' and I- A'. Let us now see that PM is complete in the absolute sense, that inconsistency arises when there is added to PM a nonderivable formula. Since we have distinguished three senses of consistency, what we now to show is: Theorem 7: PM is absolutely complete with respect to negation consistency, absolute consistency, and in Post's of consistency. Proof: Let A be any formula that is not provable in PM. Let A' be ONF of A. Since A' is not provable, it is not a tautologous formula, in proving metatheorem 6 we have proven that all tautologous formulas of PM are theorems of PM. Since A' is not tautologous, it

a conjunction B containing no mutually contradictory comLet us suppose B is P V r--'Q V R V r--..JS

we substitute r"-'P for all the negated variables and a P for the nonvariables, then we obtain:

.,,,,llUAc'U

P V r--' ,.....,P V P V ,......, r--'P

DN and Al we obtain from this the wff P. Notice that if B were a contained a sentential variable both negated and unnegated, we could not make this kincl of substitution. It is now a simple matter to show that PM + A is inconsistent in all senses. It is negation inconsistent since for P we can substitute /I r-'P. It is absolutely inconsistent since by substitution for P every would be a theorem. And it is inconsistent in Post's sense since P a theorem. "Ve may note that one can also show that PM minus Axiom 1 minus anyone of the other independent axioms) is both relatively absolutely incomplete. We have already established that Axiom 1 an independent axiom. Thus Al cannot be derived from PM minus AI. Al is a tautology, it follows that PM - Al is not relatively complete. Since Al cannot be derived from PM - AI, its addition as a would not produce a contradiction in any of the three senses. the addition of Al would not result in an inconsistency, PM - Al not absolutely complete.

Show that P-+(Q-+p/\Q)

is a thesis of PM by; (a) transforming it into CNF, (b) proving each conjunct and then using Conj to get the CNF as a theorem, (0) reversing the transformation of CNF to obtain it. Show that PM plus (P-+Q)-+ (Q-+P)

is inconsistent

9Y using CNF and substitution.

If A is added to PM and PM

+ A is inconsistent, is ,......,A a theorem of PM?

An absolutely complete system has been described as a balloon filled to capacity that will explode if any attempt is made to enlarge it. Why is the metaphor apt?

224

Axioms for the Sentential Calculus

5. Prove that PM is absolutely complete if there is no wff of PM that is inde_ pendent of the axioms of PM. 6. Oan a theory be vacuously relatively

complete~

Answers 3. Not necessarily; e.g., consider adding P to the axioms of PM. 4. An inconsistent system is like a broken balloon-both are useless.

9.8

e go about proving the soundness, consistency, and completeS HowwOUld on . 1 . neSS 0 f tIle tree-method for the sententIal calculus. w that these axioms for the sentential calculus make up 4. HoW wou ld you sho a complete set: Al P -+ (Q -+ P) A2 (P -+ (Q -+ R)) -+ ((P -+ Q) -+ (P -+ R)) A3 (~P -+ ~Q) -+ (Q -+ P)

5. 9.8

Completeness of SC

One of the matters that have been postponed from Part I is the proof of the completeness of the natural deduction for sentential logic SO. We are now in position to prove that SO is both relatively and absolutely complete. The simplest way now at hand to prove SO relatively and absolutely complete is to prove that all the theorems of PM are theorems of SO. This can be quickly disposed of. All the theorems of PM are theorems of SO since all the axioms of PM can be proven as theorems in SO; the rule of inference in PM, MP, is a rule in SO; and, finally, the rule of substitution in PM is in effect presupposed in SO since the rules are formulated in terms of symbols that are variables for any wff of sententiallogic. The direct way to prove the relative and absolute completeness of SO can proceed along lines strictly analogous to the completeness proof for PM. A review of the theorems of SO proved in Ohapter 4 reveals that Lemma I holds for SO if it holds for PM. The proof of Lemma 2 for SO is the same as the proof for PM since the three steps outlined in the last section can be followed in SO. Finally, the proof that SO is absolutely complete again can proceed along the same course as the proof that PM is absolutely complete. Exercises 1. Fill in the details of a direct proof of the completeness of SO. 2. Use mathematical induction in constructing a proof of the soundness of SO. Let AI' ... , An be a proof in SO. Prove by mathematical induction on the number of sentences in a proof. Show that the first sentence is a tautological consequence of its assumptions, and that if every sentence appearing on lines through n is a tautological consequence of its assumptions, then the sentence line n I is a tautological consequence of its assumptions. If aJl sentences appearing on a line are tautological consequences of their assumptions, then a theorem must be a tautology since it is a tautological consequence of no assumptions.

+

~'h

225

Completeness of SC

With MP and the rule of substitution added to the axioms in (4) derive a few theorems.

oU show that A3 in (4) could or could not be replaced by 6 How would Y •

(~P -+ ~Q) -+ ((~P -+ Q) -+ P)

without altering the class of theorems~ Oan it replace A3~

7. The following is a set of axioms for intuitionistic sentential calculus. 1. P-+(Q-+P) 2. (P -+ (Q -+ R)) -+ ((P -+ Q) -+ (P -+ R))

3. P /\ Q -+ P 4. 5. 6. 7. 8. 9. 10.

P /\ Q-+Q P-+(Q-+P/\Q) P-+ P V Q Q-+PVQ (P -+ R) -+ ((P -+ R) -+ (P V Q -+ R)) (p-+Q)-+ ((P-+~Q)-+~P) ~P-+ (P-+Q)

.

(a) Show that the two theorems below are not theorems of thIS system. PV~P

,......,P-+P (b)

How would one show that the intuitional calculus is a subsystem of PMall the theorems of intuitional calculus are theorems of PM, but not the other way around ~

10.1

Preliminary Discussion of LPC

1 aXioms for the predicate calculus with identity

Any system that is consistent and that has all tautologies and valid predicate formulas and only tautologies and valid predicate formulas a~ theorems may be called a predICate calculus. The natural deduction system PC found in Chapt 6 . er IS such a calculus. In this chapter we wish to formulate and partially develop the predicate calculus as an axiomatic system. To do this we shall add t~ PM a r~le corresponding to the ulllversal mtroduction and two new axioms, one related to changing the scope of a universal quantifier and the other corresponding to universal elimination. Later in ~his chapter the resulting system WIll be proven consistent and an outline of the proof for soundness and completeness will be given. It will prove convenient to make some other changes in PM. The rule of substitution will be eliminated. This will not affect the derivation of theorems since the axioms of PM will be restated in terms of what are called axiom schemes. The two new axioms will also be stated as axiom schemes. In effect, this change will result in a system that has an infinite number of axioms rather than a finite number. How this is achieved and why this is desirable is discussed below. The system for the predicate calculus will be given the name LPG (after lower p1'edicate calculus).

226

Two axiom schemes corresponding to identity elimination and introduction will be added after a partial development of the predicate calculus. The result is an axiomatic system for predicate logic with identity.

10.1

Preliminary discussion of LPC

In the axiom system for the sentential calculus, the first axiom is PVP~P

Using A t.o stand for any well-formed formula in sentential logic, the axiom scheme corresponding to this axiom is AvA~A

This scheme determines an infinite number of axioms, namely all those sentential formulas that are instances of A V A ~ A. For example, instances of this scheme would be PVP~P

QvQ~Q

(P /\ Q) V (P /\ Q)

~

P /\ Q

and each will count as an axiom of a system that has A V A --+ A as an axiom scheme. Thus in setting down the basis of a system if A V A --+ A is given as an axiom scheme, then this would mean that the system had an infinite number of axioms, namely all instances of the scheme. A V A ~ A will in fact be one of the axiom schemes of LPC. In LPC, however, A, B, and G will be metavariables for formulas in both sentential logic and predicate logic. Of course other axiom schemes besides A V A --+ A are needed to achieve a complete sentential calculus. But each such scheme will determine an infinite number of axioms. One reason why axiom schemes are preferred for the predicate calculus is that having schemes does away with the need for a rule of substitution that is cumbersome to state for predicate logic. Formulas-in LPC will be made up of sentential variables, statement connectives, predicate variables, individual variables, and quantifiers. The formation rules for a well-formed formula will be almost the same as those set down in the last section of Chapter 5. We will not need

Axioms for the Predicate Calculus with Identity

individual constants. As will be indicated in a moment, forgoing the Use of individual constants will make necessary provisos so that individual variables do not become accidentally bound. Term now is exclusively used to indicate an individual variable. We will use some of the notation used in Chapter 6. Let A be a wff, va variable, and t a term. Then A (t/v) is the formula that results when each free occurrence of v in A is replaced by an occurrence of t (note the addition of 'free'). Suppose we are given this scheme which corresponds to UE in PC:

(1) 'IvA

-+

A (tfv)

The universal quantifier 'Iv, appearing in the antecedent, has been dropped in the consequent and a term is substituted for v in A. Instances of this scheme are 'v'xFx -+ Fx (2) 'v'x3yFxy -+ 3yFyy

Scheme (1) as it stands is not a valid scheme. This means that some instances of (1) are not valid formulas, e.g., (2) above. The invalidity of (2) is demonstrated given this interpretation: V: positive integers, Fxy: x < y. Another instance of (1) that is invalid is:

10.1

229

Preliminary Discussion of LPC

We are now in position to add our restriction to (1) so that it will become a valid scheme. Below is the needed valid scheme:

(4) 'IvA

-+

provided t is free for v in A

A(tfv)

(4) is in fact one of the axiom schemes we will use, and now neither (2)

nor (3) is an instance of (4). . Another axiom scheme of LPC wIll be (5) 'v'v(A

-+

B)

-+

(A

-+

'v'vB)

provided v is not free in A

Instances of (5) are 'v'y(Fx -+ Gy) -+ (Fx -+ 'v'yGy) 'v'x(P -+ Fx) -+ (P -+ 'v'xFx) 'v'y(Fx -+ 'v'z(Gxy -+ Hy)) -+ (Fx

-+

'v'y'v'z(Gxy -+ Hy))

Note that P here is either Tor F, so there can be no free variables in P. The next formula is not an instance of (5) since v is free in A: 'v'x(Fx

-+

Fx)

-+

(Fx

-+

'v'xFx)

(3) 'Ix I " - ' 'v'yFxy -+ ,......,'vyFyy

If we suppose V: {I, 2} and Fxy: x is identical to y, (3) expresses a false statement. To rule out such invalid instances we need to add a proviso to (1) to make it a valid predicate scheme. In these last two examples, when the term is introduced for v it becomes bound by a quantifier in A. In PC such accidental binding of individual variables could not occur since in the quantifier rules use is made of individual constants; t in PC is a variable for individual constants. Now t is a variable and only a variable for individual variables, and accidental binding becomes a menace. To introduce the proviso necessary so that individual variables do not become accidentally bound we first must introduce the notion of a term being free for some variable. We can say y is free for some variable, say x in A, if and only if no free occurrence of x in A falls within the scope of a quantifier containing y. In other words, y can be substituted for x in A without y coming under the scope of a quantifier in A. For example, x is free for x in (3), but y is not free for x. In this formula 'v'x(Fx

V

Gxy)

V

Fz

z is free for y, x is not free for y, x is free for z, and y is free for y.

The invalidity of this last formula is demonstrated with the following interpretation, supposing that Anderson is a poor student: V: human beings, x: Anderson, Fx: x is poor. The definition for a theorem is the same as in PM. The universal quantifier will be taken as primitive and the existential quantifier will be introduced by definition. The axi~m schemes for the propositional calculus will be the schemes correspondmg to the first four axioms of PM. Though we will rule out as well-formed formulas, vacuously quantified formulas we will allow formulas such as 'v'x(Fx -+ 'v'xFx). One reason these ~ere excluded from the class of well-formed formulas in PC is that their exclusion helped to simplify the rules of inference of PC. Their exclusion will not now help to simplify the rules of LPC, and since the only fault of these formulas is that they contain idle quantifiers, they will be allowed. The symbols "'-', V, A, -+, +-+, 'I, 3, and = will be used in a selfreferring manner in the formation rules (as they have been used all along) and in the axiom and theorem schemes. They ,:ill be us~d to refer to object-language symbols of the same type. That IS, they WIll be metaconstants in a way analogous to A, B, and 0, which have been used

230

231

Axioms for the Predicate Calculus with Identity

as metavariables for wffs. The object-language in this case is, of course the predicate language, and the metalanguage is what we use to t n' about the object-language in the formation rules and what we use refer to the object-language in axiom and theorem schemes.

DEFINITIONS:

Exercises

AXIOM SCHEMES:

A

at~

A

3. Let (a) (b) (c) (d)

\IvA be \lx\ly\lz(Fxy /\ Fyz x is free for x in A Y is free for x in A z is free for x in A w is free for x in A

-+

Fxz). Which of the following are true?

Answers 2. (i), (a) and (c); (ii), (a) and (b), (c) and (d), and (a) and (c). 3. (a) and (d) are true.

df

df

(,.....,A

V

,.....,C.....,A

B) V

Af-tB = df (A-+B) 3vA = df~\lv~A

,.....,B) II.

(B-+A)

AvA-+A A-+BvA AvB-+BvA (A -+ B) -+ (C V A -+ C V B) Al \lv(A -+ B) -+ (A -+ \lvB) provided v is not free in A A2 \IvA -+ A (t/v) provided t is free for v in A

I. Give. s~me more instances of (I), (4), and (5), being sure to observe an restrIctIOns. Are all your instances of (I) valid? y

2. Let \IvA be \lx3yFxy. (i) In which example below is t free for v in A? (ii) Which is A(t/v), A(x/y), and A(x/x)? (a) 3yFxy (b) 3yFyy (c) 3yFxx (d) 3yFyx

=

-+ B II. B =

RULES OF INFERENCE:

Rule of detachment (MP): If I- A, I- A -+ B, then I- B. Rule of universal introduction (UI): If I- A, then I- \I vA. Exercises 1. Make the changes necessary to the primitive base so that 3 becomes a primitive symbol and \I a defined symbol. 2. How many theorems does universal introduction provide? 3. The number of axioms in LPC is infinite. How can the notion of an axiom be effective? 4. What needs to be discarded so as to have only the sentential calculus? 5. Since the number of axioms is infinite, could a new axiom be added? 6. Why isn't a rule of substitution found in LPC?

10.2

System of LPC'

PRIMITIVE SYMBOLS:

1. 2. 3. 4.

Sentential and predicate variables: F, G, H, ... Individual variables: x, y, z, ... Quantifier: \I Statement connectives: ,....., and V

7. Which of the following are definitional transformations? (a) ~~\lx~Fx ~3xFx

(b) 3x~Fx ~\lxFx

(c)

~\lx~Fx ~,.....,3x~~Fx

(d) 3x~Fx ~\lx~,-...JFx

FORMATION RULES:

1. 2. 3. 4.

All atomic formulas are wffs. If A is a wff, then ,.....,A is a wff. If A and Bare wffs, then (A V B) is a wff. If A is a wff which contains a free individual variable v then \IvA is a wff. '

The calculus described in this section is an adaptation of FI in Alonzo Church Introduction to 1Ylathematical Logic, Princeton, N.J.: Princeton University Press: 1956.

(e)

~3x~~Fx

\Ix ~ Fx (f) r-;"\fx r--.J Fx 3xFx

Answers/ __ I. Needless to say, one thing needed is a definition such as: \IvA

=

df

~3v"""" A.

1

2. Infinite number. UI can be used on each axiom provided by the axiom schemes.

232

233

Axioms for the Predicate Calculus with Identity

3. A is an axiom iff it has one of six recognizable forms.

5. Yes, e.g., P. Any axiom that does not have one of the six recognizable forms. 7. Only (a), (d), and (f) are correct definitional transformations.

liberty of putting (i) in a nonscheme form. From now on, in using tautologies we will omit their introduction in the proof and omit the use ofMP. T3 A(tjv) --+ 3vA provided t is free for v in A (i) (P --+ ",-,Q) --+ (Q --+ ",-,P)

10.3

1. 'iv",-,A--+",-,A(tjv) 2. A(tjv) --+ ",-,'iv "'-/ A 3. T3

Development of LPC

Our unnumbered axioms plus MP constitute a complete sentential calculus (axiom schemes making the use of a rule of substitution unnecessary). In the interest of brevity no theorem schemes ofthe sentential calculus will be proved. If a theorem is needed from the sentential calculus, it will be entered without proof. Standard truth-table procedures can verify for us if a formula is a tautology. TI ,-..;'iv "'-' A+-+ 3vA 1. ",-,'iv",-,A+-+",-,'iv",-,A 2. ~'iv "'-' A+-+ 3vA

Identity df

TI is a metatheorem. It is the statement: 'For any A, "'-''iv "'-/ A +-+ 3vA is a theorem'. The proof of TI is in the metalanguage. It is a blueprint for any proofof any instance of T1. A proof of a theorem scheme gives a-uniform and effective procedure for constructing a formal proof in LPC for any instance of the theorem scheme. For the record let us derive a theorem using the proof pattern above, thus obtaining ~ proof of a theorem in LPC which is an instance of Tl. 1. ",-,'ix",-, Fx +-+ ",-,'ix "'-' Fx 2. ",-,'ix",-, Fx +-+ 3xFx

Identity df

The formula on line 2 is a theorem of LPC. The proofs to follow will be in the metalanguage. In effect, in constructing proofs in the metalanguage, A, B, and 0 in the axioms and rules take as instances wellformed formulas that are metaformulas. What counts as a well-formed metaformula we can tell directly from the formation rules. T2

'iv(A

(i) ((P

--+

--+

B)

(Q

--+

--+

--+

--+

3. (1) /\ (2) 4. T2

(4)

(A

--+

B)

A2 (t is free for v in A --+ B) A2 (t is free for v in A) (i) 1, 2, 3, lVIP (2)

In 'ivA --+ A, A is always A (v/v). Thus in both uses of A2 above, t is free for v in A since t is v and v is always free for v. We can easily see that lines 1 and 2 are metainstances of A2. Line 3 is obtained by employing the t~utology from the sentential calculus indicated at the beginning of the proof and labelled (i). For the sake offamiliarity, we have taken the

(i)

df

The proviso of A2 needs to be carried on to the statement of T3. Note that substitution of a t for a v in the antecedent is involved in T3. It is a useful exercise to provide nonvalid wffs of LPC that would be instances of T3 if the restriction were removed. T4 'ivA

--+

3vA

We will now provide ourselves with some useful derived rules. DRI If I- A --+ E, then I- A --+ 'ivB provided v is not free in A 1.A--+E A 2. 'iv(A --+ E) DI 3. 'iv(A --+ E) --+ (A --+ 'ivE) Al provided v is not free in A 4. A --+ 'ivE 3, 2, lVIP This proof indicates that whenever DRI is used it may be deleted and a formal proof constructed using DI, AI, and MP. DRI is a rule involving the changing of the scope of a quantifier, and the Al proviso that deals with scope change needs to be carried over. We note that no proviso needs to be given concerning accidental binding since in the proof of DRI no axiom or rule having this proviso is used. Also note that A --+ B in line 1 is assumed to be a theorem. We must have I-A --+ B to use DI. As in PM, the use of previously proven theorems and derived rules will indicate proof outlines and not formal proofs. DR2

B)

E)) /\ (S --+ Q)) --+ (P --+ (S --+ E))

1. 'iv(A --+ B) 2. 'ivA --+ A --+

('ivA

A2 provided t is free for v in A

If I- A

--+

E, then I- 'ivA

--+

B

'iv(A --+ E) --+ ('ivA --+ 'ivE) 1. 'iv(A --+ E) --+ ('ivA --+ E) 2. 'iv(A --+ E) --+ 'iv('ivA --+ E)

T5

3. 'iv('ivA

--+

E)

--+

('ivA

--+

'ivE)

-4. T5

DR3 If I- A --+ E, then I- 'ivA l.A--+B A 2. 'iv(A --+ E) DI 3. 'ivA --+ 'ivB T5, lVIP

--+

'ivE

T2 DRI (v is not free in 'iv(A --+ E)) Al (v is not free in 'ivA) 2,3, Trans

Axioms for the Predicate Calculus with

234

T6

\1'vA

-+

\1'uA(ujv)

1. \1'vA 2. T6

-+

A (ujv)

T7

1. 2. 3. 4. 5. 6.

provided u is free for v in A and u is not free in \1'vA A2 provided u is free for v in A DRI provided u is not free in \1'vA

\1'u\1'vA -+ \1'v\1'uA \1'vA -+ A \1'u(\1'vA -+ A) \1'1t\1'vA -+ \1'uA \1'v(\1'u\1'vA -+ \1'uA) (4) -+ (6) T7

A2 (t is free for v in A) VI T5, MP VI Al (v is not free in \1'u\1'vA) 5,4, MP

T8 ",,3vA -+ \1'v "" A (i) "" ""P +-7 P 1. "" ",,\1' v "-' A +-7 \1' v "" A 2. T8

T9

\1'v"" A

+-7

Development of LPC

TIl

\1'v(A -+ B) -+ (3vA -+ B) (i) ""p -+ ""Q+-7 Q -+ P (ii) "" P -+ Q +-7 "-'Q -+ P 1. \1' v(,,-,B -+ ,,-,A) -+ (""B 2. \1'v(A 3. Vv(A 4. TIl

B) B)

-+ -+

\1' v ,,-,A)

(""B -+ Vv ,,-,A) (""Vv ""A -+ B)

Al provided v is not free in B (i), R (ii), R

df

It should be noted that the replacement theorem of PM could be carried over into LPC, but then replacement could only be done with wfl's of

PM. It would sanction line 3. We wish to have a replacement rule that applies to quantified formulas. It is the replacement theorem (tbove

(i)

TI2 TI3 TI4

df

Vv(A -+ B) -+ (3vA -+ 3vB) Vv(A A B) +-7 VvA A B provided v is not free in B 3v(A A B) +-7 3vA A B provided v is not free in B

,.....,,3vA

The statement of the metatheorems corresponding to the other major theorems of the predicate calculus (Section 6.6) and their proofs will be left for exercises. Note that none of these metatheorems carry provisos or need to carry them, since such provisos would be idle and would not do any work; if sentential variables do occur in theorems, then the corresponding metatheorem may need the addition of a restriction. To illustrate, the scheme corresponding to T32 in Section 6.6 (Vx(P A Fx) +-7 P

A

VxFx) is Vv(A

THE EQUIVALENCE THEOREM:

\1'Vl'" \1'vn (M+-7N)-+ (AM +-7 AN)

The proof is by induction on the number n of symbols in AM, counting each occurrence of "", v, or \1' as a symbol. The cases are the same as those in Section 9.4, with the added case where AM is \1'vBM. T5 from LPC is needed in this case. We leave the details as an exercise. Let M, N, AM, and AN be as in the equivalence theorem, then (R): Nand f- AM, then f- AN

THE REPLACEMENT THEOREM +-7

-+

-+

that sanctions line 2.

The equivalence theorem for LPC can make use of all the notation of the equivalence theorem in PM with a few additions. Let AM be a formula in which M occurs in A. Let AN be the result of replacing one or more occurrences of M in AM by occurrences of N. Let every variable that is free in M or N and bound in AM be in the list VI> ••• , v,., then

If f- M

-+

provided v is not free in B

Some uses of the replacement theorem are illustrated in the next proof.

A

B) +-7 A

A

VvB

Since A and B above can take open sentences as instances, v might be free in A and the result be an invalid formula. To insure that the above is a valid scheme, we need but add the restriction that A contain no free occurrences of the individual variable v. Exercises 1. Prove the unproven derived rule and theorems through TI4, found in this section. Some of the proofs may be aided by first proving relatively simpler theorems.

2. Write down some actual theorems of LPC. --

------

3. A proof of a theorem scheme gives a uniform and effective procedure for constructing a formal proof of any instance of the theorem. Vsing some scheme's proof, construct a proof of a theorem of LPC. 4. Prove schemes corresponding to the numbered theorems found in Chapter 6.

236

5. Give the proof by induction for the equivalence theorem. B)~

6. Show that V'v(A A in A.

T3

V'xV'yV'z(x = y -+ (y = z -+ x = z)) (i) (P --+ Q) -+ ((R -+ P) --+ (E -+ Q)) 1. Y = x -+ (y = z -+ x = z)

A A V'vB can have an invalid instance if v is free

2. x

= Y --+ Y

3. (1) 10.4

237

LPC with Identity

We can obtain a first order calculus with identity by adding to LPC a new two-place predicate constant Ixy that may be read, as before, "x is identical with y". With the addition of this predicate, the formation rules will be those in LPC and I followed by two individual variables is a wff.

To obtain significant theorems employing I we need two additional axiom schemes:

12

Ivv Ivu

-+

(A

-+

A (u/v)) provided u is free for v in A

Here I now acts as a metaconstant. The first axiom, II, established the reflexivity of I, and the second axiom scheme established the substitutivity of equivalents. The more familiar notations = and 0/= may be introduced by definition as follows: v

=

=df

-+

(i)

(4))

1,2,3, MP (2), DI (3)

V'xV'y(Fx A x = Y -+ Fy) (i) (P -+ (Q -+ E)) -+ (Q A P 1. x = y-+ (Fx-~ Fy) 2. Fx A x = Y -+ Fy

T4

--+

R)

12 (i)

DI (2)

3. T4 T5

T6 T7 T8

3y(x = Y A Fy)) V'y(x = y -+ Fy)) V'x3y(x = y) V'xV'yV'z(y = x A z = x --+ y V'x(Fx~ V'x(Fx~

=

z)

Some useful definitions may be introduced, and theorems employing these new terms may be derived. As we saw in Chapter 7 There is exactly one x such that Fx. can be translated as (1) 3xFx A V'xV'y(Fx A Fy

u =dfIvu

v 0/= u

((2)

12 T2, A2 (t is free for v in A)

X

4. T3

LPC with identity

II

-+

=

-+

x

=

y).

,....,Ivu (1) may be abbreviated as

The resulting system is a lower predicate calculus with identity, and it will be denoted by LPGI. We can prove the schemes corresponding to theorems but the provisos would be tedious to state. So we prove theorems rather than metatheorems. T1

V'x(x = x) 1.x=x 2. T1

T2 (i) 1.

2. 3.

V'xV'y(x = y -+ y = x) (P -+ (Q -+ E)) -+ (Q -+ (P x = y -+ (x = x -+ y = x) x = x (1) -+ ((2) -+ (4))

4. T2

(2) 3!xFx

(2) is called the numerically definite quantifier. If (2) =df (1) is introduced into LPC with identity, then this allows us to prove a theorem such as the next theorem.

II

II (i)

V'x3!y(y = x) 1.x=x 2. 3y(y = x) 3. V'xV'yV'z(y = x A z = x -+ y = z) 4. V'yV'z(y = x A z = x -+ y = z) 5. (2) A (4) 6. 3!y(y = x)

1, 2, 3, MP (2), DI (2)

7. T9

DI

-+ E))

12

T9

II

T3 (t is free for v in A) T8 A2 (t is free for v in A) 2,4, Conj df DI

238

Axioms for the Predicate Calculus with Identity

Exercises

10.6

Soundness of LPC with identity

239

Since A is its own transform and not a tautology, LPO is absolutely consistent. Oonsistency in Post's sense is assured by the same fact.

1. Give some instances of II and 12.

2. Prove the unproven theorems in this section and the numbered theorems found in 7.3.

Exercises

3. Prove

1. Show that P is not a theorem of LPCI and thus prove that LPCI is negation consistent.

(a) Vx(Fx~ Ox) --+ (3!xFx~ 3!xOx) (b) 3!xFx~ 3xVy(x = y~ Fy)

(c)

3!xFx~

3x(Fx A Vy(Fy --+ x

= y))

2. Use induction to prove every theorem of LPCI has a tautological transform. 3. Do the same (as in 2) for PC.

10.5

Consistency of LPC with identity

To prove the predicate calculus with identity consistent, we do something analogous to transforming formulas into associated sentential formulas (see Sections 6.7 and 8.4). In this case, each formula will be transformed as follows: First, every occurrence of Vv for every variable v will be deleted. Second, each occurrence of every individual variable will be replaced by x. Third, each occurrence of x = x will be replaced by an occurrence of x = x --+ x = x. A and B are regarded as transforms ofthemselves. Following this procedure, the transforms of the axioms of LPOI are: AvA--+A A--+BvA AvB--+BvA (A --+ B) --+ (0 V A --+0 V B) (A --+ B) --+ (A --+ B) A--+A x=x--+x=x (x = X --+ X = x) --+ (Ax --+ Ax)

The transform of every axiom is a tautology. Modus ponens applied to two tautologies always gives a tautology. If UI is applied to a formula whose transform is a tautology, then the transform of the resulting line is a tautology. Since a proof consists of axioms or lines derived by MP or UI, all the lines of a proof have tautologies as transforms, and thus do all the metatheorems. Finally, if a metatheorem has a tautological transform, then all its instances have tautological transforms since they have the same form; thus all theorems of LPO with identity have tautological transforms. Since for any formula A, A A ,.....,A cannot have a tautological transform, we obtain the result that LPO with identity is negation consistent.

4. Let T be an axiom system that has only contingent sentential formulas as theorems. (a) Is this theory negation consistent~ (b) Is it absolutely consistent ~

Answers 1. LPCI is negation consistent iff it is absolutely consistent.

4. (a) No, since P and ,.....,p are theorems; (b) Yes, since P V ,.....,p is not a theorem.

10.6

Soundness of LPC with identity

To show LPOI is sound is to show that every theorem is valid. To speak of a wff in LPOI as valid is to say that it is true under any interpretation. As described in earlier sections, one obtains an interpretation of a formula in LPOI when he: 1. Indicates a nonempty domain 2. Assigns to each sentential variable the value Tor F 3. Assigns to each free individual variable a name for an element in the domain 4. Assigns to each n-place predicate a predicate in the domain 5. Assigns to = the identity predicate in the domain.

To illustrate, consider (1) Fx A Vy(Fy --+ P) A x

= Y

and let the domain be show biz people, Fx: x is fat, P be T, x be Oliver Hardy, y be the partner of Stan Laurel. In this interpretation (1) is true. Af~rmula in LPOI is valid if and only if it is true on every interpretation. We again note that if an interpretation of a formula A yields a true proposition, this interpretation is called a model for A. Thus if a formula in LPOI has models for any interpretation, then it is a valid formula.

240

Axioms for the Predicate Calculus with Identity

We now begin a somewhat intuitive commentary concerning the proof that every theorem of LPOI is valid. Truth-tables confirm that the first four axiom schemes only have valid formulas as instances. To prove that the axioms from Al and A2 of LPOI are valid, We suppose that what is true of one axiom from the scheme is true of all its axioms since form is preserved. We will take as our instance of axiom scheme one 'ix(Fy -)- Fx) -)- (Fy -)- 'ixFx) and as our instance of axiom scheme two 'ixFx -)- Fx Both of these formulas are made up of one and only one predicate, the one-place predicate Fx. There are no other predicat~s and there are no sentential variables. Interpreting these two formulas is thus restricted to (1), (3), and (4) above. A one-place predicate such as Fx can on interpretation (l) give only true statements for any domain, (2) give only false statements for any domain, (3) give only true statements in some domains and only false statements in other domains, or (4) give some true and some false statements in a domain. If in each of these cases the above axioms are true, then the axioms are valid. Let us then consider each axiom and these four cases, beginning with the second axiom. OASE 1: The predicate represented by Fx has only true instances in any domain of discourse. If this is so, 'ixFx -)- Fx must be true since Fx can never have a sentence expressing a false statement as an instance. OASE 2: The predicate represented by Fx has only false instances in any V. If this is so, then 'ixFx has only sentences expressing false statements as instances; thus 'ixFx -)- Fx is always true. OASE 3: If Fx has only true instances in a domain, the considerations in case 1 apply, and if it has only false instances in a domain, case 2 applies. OASE 4: Fx has in a domain some instances that are true and some that are false. In this case 'ixFx can have only sentences expressing false statements as instances, in which case 'ixFx -)- Fx is true. Turning next to the first axiom and the four cases, we again consider each. O~SE 1: If Fx has only true instances in any domain of discourse,

10.6

241

Soundness of LPC with identity

then 'ixFx is always true; thus the consequent of 'ix(Fy -)- Fx)-+ (Fy -)- 'ixFx) will always be Fy -)- T, which is true, and thus the axiom is true. OASE 2: If Fx has only false instances in any domain of discourse, then 'ixFx only has false instances. If'ixFx has only false instances, then Fy can only be false. If Fy is false in 'ix(Fy -)- Fx) -)- (Fy -)- \;fxFx) , both the antecedent and the consequent are true; thus the axiom is true. OASE 3: If Fx has only true instances in a domain, case 1 applies, and if it has only false instances, case 2 applies. OASE 4: If Fx is sometimes true in a domain, then in the case in which Fy is true when a name is assigned, the antecedent will be false. In the case in which Fy is false when a name is assigned, the antecedent is true; but then the consequent Fy -)- 'ixFx is also true. So the axiom is true. We leave the account of the validity of the identity axioms as an exercise. Rl: Modus ponens is a valid rule. R2: If 'If ~ A, then I- 'ivA' is a valid rule, then when it is applied to a valid line, it only permits the inference of a valid line. If Fx is valid, then it immediately follows that 'ixFx is valid. Since the axioms of LPOI are all valid, and since Rl and R2 only allow the derivation of valid lines from valid lines, every theorem of LPOI is valid. LPOI can again be proven negation consistent. For no A can both A and ,.....,.,A be a theorem, for A and ,.....,.,A cannot both be valid. Exercises 1. Provide a model for each of the following: (a) 'ix(Fx+--t 3y(y = x /\ Fy)) (b) 'ix'iy3zFxyz (c) 3y'ix(Fx+--t x = y) (d) 3xFx /\ (Fa -)- 3x3y(x *- y /\ (Fx /\ Fy))) 2. Demonstrate the validity or invalidity of each of the following: (a) 'iv(A V A) -)- 'ivA V 'ivA (b) 'ivA -)- A (c) A -)- 'ivA (d) 3v'iuA -)- 'iv3uA (e) A -)- 3vA (f) ('ivA -)- 'ivB) -)- 'iv(A -)- B) 3. Prove that the identity axioms are valid. 4. DI of LPC valid in LPC but invalid in PC?

WhYIs

Answers 1. (b) V: positive integers, Fxyz: x

+y=

z.

242 10.7

Axioms for the Predicate Calculus with Identity

The sentential calculus is complete in both the relative sense (all tauto_ logical formulas are derivable) and in the absolute sense (a nonderivable wff conjoined to the axioms results in an inconsistent axiom set). The predicate calculus, however, is not absolutely complete but is relatively complete. The first is easily shown. There are nonderivable formulas that will not create inconsistent axioms if added to the axioms of the predicate calculus LPC. Examples of such formulas are: 3xFx

A

3xGx -+ 3x(Fx

Completeness of Predicate Logic Theories

243

From Theorem 1 and

Completeness of predicate logic theories

3xFx -+ 'ixFx,

10.7

A

Gx),

Theorem 2: If LPC has a model, then LPC is consistent. Theorem 3 below follows: Theorem 3: LPC is consistent if and only if it has a model. Theorem 4: Let A be a well-formed predicate formula of LPC. Let LPC' be the result of conjoining A to LPC. If LPC' is inconsistent, then "",A is a theorem of LPC.

Fx-+ 'ixFx

Their transforms can be seen to be tautologies. Consequently, the addition of these to the axioms of LPCI would thus not alter the consistency of LPCI. The above predicate formulas are not valid. Since every thesis of LPCI is valid, these formulas are not derivable. Hence LPCI is not absolutely complete. The proof of the relative completeness of the predicate calculus is complicated and requires quite a few preliminaries. We will not attempt to give the proof in this book. Many feel that the demarcation point between introductory logic and advanced logic is the completeness proof for the predicate calculus. The completeness proof for the predicate calculus was first given by Kurt Godel in the early thirties and makes use of a normal form for wellformed formulas in predicate logic. The point is to show that machinery is available in the predicate calculus to grind out all the valid formulas in the normal form. The completeness proof in wide use today is Leon Henkin's 1949 proof. The important theorem of this proof is the theorem that every consistent first order theory, of which the predicate calculus is one, has a countable modeJ.2 We have by this time some experience in providing models for formulas-interpretations that give true statements. There can also be a model for sets of formulas and a model for the set of formulas that make up the predicate calculus. In general M is a model for a theory T iff M is a model for the set of theorems of T. A model is countable when its elements can be enumerated. Let us set out, then, the crucial theorem: Theorem 1: Every consistent first order theory, LPC being a first order theory, has a countable model. 2 This outline is adopted from Section 3.27 of Angelo Margaris, Fil'St Order 1I1athematical Logic (Waltham, Mass.: Blaisdell Publishing Co., 1967).

Proof: If LPC' is inconsistent, then there is a proof in LPC that ~ A -+ "",A. Since ~ (A -+ "",A) -+ "",A, "",A is a theorem. Theorem 5: Let B be a predicate formula of LPC. If B is true in every model for LPC, then B is a theorem of LPC. Proof: Suppose that B is true in every model for LPC. Let LPC' be the result of conjoining "",B to LPC. If M is a model of LPC', then both Band ,--.,B are true in M, which is impossible by Theorem 3. Therefore LPC' has no model. Then LPC' is inconsistent by Theorem 3 (and 1). Then"", "",B is a theorem of LPC by Theorem 4. Hence B is a theorem ofLPC by DN. Theorem 6: Every valid predicate formula is a theorem of LPC. Proof: Suppose A is valid. Then A is true in every model for LPC. Hence, by Theorem 5, A is a theorem. There is an effective decision procedure to determine validity for sentential formulas and for certain classes of predicate formulas (see Section 5.6).3 There is, however, no effective decision procedure that suffices to determine for an arbitrarily selected wff of LPC whether or not it is valid and thus a theorem. The American logician Alonzo Church proved that such a procedure could not be obtained for the first-order predicate calculus. His proof was based on a 1931 paper by Kurt Godel that showed that no matter what consistent axiom system we select for the arithmetic of positive integers, we can formulate an acceptable statement of arithmetic that is not provable in the system-in other words, axiom systems for number theory cannot be relatively complete. However, even though there is no effective decision procedure for the va1i.sJ:ity of the formulas in LPC, there are mechanical procedures for 3

See Church, Introduction, Section 46.

244

Axioms for the Predicate Calculus with Identity

constructing proofs for any valid predicate formula (see, for example, Quine's Methods and Jeffrey's Formal Logic). Earlier the question was raised whether the natural deduction system for predicate logic set down in Chapter 6 is complete-whether all valid wffs of PC are theorems. Here again we can prove this in two ways. First, prove that all the theorems of LPC are theorems of PC by showing that all instances of the axiom schemes of LPC can be proven by using PC and that the rules of LPC are to be found in PC. Since LPC is complete, PC must be complete. Second, produce a proof on lines analogous to the completeness proof for LPC. Exercises 1. Prove from the theorems given that if LPC has a model, then it has a

countable model. 2. Prove theorem 2. 3. Prove that if any theory has a model, then it is consistent if exactly one of the pair (P, ,--,P) can be true in M. 4. If A is true in every model for the predicate calculus, then is A valid 1 5. A system is negation complete if and only if for every wff A either A or ,--,A is a theorem. Prove that PM and LPC are not negation complete. 6. A statement is a logical truth (or is analytic) if and only if the way that '--', V, 'if, and = enter into its construction makes it true regardless of how its other terms are understood. (a) Can the notion of a logical truth be an effective notion 1 (b) Is it possible to tell in all cases whether a statement is a logical truth 1 7. Why would we not want the predicate calculus to be absolutely complete1

further readings

The following list contains a few of the books that will prove useful to those who wish to pursue some aspects of logic set forth in this text. Books of the introductory type that provide lucid alternative treatment of the topics in this book are Copi and Mates. Kalish and Montague, Lemmon, and Suppes give alternative, up to date, introductory treatment of natural deduction systems and include topics not considered in this book. Hughes and Londey provides an introductory treatment of finite logistic systems, including syllogistics. Jeffrey gives a detailed introductory treatment of the tree method and even has a chapter on Godel's incompleteness theorem. Quine's Methods also provides an introductory account for natural deductions, along with truth-table methods, normal forms and discussion of some philosophical issues. Advanced books that provide amplification of some of the topics treated here are Church, Kleene, Margaris, Mendelson, and Quine's Mathematical Logic. Church provides the most comprehensive treatment of sentential and predicate axiom systems. Kleene, Margaris, and Mendelson cover many topics outside the range of this book, for example, the formal development of number theory. Hilbert and Ackermann is a classical compact account ofaxiomatic logical systems. Keene's book 245

Further Readings

246

Quine, W. V. 0., Mathematical Logic. Cambridge, Mass.: Harvard University Press, 1951. Quine, W. V. 0., Methods of Logic. New York: Holt, Rinehart & Winston, Inc., 1950. Robison, Gerson B., An Introduction to Mathematical Logic. Englewood Cliffs, N.J.: Prentice-Hall, Inc., 1969.

a condensed exposition ofaxio_ matic predicate calculus. Kneale and Kneale covers the history of logic, while-Bukasiewicz's book provides the pioneer treatment of axiomatic syllogistics. Beth's book is an advanced and condensed Bxposition of the topics in this book including the tree method. It als~ covers topics not found in this book for example, machines which prov~ theorems in logic. Robison is an easily understood introductory book that concentrates on the application of logic to mathematical theories. IS

Suppes, P., Introduction to Logic. Princeton, N.J.: D. Van Nostrand Co., Inc., 1957.

Beth, E. W., Formal Method. New York: Gordon & Breach Science Publishers Inc., 1962. ' Church, A., Introduction to Mathematical Logic. Princeton, N.J.: Princeton University Press, 1956. Copi, 1. M., Symbolic Logic. New York: The Macmillan Company, 1967. Hilbert, David, and W. Ackermann, Principles of Mathematical Logic. New York: Chelsea Publishing Co., 1950. Hughes, G. E., and D. G. Londey, The Elements of Formal Logic. London: Methuen & Co. Ltd., 1965. Jeffrey, R., Formal Logic: Its Scope and Limits. New York: McGraw-Hill Book Company, 1967. Kalish, Donald, and Richard Montague, Logic: Techniques of Formal Reasoning. New York: Harcourt, Brace & World, Inc., 1964. Keene, Geoffrey, First-Order Functional Calculus. New York: Dover Publications, Inc., 1964. Kleene, S. C., Mathematical Logic. New York: John Wiley & Sons, Inc., 1967. Kneale, W., and M. Kneale, The Development of Logic. Oxford: The Clarendon Press, 1962. Lemmon, E. J., Beginning Logic. London: Nelson, 1965 . .l:;ukasiewicz, J., Aristotle's Syllogistic from the Standpoint of Modern Formal Logic. Oxford: The Clarendon Press, 1957. Margaris, A., First Order Mathematical Logic. Waltham, Mass.: Blaisdell Publishing Co., 1967. Mates, B., Elementary Logic. New York: Oxford University Press, Inc., 1965. Mendelson, E., Introduction to Mathematical Logic. Princeton, N.J.: D. Van Nostrand Co., Inc., 1964.

247

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Index

I

index

Absolute completeness, 197, 221 Absolute consistency, 99, 214 Absorption (Abs), 48, 95 Accidental binding, 228 Addition rule (Add), 74 Analytic statements, 12, 131, 168 Antecedent, 7 sentential, 44 valid, 5 valid patterns of, 41 Argument pattern, 12 Aristotelian logic, 107, 190-193 Associated sentential formula (asf) for CS, 196 for LPCI, 238 for PC, 155 Association (Assoc), 48 Assumption discharged, 77 rule (A), 71 scope of, 78 Atomic formula, 130 Atomic statement, 20 Axiomatic method, 186 Axiomatic theory, 188 Axiom scheme, 227 Axiom systems, 205

Church, Alonzo, 157, 230n, 243 Circuits, simplifying, 61 Commutation (Com), 48 Completeness absolute, 197,221 relative, 197, 220-221 Compound statement, 20 Compound statement former, 20 Conclusion, 3 Conditional formula, 20 Conditional proof rule (RCP), 78 Conditional sentence, 7 Conjunct, 16 Conjunction, 16 rule, 73 Conjunctive normal form, 57 Connectives main, 37 ranking, 33 scope of, 23 truth-functional, 21 Consequence, tautological, 68 Consequent, 7 Oonsequentia mimbilis, 94 Consistency absolute, 99, 214 of CS, 97 of LPCI, 238 negation, 97, 214 o~PM, 216-217 Post's sense of, 99, 214 of SC, 97 Constants, individual, 134 Constants, sentential, 23 Contingent formula, 38 statement, 6

Basic components of a formula, 135 Basic truth-table, 17 Biconditional formula, 20 Bound individual variable, 110 Calculus, predicate, 6, 180 see also predicate calculus Calculus, sentential, 67 Categorical propositions, 107

\

248

Contradiction, 80 Contradiction law, 95 Contraposition (Con), 48 Corresponding conditional, 7 Counterexample, 75 abstract and concrete, 117 CS associated sentential formula for, 196 completeness of, 196-197 consistency of, 195-196 soundness of, 196 syllogistics system, 190-193

Decision procedure, 35 normal form, 57 predicate formula, 116-117 truth table method, 36 Deduction, see system Deduction theorem, 214 Definite descriptions, 175-178 Definitions, 19 De Morgan, A_, 48 theorem (DM), 48 Derived rules, 91, 210 Dilemma, constructive, 95 Dilemma, destructive, 95 Discharged assumption, 77 Discourse, universe of (V), 105 non-empty, 114 Disjunction, 19 rule (Dis), 79 Disjunctive normal form, 59 Disjunctive syllogism (DS), 91 Disjuncts, 19 Distribution (Dist), 48 Domain of discourse (V), 105 Double negation (DN), 48, 73 Dropping inconsistencies (DI), 52 Dropping tautologies (DT), 52 Duns Scotus' law, 92

Effective notion for a proof, 188 for a tautology, 40 for a theorem, 188 for a well-formed formula (wff), 32

249 Elimination, universal (UE), 137 Empty term, 114 Equivalence (Equiv), 48 logical, 47 Excluded middle law, 90 Exclusive use of 'or', 25 Existential elimination (EE), 140 Existential quantifiers, 107 Export-import law (EI), 95

Formal language, 13 Formal system, 188 Formation rules, 32 with individual constants, 143 for LPC, 230 for PC, 150-151 for predicate language, 130 for sentential language, 32 Formulas, 9 atomic, 130 basic components of, 135 bi-conditional, 20 conditional, 20 contingent, 38 inconsistent, 38 interpreting, 113, 134, 166, 239 in logic, 131, 168 model for a predicate, 115 sentential, 10, 50 valid, 10, 38, 113 well-formed (wff) , 32 Free individual variable, 101

General name, 193 Godel, Kurt, 197,242,243 Group theory, 179

Henkin, Leon, 242

Idempotent (Idem), 48 Identity, 160-161 axioms for, in LPC, 236 elimination (IE), 167 introduction (II), 167 rules for, in PC, 167

250 Implication (Imp), 48 transitivity of, 90 Inclusive use of 'or', 25 Inconsistencies, dropping (DI), 52 Independence of CS, 194 of PM, 217-219 of SC rules, 95 Individual constants, 134 Individual variables, 102 Induction, 212 Inference, rules of, 69-70 Interpretation of identity sign, 161 principal, 192-193, 218 of quantifiers, III of statement connective, 16-20 Interpreting a formula, 113 with identity, 166 with individual constants, 134 in LPC!, 239 Introduction, existential (EI), 138 Introduction, universal (UI), 142 Intuitional logic, 225

Kleene, S. C., 28n

Language formal, 13 meta, 32, 229-230 object, 32, 229-230 predicate, 130 Leibniz, G. W., 6 Limited term, 140, 145 Logic, see intuitional, predicate, sententiallogic Logical equivalence, 47 Logically imply, 11, 131, 168 Lower/higher predicate calculus, 180 LPC axioms for identity in, 236 axiom system, 230-231 completeness of, 242-243 predicate calculus system, 230-231 replacement rule in, 234 LPCI axiom system, 236 consistency of, 238 "",.,

Index interpreting a formula in, 239 predicate calculus system, 236 soundness of, 239-241 ..B"ukasiewicz, Jan, 193

Main connective, 37 Mathematical induction, 212 Mention, see use/mention distinction Metalogical questions, 97 Metalogical variables, 32 Model for a predicate formula, 1I5 Modus ponens (MP), 69 Modus tollens (MT), 74 Monadic predicate, 120

Name, general, 193 Natural deduction systems, 67, 189 Necessary condition, 28 Necessary statements, 6 Negation, 17 consistency, 97, 214 double (DN), 48, 73 Normal forms, 57 conjunctive, 57 disjunctive, 59 Numerically definite quantifier, 237

Object language, 32, 229-230 '01", inclusive use of, 25 exclusive use of, 25 Paraphrase, principle for correct, 42 PC completeness of, 155-157, 244 predicate calculus system, 150-151 replacement rule in, 153 rules for identity in, 167 soundness of, 155-157 system, 134, 150-151 PC!, 167 PM axiom system, 206 completeness of, 220-223 consistency of, 216-217 independence of, 217-219 replacement rule in, 213 rule of substitution in, 206 soundness of, 215-217

251

Index Polyadic predicates, 120 Post's sense of consistency, 99, 214 Predicate, monadic, 120 Predicate calculus system LPC, 230-231 system LPC!, 236 system PC, 150-151 system PC!, 167 Predicate formula, 116-117 model for a, 115 Predicate language, see formation rules Predicate logic, 58 with identity, 180 Predicate term, 104 n-place, 104 Predicate variables, 128 Premise, 3 Principal interpretation, 192-193, 218 Principia Mathematica, 203 Proof by cases, 95 by contradiction, 95 effective notion for, 188 see also system Propositions, categorical, 107

Quantification, vacuous, 129 Quantifiers, 106 existential, 107 interpretation of, 110 numerically definite, 237 scope of, 108 universal, 106

Ranking connectives, 33 Reductio ad absurdum rule (RAA), 80 truth-table test, 45 Relations, properties of, 162-165 Relative completeness, 196-197, 220221 Replacement rule, 49 in LPC, 234 in PC, 153 in PM, 213 in SC, 94

Rules of inference, 69, 70 valid, 70 Russell, Bertrand, 179, 203

Satisfying a predicate, III SC completeness of, 97, 224 consistency of, 97 replacement rule in, 94 rules, independence of, 95 soundness of, 97 system, 67, 86-88 turnstile sign in, 89 Scope of a connective, 23 of a quantifier, 108 Semantics of a theory, 188 Sentence, singular, 110 Sentential argument, 44 Sentential calculus, 67 Sentential constant, 23 Sentential formulas, 10 transformation of, 50 Sentential language, 16 see also formation rules Sentential logic, 99 Sheffer stroke, 55 Simplification (S), 69 Simplifying circuits, 61 Singular sentence, 110 Singular term, 103 Soundness of OS, 196 of LPC!, 239-241 of PO, 155-157 of PM, 215-217 of SC, 97 Statement variables, 31 Substitution, rule of, 49 in PM, 206 Sufficient condition, 20 Syllogism, disjunctive (DS), 91 Syllogistics, system OS, 190-193 Syntax of a theory, 188 System axiom system LPO, 230-231 axiom system LPCI, 236 axiom system PM, 206 OS, 191

252

Index

System-(cont.) PO, 134, 150-151 POI,167 SO, 67, 86-88

Tautological consequence, 68 Tautology, 39, 135 dropping (DT), 52 effective notation for, 40 Term empty, 114 limited, 140, 145 Theorem of OS, 191 deduction, 214 effective notion for, 188 of LPO, 229 of PO, 151 of PM, 205 of SO, 88 Theory, 187 Transformation of sentential formulas, 50 Transitivity of implication (Trans), 90 Tree (or tableau) method for predicate logic, 158 for predicate logic with identity, 180 for sentential logic, 99 Truth-functional connectives, 21

Truth-table, 17 basic, 17 method fo~ one-place predicate formulas, 119 test for reductio ad absurdum, 45 Truth-value, 6 Turnstile sign, 70 in axiom systems, 205 in SO, 89

Universal elimination (UE), 137 Universal introduction (UI), 142 Universal quantifiers, 106 Universe of discourse (V), 105 non empty, 114 Use/mention distinction, 14

Vacuous quantification, 129 Valid argument, 5 Valid argument patterns, 41 Valid formula, 10, 38, 113 Variable free, 101 individual, 110 metalogical, 32 statement, 31

Well-formed formula (wff), 32 see also formation rules

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