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SVMBOLIC LOGIC FOURTH EDITION

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### Transcript of [Irving M. Copi] Symbolic Logic

SVMBOLICLOGIC FOURTH

EDITION

Copyright @ 1973, Irving M. Copi

Far Amelia

PREFACE

viii

Preface

Jr., of Wichita State University, Murray Braden of Malcalester College, Lorin

Browning of the College of Charleston, Mario Bunge of McGill University,David R. Dilling of Grace College, Earl Eugene Eminhizer of YoungstownState University, Barry R. Gross of York College, City University of New York,Herbert Guerry of Idaho State University, James N. Hullett of Boston Univer-

sity, R. Jennings of Simon Fraser University, Robert W. Loftin of Stetson

University, Warren Matthews of Old Dominion College, Robert W. Murungiof the University of Dubuque, Jean Porte of the Centre National de laRecherche Scientiflque, Samuel A. Richmond of Cleveland State University,Donald Scherer of Bowling Green University, Anjan Shukla of the Universityof Hawaii, Leo Simons of the University of Cincinnati, Frederick Suppe ofthe University of Illinois, Norman Swartz of Simon Fraser University, William

J. Thomas of the University of North Carolina at Charlotte, William C. Wilcoxof the University of Missouri, and Jason Xenakis of Louisiana State University.

I should like to express my thanks to my daughter Margaret Copi and to

Miss Karen Lee for help in reading proof.Most of all I am deeply grateful to my wife for her help and encourage-

ment in preparing this new edition.

CONTENTS

Contents

5 The Logic of Relations 112

5.1 Symbolizing Relations 112

5.2 Arguments Involving Relations 126

5.3 Some Attributes of Relations 1305.4 Identity and the Definite Description 1365.5 Predicate Variables and Attributes of Attributes 145

S Deductive Systems 152

6.1 Definition and Deduction 152

6.2 Euclidean Geometry 153

6.3 Formal Deductive Systems 157

6.4 Attributes of Formal Deductive Systems 159

6.5 Logistic Systems 161

7 A Propositional Calculus 165

7.1 Object Language and Metalanguage 1657.2 Primitive Symbols and Well Formed Formulas 1677.3 Axioms and Demonstrations 1787.4 Independence of the Axioms 1827.5 Development of the Calculus 1887.6 Deductive Completeness 201

B Alternative Systems and Notations 210

8.1 Alternative Systems of Logic 2108.2 The Hilbert-Ackermann System 211

8.3 The Use of Dots as Brackets 227

8.4 A Parenthesis-Free Notation 231

8.5 The Stroke and Dagger Operators 2328.6 The Nicod System 234

xii 9 A First-Order Function Calculus 242

9.1 The New Logistic System RS 1 242

9.2 Development of RS I 2489.3 Duality 2549.4 RS I and the ',Natural Deduction' Techniques 2599.5 Normal Forms 2639.6 Completeness of RS I 2709.7 RS I with Identity 280

Contents

Appendix A: Normal Forms and Boolean Expansions 283

Appendix B: The Algebra of Classes 290

Appendix C: The Ramified Theory of Types 300

Solutions to Selected Exercises 311

Special Symbols 337

Index 339

Introduction:

Logic and Language

Introduction: Logic and Language

Sec. 1.2] The Nature of Argument

from the others, which are regarded as grOlmds for the truth of that one. In

ordinary usage the word 'argument' also has other meanings, but in logic it

has the technical sense explained. In the following chapters we will use theword 'argument' also in a derivative sense to refer to any sentence or collectionof sentences in which an argument is formulated or expressed. When we dowe will be presupposing that the context is sufficiently clear to ensure that

unique statements are made or unique propositions are asserted by the utter-

ance of those sentences.

Every argument has a structure, in the analysis of which the terms 'premiss'and 'conclusion' are usually employed. The conclusion of an argument is that

proposition which is affirmed on the basis of the other propositions of the

argument, and these other propositions which are affirmed as providinggrounds or reasons for accepting the conclusion are the premisses of that

argument.We note that 'premiss' and 'conclusion' are relative tenns, in the sense that

the same proposition can be a premiss in one argument and conclusion in

another. Thus the proposition All men are mortal is premiss in the argument

Introduction: Logic and Language

See. 1.4]

Introduction: Logic and Language [Ch. 1

complicates our problem. Argwnents formulated in English or any other

natural language are often difficult to appraise because of the vague and

equivocal nature of the words in which they are expressed, the ambiguityof their construction, the misleading idioms they may contain, and their

pleasing but deceptive metaphorical style. The resolution of these difficultiesis not the central problem for the logician, however, for even when they are

resolved, the problem of deciding the validity or invalidity of the argwnentremains.

To avoid the peripheral difficulties connected with ordinary language,workers in the various sciences have developed specialized technical vocabu-laries. The scientist economizes the space and time required for writing his

reports and theories by adopting special symbols to express ideas which wouldotherwise require a long sequence of familiar words to formulate. This hasthe further advantage of reducing the amount of attention needed, for when

a sentence or equation grows too long its meaning is more difficult to grasp.The introduction of the exponent symbol in mathematics permits the expres-sion of the equation

See. 1.4J

Arguments ContainingCompound Statements

See. 2.1J

Arguments Containing Compound Statements

Sec. 2.1)

Arguments Containing Compound Statements

Sec. 2.1J

Arguments Containing Compound statements

See. 2.2]

Arguments Containing Compound Statements

See. 2.2J

Arguments Containing Compound Statements

See. 2.3J

Arguments Containing Compound Statements [CII. 2

and conclusions have different truth values, by considering all possible ar-

rangements of truth values for the statements substituted for the distinctstatement variables in the argument form to be tested. These can be set forthmost conveniently in the form of a truth table, with an initial or guide columnfor each distinct statement variable appearing in the argument form. Thusto prove the validity of the Disjunctive Syllogism form

See. 2.3]

and the

Arguments Containing Compound Statements

See. 2.3]

Arguments Containing Compound Statements

Sec. 2.4J

Arguments Containing Compound Statements

See. 2.4]

Arguments Containing Compound Statements

Sec. 2.4]

The Method of Deduction

See. 3.1J Formal Proof of 'aUdi"

Disjunctive Syllogism. From the second and fourth premisses, -C :> (D :> E)and -C, we validly infer D :> E by Modus Pooens. And finally, from theselast two conclusions (or subconclusions), B :> D and D :> E, we validly inferB :> E by a Hypothetical Syllogism. That its conclusion can be deduced fromits premisses using valid arguments exclusively proves the original argumentto be valid. Here the elementary valid argument forms Modus Pooens (M.P.),Modus Tol/ens (M.T.), Disjunctive Syllogism (D.S.), and Hypothetical Syllogism(H.S.) are used as Rules of Inference by which conclusions are validly deducedfrom premisses.

'

A more formal and more concise way of writing out this proof of validityis to list the premisses and the statements deduced from them in one column,with the latter's "justifications" written beside them. In each case the "justifi-cation" for a statement specifies the preceding statements from which, and

the Rule of Inference by which, the statement in question was deduced. It, is convenient to put the conclusion to the right of the last premiss, separated

from it by a slanting line which automatically marks all of the statements

above it to be premisses. The formal proof of validity for the given argumentcan be written as

The Method of Deduction

See. 3.1J

*1. (A ::> -B)'(-C ::> D)".A::>-B

The Method of Deduction

See. 3.1)

7. (-HvI)::> (J::> K)(-L'-M) ::> (K ::> N)(H ::> L)'(L ::> H)(-L'-M)'-O... J::> N

The Method of Deduction

See. 3.2J

The Method of Deduction

14. Double Negation (D.N.):

15. Transposition (Trans.):

16. Material Implication (Impl.):17. Material Equivalence (Equiv.):

See. 3.2]

The Method of Deduction

Sec. 3.2)

9. (G v H)'(I v J).'. [(G v H)' 1] v [(G v H)'J]

*10. (K'L) � {M'[(N'O)'P]}.'. (K'L) � {M'[(O.N).P]}

11. -{Qv -[(R'-S)'(Tv -U)]}.'. -{Qv [-(R'-S) v -(Tv -U)]}

12. -V � {W � [-(X'Y) � -Z]}.'. -V � ([W'-(X'Y)) � -Z}

13. [A v (B v C)) v [(D v D) v E). '. [A v (B v C)) v [D v (D v E))

14. (F � G)'{((G � H)'(H � G)) � (H � In.'. (F � G)'{(G = H) � (H � In

*15. J = -(((K'-L) v -M)'[(K'-L) v -N]}.'. J = -{(K'-L) v (-M'-Nn

16. 0 � [(P'-Q) = (P'--R)].'. 0 � [(P'-Q) = (--P'--R))

17. -S = {--T � [---Uv (-T'S)]}.'. -5 = {---Tv [---Uv (-T'S)]}

18. V� {(-W � --X) v [(-Y � Z)v(-Z � -Y)]}.'. V � {(-X � W) v [(-Y � Z) v (-Z � -Y)]}

19. (A'-B) � [(C'C) � (C � D)).'. (A'-B) � [C � (C � D))

20. (E'-F) � [G � (G � H)].'. (E'-F) � [(G'G) � H))

The Melhod of Deduction

See. 3.2)

11. E ::> F

E::>G... E::> (F.G)

12. H ::> (I v J)-I

".H::>/13. (K v L) ::> -(M. N)

(-Mv -N) ::> (0 = P)(0 = P) ::> (Q.R)... (Lv K) ::> (R.Q)

14. S ::> T

Sv T

".T

*15. (-Uv V)'(Uv W)-x::> -W

".vvx

16. A ::> (B ::> C)C::> (D. E)... A::> (B::> D)

The Method of Deduction

See. 3.3]

The Method of Deduction

Sec. 3.4]

10. W = (Xv Y)X = (Z ::> Y)Y = (Z = -A)Z = (A ::> B)A = (B = Z)Bv-W... W = B

The Method of Deduction

See. 3.4] Incompleteness of the Nineteen Rules

p q r p=>q q=>r p => r

0 0 0 0 0 00 0 1 0 1 10 0 2 0 2 2

0 1 0 1 0 00 1 1 1 1 10 1 2 1 1 2

0 2 0 2 0 00 2 1 2 0 10 2 2 2 0 2

1 0 0 0 0 01 0 1 0 1 11 0 2 0 2 11 1 0 1 0 01 1 1 1 1 11 1 2 1 1 11 2 0 1 0 01 2 1 1 0 11 2 2 1 0 12 0 0 0 0 02 0 1 0 1 02 0 2 0 2 02 1 0 0 0 02 1 1 0 1 02 1 2 0 1 02 2 0 0 0 02 2 1 0 0 02 2 2 0 0 0

Only in the first, tenth, nineteenth, twenty-second, twenty-fifth, twenty-sixth,and twenty-seventh rows do the two premisses 'p => q' and 'q => r' have thevalue 0, and in each of them the conclusion 'p => r' has the value 0 also. Even

larger tables would be needed to show that having the value 0 is hereditarywith respect to Constructive Dilemma and Destructive Dilemma, but theyare easily constructed. (It is not absolutely necessary to construct them,however, because the alternative analytical definitions on page 48 can be usedto show that having the value 0 is hereditary with respect to the Dilemmas,

49as on page 50.)'When we construct three-valued tables to verify that having the value 0

is hereditary with respect to replacement of statements by their logicalequivalents, we notice that although the biconditionals themselves need not

have the value 0, the expressions flanking the equivalence sign necessarilyhave the same value. For example, in the table appropriate to the first ofDe Morgan's Theorems,

The Method of Deduction [th.3

P q -p -q p.q -(p.q) -p v -q -(p.q) = (-pv -q)0 0 2 2 0 2 2 00 1 2 1 1 1 1 10 2 2 0 2 0 0 01 0 1 2 1 1 1 1

1 1 1 1 1 1 1 1

1 2 1 0 2 0 0 02 0 0 2 2 0 0 02 1 0 1 2 0 0 02 2 0 0 2 0 0 0

See. 3.5]

The Method of Deduction

See. 3.6J

The Method of Deduction

See. 3.6]

The Method of Deduction

See. 3.7]

The Method of Deduction

See. 3.8J

The Method of Deduction

See. 3.8] The Strengthened Rule of Conditional Proof

I. (A v B) ::> [(C v D) ::> E] /:. A ::> [(C' D) ::> E]2. A

3. AvB4. (Cv D) ::> E

p: rD8. E

9. (C, D) :J E

10. A::> [(C' D) :J E]

TIle Method of Deduction

See. 3.9] Shorter Truth Table Technique-Reductio Ad Absurdum Method

This method of proving the validity of an argument is a version of the reductioad absurdum technique, which uses truth value assignments rather than Rulesof Inference.

It is easy to extend the use of this method to the classification of statements

(and statement forms). Thus to certify that Peirce's Law [(p ::> q) :> p] ::> Pis a tautology, we assign it the truth value F, which requires us to assign T

to its antecedent [(p :> q) ::> p] and F to its consequent p. For the conditional

[(p ::> q) ::> p] to be true while its consequent p is false, its antecedent (p :> q)must be assigned the truth value F also. But for the conditional p :> q to be

false, its antecedent p must be assigned T and its consequent q assigned F.

However, we were previously forced to assign F to p, so assuming Peirce'sLaw false leads to a contradiction, which proves it a tautology.

If it is possible to assign truth values consistently to its components on the

assumption that it is false, then the expression in question is not a tautology,but must be either contradictory or contingent. In such a case we attemptto assign truth values to make it true. If this attempt leads to a contradictionthe expression cannot possibly be true and must be a contradiction. But iftruth values can be assigned to make it true and (other) truth values assignedto make it false, then it is neither a tautology nor a contradiction, but is

contingent.The reductio ad absurdum method of assigning truth values is by far the

quickest and easiest method of testing arguments and classifying statements.

It is, however, more readily applied in some cases than in others. If F is

assigned to a disjunction, F must be assigned to both disjuncts, and whereT is assigned to a conjunction, T must be assigned to both conjuncts. Herethe sequence of assignments is forced. But where T is assigned to a disjunctionor F to a conjunction, that assignment by itself does not determine which

disjunct is true or which conjunct is false. Here we should have to experimentand make various 'trial assignments', which will tend to diminish the advantageof the method for such cases. Despite these complications, however, in thevast majority of cases the reductio ad absurdum method is superior to anyother method known.

Propositional Functionsand Guantifiers

See. 4. t]

Propositional Functions and Quantifiers

Then we can use the notation already introduced to rewrite it as

Sec. 4.1)

Propositional Functions and Quantifiers

See. 4.1)

Propositional Functions and Quantifiers

See. 4.1]

Propositional Functions and Quantifiers

See. 4.2] Proving Validity: Preliminary Quantification Rules

letter 'y' will be used to denote any arbitrarily selected individual. In this

usage the expression '«Py' is a substitution instance of the propositional hmction'«Px', and it asserts that any arbitrarily selected individual has the property«P. Clearly '«Py' follows validly from '(x)«Px' by UI, since what is true of all

individuals is true of any arbitrarily selected individual. The inference is

equally valid in the other direction, since what is true of any arbitrarilyselected individual must be true of all individuals. We augment our list ofRules of Inference further by adding the principle that the universal quantifi-cation of a propositional hmction can validly be inferred from its substitution

instance with respect to the symbol 'y'. Since this rule permits the inferenceof general propositions that are universal quantifications, we refer to it as the

'principle of Universal Generalization', and abbreviate it as 'UG'. OUT symbolicexpression for this second quantification rule is

Propositional Functions and Quantifiers

See. 4.2]

5. Cw6. Dw

7. CWo Dw

8. (3x)[Cx. Dx]

Propositional functions and Quantifiers

See. 4.2]

Propositional Functions and Quantifiers

Sec. 4.3]

Propositional Functions and Quantifiers

See. 4.3]

Propositional Functions and Quantifiers

See. 4.4)

Propositional Functions and Quantifiers

See. 4.4]

Propositional Functions and Quantifiers [Ch. 4

propositional function with respect to the variable 'x'" or "the universal (orexistential) quantification of a propositional function with respect to thevariable 'y'" and so on.

It should be clear that since '(x)[Fx :J Gx)' and '(y)[Fy :J Gy)' are alternativetranslations of the proposition 'Everything which is an F is also a G', the

universal quantification of 'Fx :J Gx' with respect to 'x' has the same meaningand is logically equivalent to the universal quantification with respect to 'y'of the propositional function which results from replacing all free occurrences

of 'x' in 'Fx :J Gx' by 'y'-for the result of that replacement is 'Fy :J Gy'.In the early stages of our work it will be desirable to have at most one

quantification with respect to a given variable in a single proposition. Thisis not strictly necessary, but it is helpful in preventing confusion. Thus thefirst multiply general proposition considered, 'If all dogs are carnivorous thensome animals are carnivorous', is more conviently symbolized as '(x)[Dx :J

Cx] :J (3y)[Ay.Cy)' than as '(x)[Dx :J Cx] :J (3x)[Ax.Cx)', although neitheris incorrect.

It has been remarked that no proposition can contain a free occurrence

of any variable. Hence in symbolizing any proposition we must take care that

every occurrence of every variable used lies within the scope of a quantifierwith respect to that variable. Some examples will help to make the matter

clear. The proposition

See. 4.4J

Propositional Functions and Quantifiers

Sec. 4.5]

Propositional Functions and Quantifiers

Sec.4.5]

Propositional Functions and Quantifiers

Sec. 4.5]

Propositional Functions and Quantifiers

Sec.4.5]

Propositional Functions and Quantifiers

See. 4.5J

Propositional Functions and Quantifiers [Ch. 4

CPv for use in applying EI where (31L)CPIL is '(3y)(Fx = -Fy)'. It should beobvious that the �g!m.l��_t ,�,� !n'lalid: it fails for a model containing some thingsthat are F and some things that are not F, which would make the premisstrue, whereas the conclusion is false for every model, being self-contradictory.

See. 4.5]

Propositional Functions and Quantifiers

.6_ 1. (3x)(y)[(Fx-Gx) ::> Hy] /:. (3x)[(Fx-Gx) ::> Hx]2. (y)[(Fz-Gz) ::> Hy]3. (Fz-Gz) ::> Hy 2, UI4. (3x)[(Fx-Gz) ::> Hy] 3, EG5. (y)(3x)[(Fx-Gy) ::> Hy] 4, UG

6. (y)(3x)[(Fx-Gy) ::> Hy] 1,2-5, EI7. (3x)[(Fx-Gx) ::> Hx] 6, UI

7_ 1. (3x)Fx2. (3x)Gx /:. (3x)(Fx-Gx)3. Fy4. Gy5. Fy-Gy 3,4. Conj.6. (3x)(Fx-Gx) 5, EG

7. (3x)(Fx-Gx) 2,4-6, EI

8. (3x)(Fx-Gx) 1,3-7, EI

8_ 1. (3x)(3y)[(Fx v Gy)- Hy] /... (x)(y)(Fy v Gx)2. (3y)[(Fxv Gy) - Hy]

[3. (Fxv Gx)-Hx

_

4. Fx v Gx 3, Simp.5. Fx v Gx 2, 3-4, EI

6. Fx v Gx 1, 2-5, EI7. (y)(Fy v Gx) 6, UG8. (x)(y)(Fy v Gx) 7, UG

.9_ 1. (3x)(Fx-Gx)2. (3x)(-Fx-Gx) /.'.(3x)(Fx--Fx)

r3.Fx -Gy

4. Fx 3, Simp.5. Fx 1, 3-4, EI

r6.-Fx -Gx

7. -Fx 6, Simp.8. -Fx 2,6-7, EI9. Fx--Fx 5,8, Conj.

10. (3x)(Fx--Fx) 9, EG

10_ 1. (x)[(Fx::> Gx)--Ga] /.'. (x)-Fx2. (x)[(Fx::> Gx)--Gy]3. (Fz::> Gz)--Gy 2, UI4. (y)[(Fy ::> Gy)--Gy] 3, UG

100 5. (Fu ::> Gu)--Gu 4, UI6. Fu ::> Gu 5, Simp.7. -Gu - (Fu ::> Gu) 5, Com.8. -Gu 7, Simp.9. -Fu 6,8, M.T.

10. (x)-Fx 9, UG

n. (x)[(Fx::> Gx)--Gy] ::> (x)-Fx 2-10, C.P.

See. 4.5J

12. (w){(x)[(Fx:> Gx)'-Gw) :> (x)-Fx)13. (x)[(Fx:> Gx)'-Ga):> (x)-Fx14. (x)-Fx

Propositional Functions and Quanlifiers

Sec.4.5]

7. (x)(Qx::> Rx)(x)(Sx ::> Tx).'. (x)(Rx ::> Sx) ::> (y)(Qy ::> Ty)

8. (3x)Ux ::> (y)[(Uy v Vy) ::> Wy](3x)Ux.(3x)Wx.'. (3x)(Ux. Wx)

9. (3x)Xx ::> (y)(Yy ::> Zy)... (3x)(Xx. Yx) ::> (3y)(Xy. Zy)

Propositional Functions and Quantifiers

See. 4.6J Logical Truths Involving Quantifiers

In demonstrating the logical truth of propositions involving quantifiers, we

shall have to appeal not only to the original list of elementary valid argumentforms and the strengthened principle of Conditional Proof, but to our quanti-fication rules as well. Thus a demonstration of the logical truth of the proposi-tion '(x)Fx ::> (3x)Fx' can be set down as

Propositional Functions and Quantifiers

Sef' 4.6] Logical Truths Involving Quantifiers

(x)Fx v (x)Gx� ; \Px

Propositional Functions and Quantifiers [Ch. 4

expressions written above, the propositions 'Ga' and '(z)Hz', and the proposi-tional functions 'Fy' and 'Gw', although lying within the scopes of the quanti-fiers '(x)', '(3y)', '(x)' and '(z)', respectively, are not really affected by them.Wherever we have an expression containing a quantifier on the variable /Lwithin whose scope lies either a proposition or a propositional function not

containing any free occurrence of p" the entire expression is logically equiva-lent to another expression in which the scope of the quantifier on p, does not

extend over that proposition or propositional function. An example Or two

will make this clear. In the following, let 'Q' be either a proposition or a

propositional function containing no free occurrence of the variable 'x', andlet 'Fx' be any propositional function containing at least one free occurrence

of the variable 'x'. Our first logical equivalence here is between the universal

quantification of 'Fx' Q' and the conjunction of the universal quantificationof 'Fx' with 'Q', which is more briefly expressed as

See. 4.6]

110

Sec.4-&]

The Logic of Relations

See. 5.1J

The Logic of Relalions

so we have the relation word 'taught' common to the propositions:

see. 5.IJ

6. Lht::J Gt7. Gt

The Logic of Relations

See. 5.1)

The Logic of Relations

A similar pair of inequivalent propositions may be written as

See. 5.1]

1. (x)[Vx::> (3y)Oyx]2. (x)( Oxa ::> Rxa]3. (x)--Rxa /... -- Va

4. Oza::> Rza 2, UI

5. --Rza 3, UI

6. --Oza 4,5, M.T.7. (y)--Oya 6, UG8. --(3y)Oya 7, QN9. Va::> (3y)Oya 1, UI

10. -- Va 9, 8, M.T.

The Logic of Relations

See. 50lJ

The Logic of Relations

�. 5.1)

tile Logic of Relations

*15. (x){[Wx'(y)[Py :> -(3z)(Nz' pxzy)]] :> Ix}16. (x){[Px'(y)(-Vxy)] :> (z)(-Cx.:)}17. (x){Vx:> (y)[Xy :> (3z)[(lz'Bzx)'-Dzy]])18. (x){[Lx'(3y)(Py' Eyx)) :> (z)(Wz :> Tgzx)}19. (x){Px:> (3y)[Py'(3z)(Bxzy)])

*20. (x){Px:> (3y)[Py.(3z)(-Bxzy)])21. (x){Px :> (y)[Py :> (z)( -Bxzy)]}22. (x){Px :> (y)[Py :> (3z)( -Bxzy)]}23. (x)[(Nx'Dx) :> (y)(Lxy :> Myx))24. (x)[Px:> (3y)(Py'Xyx)).(3u)[Pu'(v)(Pv :> -Xuv))25. (x){[Qx,(y){[(py'Wyx)'(z)(-Kyz)) :> By}] :>

(u){[(Pu'Wux)'(v)(Kuv)) :> Ou}}

See. 5.1]

The Logic of Relations

4. A wise son maketh a glad father. (Proverbs 10:1)*5. He that spareth his rod hateth his son. (Proverbs 13:24)

6. The borrower is servant to the lender. (Proverbs 22:7)7. Whoso diggeth a pit shall fall therein: and he that rolleth a stone, it will

return upon him. (Proverbs 26:27)8. The fathers have eaten sour grapes, and the children's teeth are set on edge.

(Ezekiel 18:2)9. The foxes have holes, and the birds of the air have nests; but the Son of

man hath not where to lay his head. (Matthew 8:20)10. '" the good that I would I do not; but the evil which I would not, that

I do. (Romans 7:19)

Sec. 5.2)

The Logic of Relations

See. 5.2)

7. (x)[Mx::> (y)(Ny ::> Oxy)](x)[Px ::> (y)(Oxy ::> Qy)].' . (3x)(Mx' Px) ::> (y)(Ny ::> Qy)

8. (x)[(Rx' -Sx) ::> (3y)(Txy' Uy)](3x)[Vx'Rx'(y)(Txy ::> Vy))(x)(Vx ::> -Sx).'. (3x)(Vx' Ux)

9. (x)(Wx::> Xx)(x)[(YX' Xx) ::> Zx](x)(3y)(Yy' Ayx)(x)(y)[(AyX' Zy) ::> Zx]

. '. (x)[(y)(Ayx ::> Wy) ::> Zx]10. (x){[BX'(3y)[Cy'DyX'(3z)(Ez'Fxz))) ::> (3w)Gxwx)

(x)(y)(Hxy ::> Dyx)(x)(y)(Fxy ::> Fyx)(x)(Ix ::> Ex)... (x){Bx::> ([(3y)(Cy'Hxy).(3z)(Iz'Fxz)] ::> (3u)(3w) GxwuJ}

The Logic of Relations

See. 5.3]

132

Sec. 5.3]

134

See. 5.3J

The logic of Relations

Sec. 5.4)

138

Sec. 5.4J

for this would entail

140

See. 5.4]

Finally, the (presumably false) statement

The Logic of Relations

See. 5.4)

144

See. 5.5]

146

See. 5.5]

148

See. 5.5]

The Logic 0' Relations [Ch. 5

uals by boldface italic capital letters 'A', 'B', 'C', . . . , to prevent their beingconfused with attributes of individuals. With this additional symbolic appara-tus we can translate into our notation such propositions as 'Unpunctualityis a fault', and 'Truthfulness is a good quality'. Here we use 'Ux', 'Tx', 'FF',and 'GF' to abbreviate 'x is unpunctual', 'x is truthful', 'F is a fault', and 'Fis good', and symbolize the two stated propositions as 'FU' and 'GT'. More

complex propositions can also be symbolized. The propositions

See. 5.5]

Deductive Systems

See. 6.2J

154

See. 6.2J

Deductive Systems

See. 6.3J

158

See. 6.4]

160

See. 6.5]

162

Sec. 6.5]

Deductive Systems

A Propositional Calculus

A Propositional Calculus

See. 7.2]

A Propositional Calculus [CII. 7

Bl(A). (A)-(D)-(- )-((AI)' (C a »B�aA7-( )( ).( ))))( (

See. 7.2]

A Propositional Calculus

See. 7.2]

A Propositional Calculus

See. 7.2] Primitive Symbols and Well formed formulas

in R.S. in its intended or normal interpretations is easily seen. First we note

that the intended or normal interpretations of -P and PQ are given by thetruth tables

P -Pand

P Q P�T F T T T

F T T F F

F T F

F F F

That RS. is adequate to express Ul!), f2(l!),f3(F), and fiP) is proved by actuallyformulating them in RS. The function f2(P) is true when P is true and falsewhen P is false, and is therefore expressible in RS. as P itself. The function

fl (F) is false when P is true and true when P is false, and is therefore expressiblein RS. as - P. The function f3(P) is false nO matter which truth value P assumes,and is therefore expressible in RS. as -PP. The function f4(l!) is true in everycase and can therefore be expressed in RS. as the negation of f3(l!), that is,as -(-Pl!). We have thereby shown that all singulary truth functions are

expressible in RS.

There are, of course, more truth functions of two arguments than of one

argument. These are defined by the following truth tables:

P Q fI(P.Q) P Q f2(P.Q) P Q f3(P.Q) P Q f4(P.Q)T T F T T T T T T T T T

T F T T F F T F T T F T

F T T F T T F T F F T T

F F T F F T F F T F F F

P Q f5(P.Q) P Q f6(P.Q) P Q f7(P.Q) P Q fs(P.Q)T T F T T F T T F T T T

T F F T F T T F T T F F

F T T F T F F T T F T F

F F T F F T F F F F F T

P Q f9(P.Q) P Q fIo(P.Q) P Q fu(P.Q) P Q fdP.Q)T T T T T T T T F T T FT F F T F T T F F T F F 173F T T F T F F T F F T TF F F F F F F F T F F F

P Q fI3(P.Q) P Q f14(P.Q) P Q f15(P.Q) P Q f16(P.Q)T T F T T T T T F T T T

T F T T F F T F F T F T

F T F F T F F T F F T T

F F F F F F F F F F F T

174

See. 7,2J

176

See. 7.2]

178

See. 7.3]

180

See. 7.3]

A Propositional Calculus

See. 7.4)

A Propositional Calculus [Ch.7

(P Q) :> P

0 0 0 0 0

0 1 1 0 00 2 2 0 01 1 0 0 1

1 2 1 0 11 2 2 0 12 2 0 0 2

2 2 1 0 22 2 2 0 2

(P :> Q) :> [- (Q R) :> (R P)]0 0 0 0 2 0 0 0 0 2 0 0 00 0 0 0 1 0 1 1 0 1 1 1 0

0 0 0 0 0 0 2 2 0 0 2 2 00 1 1 0 1 1 1 0 1 2 0 0 0

0 1 1 0 0 1 2 1 1 1 1 1 0

0 1 1 0 0 1 2 2 0 0 2 2 0

0 2 2 0 0 2 2 0 2 2 0 0 0

0 2 2 0 0 2 2 1 1 1 1 1 00 2 2 0 0 2 2 2 0 0 2 2 01 0 0 0 2 0 0 0 0 1 0 1 1

1 0 0 0 1 .0 1 1 0 0 1 2 1

1 0 0 0 0 0 2 2 0 0 2 2 11 0 1 0 1 1 1 0 0 1 0 1 1

1 0 1 0 0 1 2 1 0 0 1 2 1

1 0 1 0 0 1 2 2 0 0 2 2 1

1 1 2 0 0 2 2 0 1 1 0 1 1

1 1 2 0 0 2 2 1 0 0 1 2 1

1 1 2 0 0 2 2 2 0 0 2 2 1

2 0 0 0 2 0 0 0 0 0 0 2 2

2 0 0 0 1 0 1 1 0 0 1 2 2

2 0 0 0 0 0 2 2 0 0 2 2 2

2 0 1 0 1 1 1 0 0 0 0 2 2

2 0 1 0 0 1 2 1 0 0 1 2 2

2 0 1 0 0 1 2 2 0 0 2 2 2

1842 0 2 0 0 2 2 0 0 0 0 2' 2

2 0 2 0 0 2 2 1 0 0 1 2 2

2 0 2 0 0 2 2 2 0 0 2 2 2

The characteristic is easily seen to be hereditary with respect to R 1 ofR.S. by consulting the table given for ':>'. In the only row in which bothP and P :> Q have the value 0, Q also has the value O. Hence if the charac-teristic belongs to one or more wjJs it also belongs to every wjJ deduced fromthem by R 1.

Finally it is readily seen that the characteristic in question does not belong

See. 7.4)

A Propositional Calculus [Ch.7

P :> (P P)0 0 0 0 01 0 1 0 12 0 2 2 2

(P :> Q) :> [- (Q R) :> (R P)]0 0 0 0 2 0 0 0 0 2 0 0 0

0 0 0 0 2 0 0 1 0 2 1 0 0

0 0 0 0 0 0 2 2 0 0 2 2 00 2 I 0 2 1 0 0 0 2 0 0 00 2 1 0 2 I 0 1 0 2 1 0 00 2 I 0 0 1 2 2 0 0 2 2 00 2 2 0 0 2 2 0 2 2 0 0 0

0 2 2 0 0 2 2 1 2 2 1 0 00 2 2 0 0 2 2 2 0 0 2 2 0

1 0 0 0 2 0 0 0 0 2 0 0 I

I 0 0 0 2 0 0 1 0 2 1 0 1

1 0 0 0 0 0 2 2 0 0 2 2 I

1 2 1 0 2 I 0 0 0 2 0 0 I

1 2 1 0 2 1 0 1 0 2 1 0 1

I 2 I 0 0 1 2 2 0 0 2 2 1

I 2 2 0 0 2 2 0 2 2 0 0 1

1 2 2 0 0 2 2 1 2 2 1 0 1

1 2 2 0 0 2 2 2 0 0 2 2 12 0 0 0 2 0 0 0 0 0 0 2 22 0 0 0 2 0 0 1 0 0 1 2 22 0 0 0 0 0 2 2 0 0 2 2 2

2 0 1 0 2 1 0 0 0 0 0 2 22 0 1 0 2 1 0 1 0 0 1 2 2

2 0 I 0 0 1 2 2 0 0 2 2 2

2 0 2 0 0 2 2 0 0 0 0 2 2

2 0 2 0 0 2 2 1 0 0 1 2 2

2 0 2 0 0 2 2 2 0 0 2 2 2

186The characteristic is easily seen to be hereditary with respect to; R 1 of

R.S. by consulting the table for ':>'. In the only roW in which both P andP :> Q have the value 0, Q also has the value O. Hence if the characteristic

belongs to one or more wffs it also belongs to every wff deduced from them

by R 1.

Finally, it is readily seen that the characteristic in question does not belongto Ax. 2. When P and Q are both assigned the value 1, (P- Q) :> P has thevalue 2 rather than 0 for (I-I) :> 1 is 0 :> I which is 2. Hence Ax. 2 is

independent.

See. 7.4) Independence of the Axioms

To prove the independence of Ax. 3 of RS. we use the same three-elementmodel and the same table for '-P'. The difference lies in the table for 'poQ',which follows, along with the derivative table for 'P :> Q'.

188

See. 7.5]

190

See. 7.5]

192

Sec. 7.5]

A Propositional Calculus

See. 7.5]

*DR 6. p:J Q, Q :J R I- P :J R

A Propositional Calculus

See. 7.5]

A Propositional Calculus

See. 7.5]

DR 15. P = QI--P = -QDR 16. P = Q, R = 5 I- PR = Q5DR 16, COR. P = Q, R = 5 I- P v R = Q v 5

A Propositional Calculus

See. 7.6] Deductive Completeness

TH. 26. f- P = pp (Tautology)TH. 26, COR. f-p = pvP (Tautology)TH. 27. f- --(PQ) = (--Pv --Q) (De Morg�'s Theorem)

TH. 28. f- --(Pv Q) = (--P--Q) (De Morgan's Theorem)TH. 29. f- (P::> Q) = (--Pv Q) (Material Implication)

*TH. 30. f-P(Qv R) = PQv PR (Distribution of '.' over V)TH. 30, COR. f- (P v Q)R = PR v QRTH. 31. f- (P = Q) = [PQ v --P--Q] (Material Equivalence)TH. 32. f- Pv QR = (Pv Q)(Pv R) (Distribution of V over ,.')

A Propositional Calculus

See. 7.6]

204

See. 7.6]

206

See. 7.6) Deductive Completeness

by DR 12 � Qj ::> -(Sl' S2)' which is I- Qj ::> -5. If to S2 then � Q; ::> -52by the f3-case assumption, and hence by DR 13 � Q; :> -(S1' S2)' which is

�Qj:> -So

A Propositional Calculus

See. 7.6]

Alternative Systems andNatations

See. 8.2)

212

Sec. 8.2]

Alternative Systems and Notations [Ch.8

P -p p Q PvQ P� Q0 2 0 0 0 0

1 1 0 1 0 1

2 0 0 2 0 21 0 0 01 1 0 01 2 1 12 0 0 0

2 1 1 02 2 2 0

To prove Postulate 2 independent we use the three-element model {O, 1, 2}of which 0 is designated, with the tables:

P -p P Q PvQ P�Q0 1 0 0 0 01 0 0 1 0 1

2 2 0 2 0 11 0 0 01 1 1 01 2 1 02 0 0 02 1 1 12 2 1 1

To prove Postulate 3 independent we use {O, 1, 2} with 0 designated andtables:

P -P P Q PvQ P�Q0 2 0 0 0 0

1 0 0 1 0 22 1 0 2 0 2

1 0 0 0

2141 1 1 01 2 0 02 0 0 02 1 2 1

2 2 2 0

To prove Postulate 4 independent we use the four-element model {O, 1,2, 3}with 0 designated and tables:

Set:. 8.2] The Hilbert-Ackermann System

p -p p Q PvQ p:J Q0 I 0 0 0 0I 0 0 I 0 I2 3 0 2 0 23 0 0 3 0 3

I 0 0 0I I I 0I 2 2 0I 3 3 02 0 0 02 I 2 32 2 2 02 3 0 33 0 0 0

3 I 3 03 2 0 0

3 3 3 0

Alternative Systems and Notations [Ch.8

P'QP Q PvQ -P -Q -Pv-Q -(-Pv -Q)0 0 0 5 5 5 0

0 1 0 5 5 5 00 2 3 5 4 5 0

0 3 3 5 1 0 50 4 0 5 0 0 50 5 0 5 0 0 5

1 0 0 5 5 5 0

1 1 0 5 5 5 0

1 2 3 5 4 5 0

1 3 3 5 1 0 5

1 4 0 5 0 0 51 5 0 5 0 0 52 0 3 4 5 5 0

2 1 3 4 5 5 0

2 2 3 4 4 5 0

2 3 3 4 1 0 52 4 3 4 0 0 52 5 3 4 0 0 53 0 3 1 5 0 53 1 3 1 5 0 53 2 3 1 4 0 5

3 3 3 1 1 0 53 4 3 1 0 0 53 5 3 1 0 0 5

4 0 0 0 5 0 54 1 0 0 5 0 54 2 3 0 4 0 5

4 3 3 0 1 0 5

4 4 5 0 0 0 54 5 5 0 0 0 55 0 0 0 5 0 55 1 0 0 5 0 55 2 3 0 4 0 55 3 3 0 1 0 5

5 4 5 0 0 0 5216 5 5 5 0 0 0 5

In this model the three elements 0, 1, 2 are designated. The characteristicof taking only designated values is hereditary with respect to the rule: From

P and _(po -Q) to infer Q: and the three H.A. formulations of the R.S. axioms

take only designated values. But for the value 2 for P, we have

Pv -P = 2 v -2 = 2 v 4 = 3 which is not a designated value. 2

2See Henry Hiz, "A Warning About Translating Axioms," American Mathematical Monthly,vol. 65 (1958), pp. 613 f.; Thomas W. Scharle, "Are Definitions Eliminable in Fonnal Systems"

Set:. 8.2]

Alternative Systems and Notations

THEOREM 4. mpv-p

Sec. 8.2J The Hilbert-Ackermann System

THEOREM 9. lux [(Pv Q) v R] ::> [Pv (Qv R)]

Alternative Systems and Notations

THE 0 REM 14. 1m: (PQ) ::> P

See. 8.2J The Hilbert-Ackermann System

THEOREM 16. luxp = (Pv P)

Proof: I. P::> (Pv P) P22. (Pv P) ::> P PI3. [P::> (P v P)][(P v P) ::> P] DR44. P = (Pv P) df.

THEOREM 17. IRA P = (PP)

Proof: I. -P = (-Pv-F) Th.I62. --p = -(-Pv-P) DR83. P = --P Th.ll4. P = -(-Pv-F) DR9

5. P = (PP) df.

DR 10. Q ::> R lux (Pv Q) ::> (R v P)

Proof: I. Q::>R premiss2. (Pv Q) ::> (Pv R) DR23. (Pv R) ::> (R v P) P34. (Pv Q) ::> (R vp) DR 1

DR II. P::> Q,R::> 8 Iux(PvR)::> (Qv8)

Proof: I.R::>8 premiss2. (P v R) ::> (8 v P) DR 10

3. P::> Q premiss4. (8 v P) ::> (Q v 8) DR 10

5. (Pv R) ::> (Q v S) DR 1 (2, 4)

DR 12. P = Q,R = 8 Iux(PvR) = (Qv8)

Proof: I.P = Q premiss2. (P::> Q)( Q ::> P) df.3. P::> Q DR6

4. Q ::> P DR7 2215. R = S premiss6. (R ::> 8)(S ::> R) df.

7. R ::> S DR6

8. S ::> R DR7

9. (PvR)::> (QvS) DR 11 (3, 5)10. (Q v S) ::> (Pv R) DR 11 (4, 8)II. [(PvR)::> (QvS)][(QvS)::> (PvR)] DR4

12. (PvR) = (QvS) df.

Afternative Systems and Notations

See. 8.2] The Hilbert-Ackermann System

THEOREM 18. \Hx(Pv Q) = (Qv P)

Proof: 1. (PvQ):> (QvP) P32. (QvP) => (PvQ) P3

3. [(Pv Q) => (Q v P)][(Q v p) => (Pv Q)] DR4

4. (Pv Q) = (Q vp) elf.

DR 13. P:> Q,P:> R luxp => (QR)

Proof: I.P=>Q premiss2. -Q:> -P DR5

3. P => R premiss4. -R :> -P DR5

5. (-Q v -R) => (-Pv-P) DRll

6. -(Pv -p) :> -(-Qv -R) DR5

7. (PI') :> (QR) elf.8. P:> (QR) MT I, Cor., Th. 17

THEOREM 19. hu [P v (QR)] => [(P v Q)(P v R)]

Proof 1. (QR) :> Q Th.142. [Pv (QR)] => (Pv Q) DR2

3. (QR) => R Th.154. [Pv (QR)] => (Pv R) DR2

5. [Pv (QR)J => [(Pv Q)(Pv R)] DR 13

DR 14. P => (Q:> R) huQ => (P => R)

Proof: 1. P => (Q => R) premiss2. -Pv (-Q v R) df.3. [-Pv(-QvR)] => [-Qv(-PvR)] Th.74. -Qv(-PvR) R'I

5. Q :> (P :> R) df.

DR 15. P => (Q :> R) IRA (PQ) :> R223

Proof: 1. P => (Q :> R) premiss2. -Pv (-Q v R) df.3. (-Pv -Q) v R MT I, Cor., Th. 12

4. --(-Pv-Q)vR MT I, Cor., Th. II

5. (PQ) :> R df.

224

See. 8.2] The Hilbert-Ackermann System

({3) Here the Metatheorem is assumed to be true for any k < n disjuncts PI'P 2 , . . . , P k . Now consider Q and R each constructed out of exactly n (> 1)disjWlcts PI' P 2 ,. .., Pn' Q is S v T and R is X v Y.

Each of the wffs Sand T contains at least one of the wffs Pi (1 :::; i :::; n).We can assume that PI is a disjunct of S, because if not we can use Th. 18and MT I, Cor. to obtain iHx Q = (S v T) where S now does contain PI as

a disjunct.Because T contains at least one of P 2 , P 3 , . . . , Pn as a disjunct, S contains

fewer than n of the disjuncts Pi' Hence either S is PI and tHA Q = (PI V T),or by the {3-case assumption fIci S = (Sl V S'), where S' is a wff that contains

all the disjuncts of S except Pl' In the latter case, by MT I, Cor. we have

226

See. 8.3J

228

See. 8.3]

Afternative Systems and Notations

See. 8.4)

Altemative Systems and Notations [Ch. 8

The Polish notation has the obvious advantage of dispensing with all specialpunctuation marks, for the order in which its symbols are written suffices to

make any formula unambiguous.

See. 8.5] The Stroke and Dagger Operators

its standard interpretation is to deny that either of the formulas P or Q is

true, which is the same as affirming that they are both false. It is defined bythe truth table

234

See. 8.6J

Alternative Systems and Notations

See. 8.6] The Nicod System

THEOREM 4. !NPIP.I.P

238

Sec. 8.6] The Nicod System

is Theorem 8 with Q in place of P, with PIP.I.P:.I:.Q/P.I.P:I:QIP.I.P in

place of Q and of R, and with Q:.I:.Q/P.I.P:I:QIP.I.P::I::Q:.I:.QIP.I.P: I :Q I P.I.P in place of S. Line 16 is the result of applying the Nicod Ruleto lines 12 and 15. Line 17 is the result of applying the Nicod Rule to

lines 14 and 16. Line 18 is Theorem 10 with QIP.I.P:I:QIP.I.P in placeof S. Line 19 is Theorem 3 with QI Q in place of S, and with Q:.I :.QIP-I.P:I :QIP.I.P::I ::Q:.I :.QIP.I.P: I :QIP.I.P in place of P. Line 20 is the resultof applying the Nicod Rule to lines 18 and 19. Line 21 is Theorem 8 with

Q:.I:.QIP.I.P:I:QIP.I.P::I::Q:.I:.QIP.I.P:I:QIP.I.P in place of P and of S,and with Q in place of R. Line 22 is the result of applying the Nicod Ruleto lines 20 and 21. Line 23 is the result of applying the Nicod Rule to lines17 and 22. Line 24 is Theorem 4 with Q:.I:.QIP.I.P:I:QIP.I.P::j::Q:.\:.QIP.I.P:I:Q/P.I.P in place of P. Line 25 is the result of applying theNicod Rule to lines 23 and 24.

240

See. B.6J The Nicod System

DR 1. P, PIP.I.PIP:/:QIQ IN Q (R' 1 of H.A.)

Proof: Line 1 is Theorem 9 with P in place of S, and with PIP in placeof P. Line 2 is the premiss P. Line 3 is the result of applying the Nicod Ruleto lines 1 and 2. Line 4 is the premiss PI P.I.PI P: I:Q I Q. Line 5 is the result

of applying the Nicod Rule to lines 3 and 4.

A First-Order Function

Calculus

Sec. 9.1] The New Logistic System RS I

3. Infinitely many capital letters from the first part of the alphabet, withand without subscripts, having right-hand superscripts '1', '2', '3'. . . .

244

Sec. 9.1]

246

Sec. 9.1)

248

Sec. 9.2J

250

Sec. 9.2]

A firsl-order Function Calculus

See. 9.2)

254

Sec. 9.3]

1. W: p.QW6: -Pv-Q

2. W: (x)(Pv Q)W6: (3x)(-P._Q)

3. W: (y)(3z)[Pv(-QvR.S)]W6: (3y)(z)[-P.Q.(-R v -S)]

A first-Order Function Calculus

Sec. 9.3]

The dual (and hence the negation) of this formula is

258

Sec. 9.4]

260

See. 9.4]

262

Sec. 9.5]

264.

See. 9.5)

266

See. 9.5) Nonnal Forms

Now the prenex normal form of (3t){ G- [D(t) ::> D(t)]} is the formula oftype R that was desired, for it is closed, is in prenex normal form, beginswith an existential quantifier, and I- F if and only if I- R. Where G is (Qxl)(Qx2) . . . (Qxn)G', then

I- (3tH G. [D(t) ::> D(t)]} or

I- (3tH [(Qxl)(Qx2) . . . (Qxn)G'J - -[D(t)- -D(t)]}

A First-Grder Function Calculus

Sec. 9.5)

270

See. 9.6) Completeness of RS I

I- F or I- -F. This kind of completeness is not desirable either, for on theirnormal interpretations, each of the following wffs

272

See. 9.6]

274

Sec. 9.6] Completeness of RS l

function calculus RS I is complete if and only if every valid cwff is provablein it as a theorem. To say that every valid cwffis provable in RS 1 as a theorem

is, by transposition, to say that for any cwff S, if S is not a tlieorem then S

is not valid.In defining the terms 'valid' and 'satisfiable' we remarked that for any wff

Seither 5 is valid or -S is satisfiable. Hence to say that S is not valid is

to say that -S is satisfiable. And so we can say that RS I is complete if and

only if, for any cwff S, if S is not a theorem then -S is satisfiable. We can

establish this result by introducing a characteristic cp such that both

A First-Grder Function Calculus

Sec. 9.6]

278

See. 9.6]

280

See. 9.7)

A First-Grder Function Calculus

APPENDIX

284

Normal Forms and Boolean Expansions

It is clear that by invoking the defining equivalences

286

Normal Forms and Boolean Expansions

Then we rearrange the terms by simply commuting or interchanging thesecond and third disjuncts, to get

288

Normal Forms and Boolean Expansions

Expansion represent all possible assignments of truth values to its variables.

Since the 2" disjuncts represent all possible assignments of truth values to its

variables, at least one of them must be true. And since it asserts only thatat least one of its disjuncts is true, any disjunctive Boolean Expansion contain-

ing n variables and 2" disjuncts is tautologous. This point is made in somewhatdifferent terms in Section 7.6 and again in Section 8.2.

It was pointed out in Chapter 2 that a truth-functional argument is validif and only if its corresponding conditional statement (whose antecedent is

the conjunction of the argument's premisses and whose consequent is the

argument's conclusion) is a tautology. Since counting the number of disjunctsof its disjunctive Boolean Expansion permits us to decide whether or not a

given form is a tautology, this provides us with an alternative method of

deciding the validity of arguments. Thus the argument form p v q, -p .'. qis proved valid by constructing the disjunctive Boolean Expansion of its

corresponding conditional [(p v q)'p] ::> q, and observing that the number ofits disjuncts is 2 2

.

Since the negation of a tautology is a contradiction, an argument is validif and only if the negation of its corresponding conditional is a contradiction.Hence another method of deciding the validity of an argument is to formthe conjunctive Boolean Expansion of the negation of its correspondingconditional and count the number of its conjuncts. H it contains n distinctvariables and has 2" conjuncts, then the argument is valid; otherwise it isinvalid.

290

The Algebra of Classes

292

The Algebra 01 Classes

a and {3: I = a{3 U ap U af3 U ap, and the product of any two of those four

products is the empty class. Similarly, any n classes will divide the universalclass into 2 11 subclasses which are exclusive and exhaustive. The class expressionwhich symbolizes such a division of the universal class, it should be observed,is a disjunctive Boolean Expansion. A disjunctive Boolean Expansion containingn different simple class terms designates the universal class if it is the sum

of 2 11 distinct products (where a mere difference in the order of their terms

does not make two products distinct). Disjunctive Boolean Expansions thus

provide us with a method for deciding whether or not any class expressiondesignates the universal class regardless of what classes are designated by the

simple class terms which it contains. Given any class expression, we need onlyconstruct its disjunctive Boolean Expansion and count the number of productsof which it is the sum.

A conjunctive Boolean Expansion is a product of distinct sums of simpleclass terms or their complements, where any simple class term which occurs

anywhere in the expression will occur exactly once in every sum. By De

Morgan's Theorem and the other equivalences already mentioned, the com-

plement of any disjunctive Boolean Expansion can be transformed into a

conjunctive Boolean Expansion which involves the same simple class terms

and which is the product of as many sums as the disjunctive Boolean Expansionis the sum of products. Since the complement of 1 is 0, a conjunctive Boolean

Expansion containing n different simple class terms designates the empty classif it is the product of 2 11 distinct sums. Hence we have a method for decidingwhether or not any class expression designates the empty class regardless ofwhat classes are designated by the simple class terms which it contains.

The notations introduced thus far permit the symbolization of the A andE subject-predicate propositions. The E proposition: No a is {3, asserts thatthe classes a and {3 have no members in common, which means that their

product is empty. The E proposition is therefore symbolized as

Appendix B

The 0 proposition: Some a is not fJ, asserts that there is at least one memberof a which is not a member of fJ, i.e., that the product of a and P is not

empty. In symbols, the 0 proposition is expressed as

The Algebra of Classes

not only to validate immediate inferences involving categorical propositions,but is capable of validating categorical syllogisms also.

The symbol' C'

for class inclusion is often used in working with the algebraof classes. The expression 'a C p' asserts that all members of a, if any, are

also members of p, and is used as an alternative symbolization of the A

proposition: All a is p. It can be defined in terms of the symbols alreadyintroduced in various ways: either as ap = 0 or as ap = a or as a U p = por as a U p = 1, all of which are obviously equivalent. The relation C is

reflexive and transitive (see pages 131-132) and has the (transposition) propertythat if a C p then pea. The latter is an immediate consequence of double

negation and commutation when 'a C p' is rewritten as 'ap = 0' and 'P C ii'

is rewritten as 'p& = 0'. Its reflexiveness is obvious when 'a C a' is rewrittenas 'aa = 0', and its transitivity has already been established in our algebraicproof of validity for categorical syllogisms containing only W1iversal proposi-tions.

The algebra of classes can be set up as a formal deductive system. Sucha system is called a Boolean Algebra, and a vast number of alternative postulatesets for Boolean Algebra have been proposed. One of them can be set forthas follows.

Special W1defined primitive symbols:

296

The Algebra of Classes

Class inclusion, equality, and inequality may be defined as follows:

298

The Algebra of Classes

(A U f) #: 1. And if it is logically true that n #: I, then it is not logicallytrue that n = I, from which it follows that the wff which designates IT is

not a provable theorem in R.S. Since we have an effective criterion for

distinguishing between theorems and nontheorems of R.S., we have therein

an effective criterion for recognizing logically true equations and inequalitiesof class algebra.

The preceding discussion should suffice to indicate the intimacy of the

connection between the algebra of classes and the propositional calculus.

APPENDIX

The Ramified Theory of Types

relation, so that's desigflates tf>' is symbolized as 'sDes<P', we begin with thedefinition

Appendix C

The Ramified Theory of Types .

according to the ramified theory of types, say that Bob has all of AI's goodqualities, which would ordinarily be symbolized as

Appendix C

but can say instead either that 'None of the first-order propositions uttered

by Smith tends to incriminate him', or that 'None of the second-order propo-sitions uttered by Smith tends to incriminate him', or etc. We would partiallysymbolize the second of these alternative propositions as

The Ramified Theory of Types

from which definition all of the usual attributes of the identity relation can

be deduced. But that definition violates the ramified theory of logical types,since in it reference is made to all functions of type 1. Were we to replaceit by the definition

Appendix C

presence of the semantical terms the paradoxes do not seem to be derivableeven with the aid of the Axiom of Reducibility.lO

It may not be out of place here to indicate briefly how the 'levels of

language' method of avoiding the semantical paradoxes ll is remarkably similarto the ramified type theory's hierarchy of orders. 12 Confining our remarks to

the Grelling paradox, we note that it does not arise in an object language(like the extended function calculus, for example) when we assume that thereare in it no symbols which designate symbols. Nor does it arise in the meta-

language of that object language. Since the metalanguage contains synonymsfor all symbols of the object language and 'names for all symbols of the objectlanguage, as well as its own variables and the name relation (which we write

as 'Des'), the symbol 'Het' can be defined in it. By definition:

The Ramified Theory of Types

In the first place, the meta-metalanguage contains two symbols for the name

relation, 'Des l

'

and 'Des z '.13 The first of these is the meta-metalanguage'ssynonym for the name relation in the metalanguage. The full sentence.

308

The Ramified Theory of Types

This is very like the theory of orders, because the contradiction is evaded

by arranging that certain symbols of the meta-metalanguage are defined over

certain ranges. Thus 'Des 1'

is defined over a narrower range than 'Des2', and'Het 1

'

is defined over a narrower range than 'Het 2 '; Des 1 and Het 1 beingsatisfied only by symbols of the object language, Des2 and Het 2 being satisfiedonly by symbols of the metalanguage, which is a wider and more inclusivelanguage. Not only is the levels of language theory remarkably analogous tothe theory of orders, but where each metalanguage is conceived as actuallycontaining the object language with which it deals,14 it can be identified withthe Russellian theory of orders as applied to symbols rather than to thefunctions they denote.

In spite of the indicated similarities, there are fundamental differencesbetween the two. Most significant is that unlike the ramified type theory, thelevels of language device for avoiding the paradoxes does not jeopardize thederivation of any parts of classical mathematics, so that no need arises forany analogue to the reducibility axiom.

SOLUTIONS TO

SELECTED EXERCISES

312

Solutions to Selected Exercises on Pages 42-45

314

Solutions to Selected Exercises on Page 58

316

Solutions to Selected Exercises on Pages 77-82

318

Solutions to Selected Exercises on Pages t0ll-103

Solutions to Selected Exercises on Page 103

Solutions to Selected Exercises on Pages 104-111

Solutions 10 Selected Exercises on Pages 111-124

8. 1. (3x)(Fx v Q) 1. -(3x)(Fx v Q)2. FxvQ 2. (x)-(Fx v Q) 1,QN3. -Q 3. -(Fx v Q) 2, UI4. Fx 2,3,D.S. 4. -Fx'-Q 3, DeM.5. (3x)Fx 4,EG 5. -Fx 4, Simp.6. -Q ::> (3x)Fx 3-5, C.P. 6. (x)-Fx 5,UG7. Q v (3x)Fx 6, Impl., D.N. 7. -(3x)Fx 6,QN8. (3x)Fx v Q 7, Com. 8. -Q 4, Simp.9. (3x)Fx v Q 1, 2-8, EI 9. -(3x)Fx' -Q 7,8, Conj.

10. (3x)(Fx v Q) ::> 10. -[(3x)Fx v Q] 9,DeM.

[(3x)Fx v Q] 1-9, C.P. 11. -(3x)(Fx v Q) ::>

-[(3x)Fx v Q] 1-10, C.P.12. [(3x)Fx v Q] ::>

(3x)(Fx v Q) 11, Trans.

12. 1. -(3x)(Fx v Gx)2. (x)-(Fx v Gx) I,QN3. -(Fx v Gx) 2, UI4. -Fx'-Gx 3,DeM.5. -Fx 4, Simp.6. -Gx 4, Simp.7. (x)-Fx 5,UG8. (x)-Gx 6,UG9. (x)-Fx'(x)-Gx 7, 8, Conj.

10. -(3x)Fx' -(3x)Gx 9,QN11. -[(3x)Fx v (3x)Gx] 10, De M.

12. -(3x)(Fx v Gx) ::> -[(3x)Fx v (3x)Gx] 1-11, C.P.13. [(3x)Fx v (3x)Gx] ::> (3x)(Fx v Gx) 12, Trans.14. (3x)(Fx v Gx)15. Fy v Gy16. -(3x)Fx17. (x)-Fx 16,QN18. -Fy 17, UI19. Gy 15, 18, D.S.20. (3x)Gx 19,EG21. -(3x)Fx::> (3x)Gx 16-20, C.P.22. (3'.\Fx v (3x)Gx 21, Impl., D.N.

23. (3x)Fx v (3x)Gx 14, 15-22, EI

32224. (3x)(Fx v Gx) ::> [(3x)Fx v (3x)Gx] 14-23, C.P.25. {13}'{24} 13, 24, Conj.26. [(3x)Fx v (3x)Gx] = (3x)(Fx v Gx) 25, Equiv.

Solutions to Selected Exercises on Pages 124-128

324

Solulions 10 Selected Exercises on Pages 130-135

Solutions to Selected Exercises on Page 135

Solutions to Selected Exercises on Page 145

Exercises on page 145:

2. 1. (3x){Px.Sx.(y)((Py.Sy) ::> x = y].Lx} /..

. (x)((Px. �x) ::> Lx]2. Pz. Sz3. Px.Sx.(y)((Py.Sy) ::> x = y].Lx4. (y)((Py.Sy) ::> x = y] 3, Simp.5. (Pz. Sz) ::> x = Z 4, UI

6. x = Z 5, 2, M.P.7. Lx 3, Simp.8. Lz. 6, 7, Id.

9. Lz 1,3-8, EI

10. (Pz. Sz) :> L::; 2-9, C.P.11. (x)[(Px. Sx) ::> Lx] 1O,UG

4. 1. (3x){Px.(y)[(Py.x ¥' y) ::> Fxy].Sx} /:. (y)((Py. -Sy) :> (3x)(Px. Fxy)]2. Py.-Sy3. Px.(y)((Py.x ¥' y) ::> Fxy].Sx4. Sx 3, Simp.5. -Sy 2, Simp.6. x¥'y 4,5,Id.7. Py 2, Simp.8. Py.x ¥' y 7, 6, Conj.9. (y)[(Py.x ¥' y) ::> Fxy] 3, Simp.

10. (Py.x ¥' y) :> Fxy 9, UI11. Fxy 10,8, M.P.12. Px 3, Simp.13. Px.Fxy 12, 11, Conj.14. (3x)(Px- Fxy) 13,EG15. (3x)(Px. Fxy) 1,3-14, EI

16. (Py.-Sy)::> (3x)(Px-Fxy) 2-15, C.P.17. (y)((Py. -Sy) :> (3x)(Px. Fxy)] 16, UG

6. 1. (x){Fx::> (y)[(Fy.Lxy) :> SxyJ}/:. (3x){Fx.(y)((Fy.x ¥' y) :> LxyJ} :> (3x){Fx-(y)[(Fy.x ¥' y) ::> SxyJ}

2. (3x){Fx.(y)((Fy.x ¥' y) ::> LxyJ}3. Fx-(y)[(Fy.x ¥' y) :> Lxy]4. Fx 3, Simp.5. (y)[(Fy.x ¥' y) :> Lxy] 3, Simp.6. Fx :> (y)[(Fy. Lxy) ::> Sxy] 1, UI7. (y)((Fy.Lxy) :> Sxy] 6,4, M.P.8. (Fy.Lxy) :> Sxy 7,UI9. (Fy.x ¥' y) :> Lxy 5,UI 327

10. Lxy :> (Fy ::> Sxy) 8, Com., Exp.11. (Fy.x ¥' y) ::> (Fy ::> Sxy) 9, 10, H.S.12. (Fy.x ¥' y.Fy) ::> Sxy 11, Exp.13. (Fy.x ¥' y) :> Sxy 12, Com., Taut.14. (y)((Fy.x ¥' y) ::> Sxy] 13, UG15. Fx-(y)((Fy.x ¥' y) ::> Sxy] 4,14, Conj.16. (3x){Fx.(y)((Fy.x ¥' y) ::> SxyJ} 15,EG17. (3x){Fx.(y)((Fy.x ¥' y) ::> SxyJ} 2,3-16, EI

18. 2 ::> 17 2-17, C.P.

Solutions to Selected Exercises on Pages 150-170

Exercises on pages 150-151:

I. 3. (x)(y)[x ¥' y :J (3F)(Fx'-Fy)]6. (3x){Fxd'(y)(Fyd :J x = y)'(G)[(Gx-FG) :J Gd]'(H)[(Hx-VH) :J -HdJ}9. (x){[Mx-(F)(VF:J Fx)] :J Vx}'(3x)[Mx-Vx-(3F)(VF'-Fx)]

II. 2. 1. (3x)(3F)Fx 1. (3F)(3x)f.x���� ����3. Fx 3. Fx4. (3x)Fx 3, EG 4. (3F)Fx 3, EG5. (3F)(3x)Fx 4, EG 5. (3x)(3F)Fx 4, EG

6. (3F)(3x)Fx 2, 3-5, EI 6. (3x)(3F)Fx 2, 3-5, EI

7. (3F)(3x)Fx 1, 2-6, EI 7. (3x)(3F)Fx 1, 2-6, EI

8. 1 :J 7 1-7, C.P. 8. 1 :J 7 1-7, C.P.

6. 1. (x)(y)(z)[(Rxy'Ryz) :J Rxz]'(x)-Rxx2. (x)(y)(z)[(Rxy' Ryz) :J Rxz] 1, Simp.3. (y)(z)[(Rxy' Ryz) :J Rxz] 2, UI4. (z)[(Rxy' Ryz) :J Rxz] 3, UI

5. (Rxy'Ryx) :J Rxx 4, UI

6. (x)-Rxx 1, Simp.7. -Rxx 6, UI

8. -(Rxy' Ryx) 5, 7, M.T.

9. -Rxy v -Ryx 8, De M.10. Rxy ::> -Ryx 9, Impl.11. (y)(Rxy :J -Ryx) 10, UG12. (x)(y)(Rxy :J -Ryx) 11, UG

13. {I} :J 12 1-12, C.P.14. (R){13} 13, UG

10. 1. (x)(y)[(x = y) = (F)(Fx = Fy)] /.'. (x)(x = x)2. Fx

3. --Fx 2, D.N.

4. Fx 3, D.N.

5. Fx :J Fx 2-4, C.P.6. (Fx :J Fx)'(Fx :J Fx) 5, Taut.

7. Fx = Fx 6, Equiv.8. (F)(Fx = Fx) 7, UG9. (y)[(x = y) = (F)(Fx = Fy)] 1, UI

10. (x = x) = (F)(Fx = Fx) 9, UI

11. [(x = x) :J (F)(Fx = Fx)]'328 [(F)(Fx = Fx) :J (x = x)] 10, Equiv.

12. (F)(Fx-

Fx) :J (x = x) 11, Simp.nx=x ��M�14. (x)(x = x) 13, UG

Solutions to Selected Exercises on Pages 172-195

Solutions to Selected Exercises on Pages 195-200

Solutions to Setected Exercises on Pages 201-227

Solutions to Selected Exercises on Pages 227-257

THEOREM 1. IP;; (P v Q) :> (Q v P)

Proof: 1. [Q:> (QvP)]:> {(PvQ):> [Pv(QvP)]}2. Q:> (Q v P)3. (Pv Q) :> (Pv (Q v P)]4. [Pv (Qv P)] :> [Qv (Pv P)]5. (Pv Q) :> [Q v (Pv P)]6. [(PvP):> P]:> {(Qv(PvP)]:> (QvP)}7. (P v P) :> P

8. [Qv(PvP)]:>(Qvp)9. (PvQ):> (QvP)

Solutions to Selected Exercises on Pages 258-296

Solutions to Selected Exercises on Page 296

Solutions to Selected Exercises on Page 296

1. 20. If a n /3 -:;6 0 and /3 n - y = 0, then any -:;6 0

SPECIAL SYMBOLS

INDEX

Index

Completenessdeductive, 160-161, 182, 201,

203-207, 215, 225-227, 270-280

expressive, 159, 170

functional, 170, 172-175,232-233of the method of deduction, 56,

208-209of the 'natural deduction' apparatus,

280

Compound statement, 8-9, 71, 170

Conclusion, 3, 5, 179Condition

necessary, 16

sufficient, 16

Conditional

corresponding, 28, 51, 289

proof, 50-53, 55, 56-57, 58-61, 201,250, 259

statement, 14-16, 28, 50-52, 56-57,68, 170-171, 289

Conj., 32

Conjunct, 8

Conjunction, 8-9, 70, 112, 297

principle of, 32, 101, 192, 195, 198,200

Conjunctive normal form, 225-227, 286Boolean, 288-289, 293

Connective, 9, 13

Consequent, 14

Consistency, 62-63, 79n., 159-161, 182,247-248, 275-279

Constant

individual, 64, 243, 276, 278

predicate, 243

propositional, 242

Constructive dilemma, 32, 49, 50, 200

Contingent, 26, 62-63

Contradiction, 26, 53-55, 57, 61, 63,148-149, 159, 166-167, see also

Appendix C

Contradictory, 9-10, 26, 62-63, 67, 69,294

co�traries, 67, 69

Conventionof association to the left, 171-172,

245

governing '-', 11, 171, 244

governing '.', 171,244

342

Fallacyof affirming the consequent, 22

of denying the antecedent, 22-23

Feigl, H., 309n.

First-order

function, 302, 305function calculus, see Chapter 9

proposition, 303-304

Form

argument, 18-23, 28, 289Boolean normal, 287-289, 292-293

elementary valid argument, 31-32

normal, 263-270, 286, 287

prenex normal, 263-265Skolem normal, 267-270

specific-of an argument, 19, 23, 32

specific-of a statement, 25

statement, 25, 27-28of valid categorical syllogism,

294-295Formal

criterion, 163deductive system, 157-161, 295,

297

definition, 147

equivalence, 305nature of validity, 18

proof of validity, 30-32, 37-40, 61,89-90

truth, 25-26

Formula, 158-159, 163, 167, 245

associated propositional, 247-248of type R, 267well formed, 162-164, 168-169, 245

Free occurrence of a variable, 84, 108,246

Freeing of bound variables, 90; see also

EI, UI

Frege, G., 142n., 149n., 188, 282n.Function

binary, 174

order of, 302-303

propositional, 65, 83-84, 89-93, 114

singulary, 172

ternary, 174

Index

L.S., 188

Lambtla, 274

Langford, C. H., 23On.

Language, 5-6, 163-164, 165-167levels of, 165-167, 306-309

object, 165-167, 306-309

Syntax, 165, 100

Least upper bound, 305

Lee, Karen, ix

Leibniz, 137Levels of language, 165-167,306-309Lewis, C. I., 230n.

generality, 120

scope of an assumption, 60-61, 75,90,96-97

Lincoln, 112, 122, 136, 141

Lobachevsky, 156, 210

Loftin, Robert W., ix

Logicdefinition, 1

science of, 152

study of, 1

symbolic, 5-7task of deductive, 3

Logicalanalogy, 18-19

equivalence, 27-28, 37

proof, 162

sum, 202, 203, 290

truth, 79n., 104-111, 116, 271, 273

Chapters 7, 8, 9 passim.Lower function calculus, 242n.

Lukasiewicz, J., lOOn., 188, 210, 231

Luke, 71

Index

Proof (cont.)of functional completeness, 172-175,

232-233of functional incompleteness,

176-177

of incompleteness of rules, 47-50

of independence of axioms, 160,182-187

indirect, 53-56, 57, 61, 201

of invalidity, 45-46, 78-81in R S., 190-192reductio ad absurdum, 53-56,57,61,

62-63

shorter, 101of tautology, 56-57of validity, 30-32, 37-40

versus demonstration, 190-192

categorical, 290, 293-294

general, 64-70, 78

multiply general, 83-87

negative, 65, 67

numerical, 140-141orders of, 303-304

particular, 67

relational, 112-122

singly general, 83

singular, 64-65, 84

subject-predicate, 67-70, 290

universal, 67

and 8

constant, 242

function, 65, 83-84, 89-93, 114

symbol, 167, 242, 272

variable, 168, 242

Proverbs, 14, 71, 126

Protasis, 14

Psalms, 14, 71, 125

Psi, 69, 90-91

Psychology, 1-2

Punctuation, 11-12, 227-232

Pythagoras, 153-154

Index

Strongdisjunction, 10, 170

induction, 176-177

Subclass, 292-293

Subcontraries, 67, 69

Subject term, 64, 84,290

Subject-predicate propositions, 67-70,118, 290

Subset, 274

Substitution instance, 18, 25, 31-32, 65,66, 68, 84-85, 90

Substitution rule for functional

variables, 269n.

'Sum, logical, 202, 203, 290

Summand, 203

Superlative, 143

Suppe, Frederick, ix

Swartz, Norman, ix

Syllogismcategorical, 294-295

disjunctive, 10-11, 19-20, 32, 200

hypothetical, 21, 22, 32, 48-49, 194,200

Symmetrical relations, 130

Syntactical, 163-164, 165

variable, 244

Syntax language, 165, 166, 178

Systemdeductive, see Chapter 6