[Irving M. Copi] Symbolic Logic

SVMBOLIC

LOGIC

FOURTH EDITION

Copyright

@ 1973,

Irving

M.

Copi

Far

Amelia

PREFACE

vii

Preface

Jr., of Browning

Wichita

David

R.

State

Herbert

sity, University, of

State

the

of

University,

College of

Dil ing

University, Guerry R. Jennings

of

State Simon

of

I should

University

Karen

Miss

in

to

all

preparing

I

am

Old A.

of my

help deeply this

new

City

Richmond

of

Carolina

at

reading grateful edition.

my

daughter

proof. to

my

wife

of

Shukla

de

University

Suppe C. State

Margaret

help

of

Wil iam

Wil iam

her

la

University, the

University,

Louisiana

for

Stetson

Murungi

Frederick

Charlot e, of

to

State

Fraser

Xenakis

of

National

Cleveland

Simon

UniverW.

Centre

Cincinnati,

York,

New

Boston

Robert

Anjan

Youngstown

Loftin

W.

the of

Jason

of

College, of

of

Hullet

Lorin

University, of

Robert

of

McGil

University

N.

Porte

thanks

College, of

Eminhizer

Dominion

North and

Missouri, in

Eugene College, James

Swartz

Norman

Malcalester

Bunge

University, University

the

of

of

Mario

University,

Jean Green

express for

Lee

of

Most ment

of

like

of Samuel

Scientiflque,

the

Fraser

Dubuque,

of Bowling Donald Scherer of Hawaii, Leo Simons the University of Il inois, of the University J. Thomas of

York

University,

Matthews

University

Recherche

of

Gross

Idaho of

Earl

College, R.

Warren

the

Charleston,

Grace

Barry

Braden

Murray of

and

Wilcox

University. and Copi encourage-

to

CONTENTS

Contents

5

S

Logic

5.1

Symbolizing

5.2

Arguments

5.3

Some

5.4

Identity

5.5

Predicate

B

Euclidean

6.3

Formal

6.4

Attributes

6.5

Logistic

A

Propositional

Definite

the

145

Attributes

153 157

Systems Formal

of

Deductive

159

Systems

161

Systems Calculus

Axioms

7.4

Independence Development

165

and

Language Symbols and

Alternative

the

Axioms

the

Calculus

A

8.5

The The

and

Stroke Nicod

9.3 9.4

RS

9.5

Normal

9.6

Completeness

I

and

with

211 227

Notation

231

232

Operators

Dagger 234

Calculus

242 RS

System

Logistic of

RS

1

242

248

I

254

Deduction'

',Natural

the

of

Identity

Techniques

263

Forms

I

210

Logic System

System

New

210

Brackets

as

Function

Development Duality

Notations of

Parenthesis-Free

The

188 201

and

8.4

167

Formulas 182

Completeness

8.3

9.2

Formed 178

of

Alternative Systems The Hilbert-Ackermann The Use of Dots

9.1

Well

of

Systems

First-Order

165

Metalanguage and

Demonstrations

Deductive

RS

152

Deduction

Geometry

7.3

9.7

of

152

Deductive

Primitive

A

136

Description

Attributes

and

and

Object

8.6

126

130

Relations

Variables

7.1

8.2

9

of

and

7.2

8.1

xii

Involving

Systems

6.2

7.6

112

Relations

Attributes

Definition

7.5

112

Relations

Deductive 6.1

7

Relations

of

The

RS

270

I

280

259

Contents

Appendix

A:

Normal

Appendix

B:

The

Algebra

Appendix

C:

The

Ramified

Solutions

to

Selected

Special Index

Symbols

Forms

and

of

Classes

Theory Exercises

Expansions

Boolean

283 290

of

Types

300 311

337 339

Introduction:

Logic

and

Language

Introduction:

Logic

and

Language

The

1.2]

Sec.

the

from

word sentences

we

also

wil

be of

those

Every and

derivative

made

has

a

the

to

is

formulated

to

to

it

the

use

collection do

we

that

ensure

asserted

are

In

one.

or

clear

sufficiently

logic

When

expressed.

propositions

in

wil

we

sentence

any

Argument

that

but

chapters or

is

of

meanings,

refer

context

truth

the

other

fol owing

sense

unique

or

has

the

argument that

are

In

for

grOlmds

as

also

of

by

the

utter-

sentences.

argument

'conclusion'

proposition

and

argument,

other

these for

reasons

or

the

on

conclusion of

basis

propositions the

the of

the

which

accepting

which

of

analysis The

employed.

affirmed

is

the

in

structure,

usually

are

which

grounds

a an

presupposing

statements

ance

in

which

in

unique

word sense

'argument'

of

are

the

usage technical

the

has

regarded 'argument' explained.

which

others,

ordinary

Nature

other

terms

'premiss'

argument

is

are

of

propositions affirmed

are

conclusion

an

the

as

that the

providing of

premisses

that

argument. We the

another.

proposition Thus

and

'premiss'

that

note same

can

the

proposition

'conclusion' be

a

All

relative

are

in

premiss men

are

one

mortal

in

tenns,

and

argument is

premiss

the

sense

conclusion in

the

that in

argument

Introduction:

Logic

and

Language

See.

1.4]

Introduction:

complicates language

equivocal of

of

but

pleasing

central

the

not

dif icult

the

words

which

in

problem problem

idioms

of

The style. however, logician, the validity

for

the

or

of

they

the

for

these

even

their

dif iculties when

they

the

argwnent

of

invalidity

or

ambiguity and

of

resolution

and

vague

contain,

may

other

any

the

expressed,

are

the

deciding

English

in

because

appraise they

to

misleading metaphorical

deceptive

the

resolved,

formulated

Argwnents

construction,

1

[Ch.

often

are

nature

their

Language

problem.

our

natural

is

and

Logic

are

remains.

avoid

To

workers

the in

laries.

The

otherwise the a

various

the

theories

by adopting long sequence of reducing

special

further

a

advantage

sentence

or

The

introduction

sion

of

the

developed

economizes

require

equation

grows of

equation

the

exponent

and

symbols amount

long symbol

its

express words of attention

in

mathematics

is

to

vocabufor

ideas

to

meaning

language,

technical

required

time

familiar

the

ordinary

specialized

space of

too

with

connected

have

sciences

scientist

and

reports

dif iculties

peripheral

the

which

would

formulate.

This has for when

needed, more

dif icult

permits

his

writing

to

the

grasp.

expres-

See.

1.4J

Arguments Compound

Containing 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

conclusions

and

have

of

rangements

for to

dif erent values

truth

variables

statement

each prove

distinct the

form

of

the

values,

the

by

a

Disjunctive

truth

all

considering substituted

statements

form of

variable

statement

validity

truth

argument

the

2

[CI .

for

the

in

in

conveniently

most

Statements

Compound

Containing

Arguments

to

be

table,

tested. with

Syllogism

These an

in

appearing

for

the

form

initial

argument

possible be

can or

ar-

distinct

the

forth

set

column

guide form.

Thus

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

Disjunctive and

-C,

last

conclusions

two

B its

by

E

:>

Modus

(H.S.)

Rules

as

valid

D

and

the

from

original

argument

(M.P.), Syllogism

Pooens

Hypothetical

conclusions

infer

deduced

Modus

and

(D.S.),

deduced

validly

are

these

validly

we

E)

:>

from

be

can

proves forms

which

E,

:>

'aUdi"

(D

:>

finally,

conclusion

argument

Syllogism by

And

D

:>

its

exclusively

arguments

elementary Disjunctive of Inference '

A

more

to

list

with

formal the

the Rule

is

convenient

from above

lat er's

Inference

by

a

to

be

writ en

the

and

slanting premisses. as

beside

the

them.

preceding

statements

which,

the

statement

in

line

which The

the

right

of

automatically proof

of

validity

to

formal

in

each

last

marks

of the

"justifi-

from

which,

was

deduced.

separated

premiss, all

for

validity column,

one

the

case

question the

of

proof

them In

the

conclusion

this

out

from

specifies by put

of writing deduced

way statements

writ en

statement

to

it

concise

more

"justifications" a

of it

be

and

premisses

for

the

can

(M.T.),

B

That

-C

premisses,

Pooens.

of

premisses.

cation"

,

the

used

are

from is

Here

fourth

Modus

by

E

:>

subconclusions), Syllogism.

valid

using

valid. Tol/ens

D

(or

Hypothetical

a

premisses be

to

infer

and

second

the

From

Syllogism. validly

we

Proof

the

given

statements

argument

and

It

The

Method

of

Deduction

See.

3.1J *1.

(A ".A: >-B

:>

-B)'(-C

:>

D)

The

Method

of

Deduction

See.

3.1) 7.

(-HvI): >

..

J::>

K)

(J::>

(-L'-M) (H : > L)'(L (-L'-M)'-O N

:> :>

(K H)

:>

N)

The

Method

of

Deduction

See.

3.2J

The

Method

of

Deduction

14.

Double

15.

Transposition

16.

Material

Implication

(Impl.):

17.

Material

Equivalence

(Equiv.):

(D.N.):

Negation (Trans.):

See.

3.2]

The

Method

of

Deduction

Sec.

3.2) 9.

*10.

(G

v

.'.

[(G

H)'(I

(K'L)

12.

13.

-V



*15. 16. 17.



([W'-(X'Y))

v

(F



G)'{((G

.'.

(F



J

[(D

C))

v

v

[D



[(P'-Q)

.'.

0



=

-S

=

-5

=

=

[(P'-Q) {--T = {---Tv

{(-W

V

V

(A'-B)



(D

E) E))

v

G))



H)  (H v -M)'[(K'-L) v (-M'-Nn (P'--R)] = (--P'--R))

(H



In

v

-N]}

(-T'S)]}

--X)

(-T'S)]} [(-Y

v

W)

[C



(C

[G



(G





[(G'G)

[(C'C)



In



[---Uv







(E'-F)

v

v

[---Uv

{(-X 

(E'-F)





(A'-B)

D)

H)'(H



-{(K'-L)

0

.'.

v

v

G)'{(G

J

-U)]} -Z]} -Z}





-(((K'-L)

=

-U)]} -(Tv

v

[-(X'Y)

C)) (B

.'.

.'.

20.



v

.'. 19.

{W

[A

.'.

18.

(B

v

'.

.

14.

-V

[A

H)'J]

v

{M'[(O.N).P]} -[(R'-S)'(Tv [-(R'-S)

-{Qv

.'.

[(G

v



-{Qv .'.

J) 1]

{M'[(N'O)'P]}



(K'L)

.'.

11.

v

H)'

v

[(-Y

v

(C





D))

H)] 

D))





H))

Z)v(-Z



Z)

v

-Y)]}



(-Z



-Y)]}

The

Melhod

of

Deduction

See.

3.2) 11.

E

F

:>

E::>G 12.

..

E::>

H

:>

(F.G) (I

J)

v

-I

".H: >/ 13.

(K

L)

v

14.

-(M.

:>

(-Mv

-N)

(0

=

..

(Lv

P)

S :>

N) :>

:>

K)

(0

(Q.R) :>

(R.Q)

T

Sv

T

".T

*15.

(-Uv

V)'(Uv

-x::>

-W

W)

".vvx

16.

A

:>

C::> ..

A::>

(B (D.

:>

C)

E) (B::>

D)

=

P)

The

Method

of

Deduction

See.

3.3]

The

Method

of

Deduction

Sec.

3.4] 10.

W

=

Z

=

A

=

(B

=

Y =

Y)

(Xv

(Z (Z (A

X

:> =

=

:>

Bv-W ..

W

=

B

Y) -A) B) Z)

The

Method

of

Deduction

See.

3.4]

Only

Incompleteness

value

and

0,

with

r

0

0

0

0

0

1

0

1

1

0

0

2

0

2

2

0

1

0

1

0

0

0

1

1

1

1

1

0

1

2

1

1

2

0

to

show

as

on

0

2

0

2

1

2

0

1

0

2

2

2

0

2

1

0

0

0

0

0

1

0

1

0

1

1

1

0

2

0

2

1

1

1

0

1

0

0 1

1

1

1

1

1

1

1

2

1

1

1

1

2

0

1

0

0

1

2

1

1

0

1

1

2

2

1

0

1

2

0

0

0

0

0

2

0

1

0

1

0

2

0

2

0

2

0

2

1

0

0

0

0

2

1

1

0

1

0

2

1

2

0

1

0

2

2

0

0

0

0

2

2

1

0

0

0

2

2

2

0

0

0

nineteenth,

each

in

would

of

them

be

needed

(It the

having 50.)' with we

have

the

value

have

the

same

De

Morgan's

conclusion

respect notice

0, the value.

Theorems,

'p

show

to

is

not

0

value

that and

absolutely analytical is hereditary

that

has

respect that

flanking the

table

hereditary but

they

them,

48

page

be

can

the

to

used

Dilemmas,

49 the

having

value

their

by

0

logical

themselves

equivalence appropriate

Even

construct

statements

biconditionals

the

0 also. 0 is

to on

verify the

value

Dilemma,

with

the

have

r'

=>

value

the

definitions

to

twenty-sixth, 'q

the

Destructive

of

in

r'

having

replacement

although

expressions example,

For

=>

twenty-fifth, q' and

=>

necessary

tables to

'p

premisses

three-valued

construct

hereditary equivalents,

the

alternative

the

is

two

Dilemma

constructed.

we

the

Constructive

to

page

twenty-second,

do

rows

because that

When

r

2

tenth,

Rules

0

0

easily

however,

p 0

0

respect

are

=>

q

tables

larger

q=>r

p

first,

twenty-seventh

Nineteen

the

0

the

in

and

p=>q

of

need

not

necessarily

sign to

the

first

of

The

Method

Deduction

of

-p

[th.3

-(p.q)

-(p.q)

(-pv

P

q

0

0

2

0

1

2

1

1

1

1

1

0

2

2

0

2

0

0

0

1

0

1

2

1

1

1

1

1

1

1

1

1

1

1

1

0

-q

p.q 2

0

-p 2

v

-q

2

=

0

1

2

1

0

2

0

0

2

0

0

2

2

0

0

0

2

1

0

1

2

0

0

0

2

2

0

0

2

0

0

0

-q)

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] I.

(A

2.

A

3.

AvB

4.

(Cv

p:

8.

9.

10.

The v

B)

:>

[(C

D)

:>

E

v

D)

rD E

(C, A::>

D)

:J

[(C'

E

D)

:J

E]

:>

E]

/:.

A

Strengthened :>

[(C'

Rule

D)

:>

E]

of

Conditional

Proof

TIle

Method

of

Deduction

See.

3.9]

This

Shorter

absurdum

of

Inference. It

to

tautology, :>

be

must

false, However,

[(p

p] assigned

q)

:>

false

Law

If

it

leads

to

be

must

assign the expression

truth

is

make

it

it

neither

is

F.

Peirce's

tautology,

a

attempt

we

case

to

contradiction

a

contradiction.

truth

if

But values

assigned but

contradiction,

a

the

on

not

a

a

nor

be

to

q

components

such

(other)

tautology

a

q)

:>

tautology. is

be

must

P T

assigned

q

leads

and

true

:>

assuming

question In

:>

(p p

its

attempt

and

true

make

to

this

If

antecedent

so

a

to

in

For

p] assign

conditional

the

p.

p,

it

:> to

us

its

Rules

statements

q)

:>

consequent to

proves

true.

be

possibly assigned then

false,

it

or

it

[(p

conditional

its

reductio than

requires

F

consistently expression contingent.

the

make

to

be

can

then

contradictory

cannot

values

values

and

assign

to

which truth

assign false,

either values

truth

to

it

T

assigned

the

of

Law

p is false, for the

But

forced

contradiction,

a

to

that

also.

F

be

must

its

to

of

classification

F, which consequent

consequent

value

previously

were

possible

is

assumption but

truth p

we

F

its

Method

rather

Peirce's

value

and

p]

:>

the

to

that

Absurdum

version

a

assignments

method

truth

while

the

antecedent

its

q)

true

value

certify

the

is

argument

an

truth

this

to

it

:>

be

to

of

use

Ad

Technique-Reductio

of uses

Thus

assign

we

antecedent

its

[(p

to

the

forms).

Table

validity

which

extend

to

easy statement

a

the

proving

technique,

is

(and is

of

method

ad

Truth

is

contingent. The

reductio

quickest It

assigned

the

to

a

disjunct of vast

other

is

make

and the

conjunction, of assignments conjunction, or

'trial

various

method

majority method

which

known.

conjunct

cases

is

reductio

wil

these ad

T

itself

Here

which

Despite

cases.

the

false.

both

absurdum

we

both

is

assigned

does

not

should

tend

to

far

the

If and

F

to

a

Here

disjunction which

determine have

diminish

to

the

experiment advantage

however, is

is

where

conjuncts.

complications, method

by

others.

in

disjuncts, to

where

by

assignment

assignments', such

for of

that

classifying

assigned

But

is

statements.

than

cases

to

be

forced.

is

some

assigned

must

values

and

arguments be

T

truth

assigning

in

must

a

true

testing

applied F

disjunction, to

sequence F

or

a

of

method of

readily

more

to

is

absurdum method

easiest

however,

is,

assigned T

ad

and

superior

in to

the any

Functions

Propositional and

Guantifiers

See.

4.

t]

Functions

Propositional Then

we

can

and use

the

Quantifiers 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 be

let er

'y'

wil

usage

the

expression

and

'«Px', «P.

it

is

valid

Rules cation instance

of

general 'principle expression

in

of

a

with

the

propositions

are

Generalization', second

quantification

is

'y'.

this

permits

quantifications, and

abbreviate rule

we

it is

as

is

arbitrarily

any

'UG'.

of

list

our

from rule

all

inference

universal

the

inferred

be

of

true

augment

that

Since

universal

of

true

property

is

The We

principle validly

can

symbol

the

that

what individuals.

the

by adding to

all

the

has

what

since

this

In

hmction

propositional

individual.

since

of

true

hmction

UI,

by

selected

direction, be

further

respect

'(x)«Px'

arbitrarily

other

propositional of Universal this for

from

any

must

Inference

of

of

true

the individual

selected

arbitrarily

validly

individual.

of

instance

Rules

Quantification

selected

arbitrarily

any

substitution

a

any

fol ows

individual

selected

is

that

asserts

individuals

denote

to

'«Py'

'«Py'

Clearly

equally

used

Preliminary

Validity:

quantifisubstitution

its

inference

the

refer

to

OUT

it

as

symbolic

the

Propositional

Functions

and

Quantifiers

See.

4.2]

5.

Cw

6.

Dw

7.

CWo

8.

(3x)[Cx.

Dw

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]

function

propositional existential) variable

and

'y'" should

It

of

the

of

'x'

In

the

the

not

first

strictly multiply

Cx] is

the

to

universal

with

(or the

to

respect

results the

of

wil

be

it

is

helpful

'(x)[Dx

desirable

:J

alternative

also

a

the

G',

meaning

same

occurrences

'Fy

is

have

:J most

at

one

the

Thus

then

carnivorous

'(x)[Dx

as

(3x)[Ax.Cx)',

Gy'. This

proposition.

dogs are symbolized

'y'

to

respect free

confusion.

all

conviently Cx]

:J

are

with

all

single

a

preventing 'If

is

the

to

in

in

more

has

quantification replacing that replacement

considered, as

F

an

'x'

to

variable

given

is

than

result it

a

but

from

Gy)'

:J

is

respect

universal

work

'(y)[Fy which

with

which

carnivorous',

are

"the

or

:J

neither

although

incorrect.

It

of

Gx'

:J

proposition

(3y)[Ay.Cy)'

:J

'Fx

to

necessary,

and

Gx)'

:J

'Everything

respect

general

animals

some

'(x)[Fx

since

proposition

with

quantification

'x'"

function

propositional

a

on.

that

of quantification equivalent logically function propositional :J Gx' in 'Fx by 'y'-for of our stages early

universal and is

is

so

variable

the

to

respect of

clear

be

translations

of

with

quantification

4

[Ch.

Quantifiers

and

Functions

Propositional

any

has

been

remarked

variable.

respect

clear.

The

in

of

occurrence

every with

that

Hence to

proposition

that

every variable.

no

proposition

symbolizing variable

used Some

contain

can

any

examples

lies

proposition

a

within

wil

free

the

scope

help

to

occurrence

take

must

we

of

make

that

care

quantifier

a

the

matter

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 CPv

and

Functions

for

in

use

that

obvious that

are

true,

whereas

F

the and

applying g!m.l_t some

the

conclusion

[Ch.

Quantifiers EI

(31L)CPIL

where

,, !n'lalid: things

that

is

it

not

are

false

is

fails

for

every

for

'(3y)(Fx a

F, which model,

=

model

would

being

should

It

-Fy)'. containing

some

make

the

self-contradictory.

4

be

things premiss

See.

4.5]

Functions

Propositional .6_

1.

2. 3. 4.

6. 7.

(3x)[(Fx-Gx)

1.

2.

(3x)Fx (3x)Gx

3.

Fy

4.

Gy Fy-Gy

7_

5.

6. 7.

8.

8_

Quantifiers :>

Hy]

(3x)[(Fx-Gx)

/:.

Hy]

:>

:>

Hy]

4, UG

:>

Hy]

1,2-5, 6, UI

Hx]

:>

(3x)(Fx-Gx)

/:.

3,4. Conj. 5, EG

(3x)(Fx-Gx) (3x)(Fx-Gx) (3x)(Fx-Gx)

2,4-6,

EI

1,3-7,

EI

3.

(Fxv

4.

Fx

v

Gx

3, Simp.

5.

Fx

v

Gx

2, 3-4,

EI

6.

Fx

v

Gx

EI

7.

(y)(Fy (x)(y)(Fy

1, 2-5, 6, UG 7, UG

v

Gy)

Gy)Hy]

Hy]

(x)(y)(Fy

/. .

.9_

1.

2.

r3.

4.

Fx

5.

Fx

r6.

-Fx

10_

Gx)

/.'.(3x)(Fx--Fx)

-Gy 3, Simp. 1, 3-4, 6, Simp.

-Fx

9.

Fx--Fx

2,6-7, 5,8,

(3x)(Fx--Fx)

9, EG

4.

(x)[(Fx::> (x)[(Fx::> (Fz::> (y)[(Fy

5.

(Fu

6.

Fu

7.

-Gu

3.

EI

-Gx

-Fx

1.

10.

v

8.

2.

100

Gx)

v

7.

10.

Gx)

Gx)-Hx

(3x)(Fx-Gx) (3x)(-Fx-Gx) Fx

v

-

_

n.

EI

(3x)(3y)[(Fx (3y)[(Fxv

8.

Hx]

2, UI 3, EG

Hy

:>

:>

Hy]

:>

2.

1.

[

(3x)(y)[(Fx-Gx) (y)[(Fz-Gz) (Fz-Gz) (3x)[(Fx-Gz)

(y)(3x)[(Fx-Gy) (y)(3x)[(Fx-Gy)

5.

and

EI

Conj.

Gx)--Ga]

/.'.

(x)-Fx

Gx)--Gy] Gz)--Gy :> :>

2, UI

Gy)--Gy]

3, UG 4, UI

Gu)--Gu Gu

:> -

(Fu

:>

5, Simp. 5, Com.

Gu)

8.

-Gu

7,

9.

-Fu

M.T. 6,8, 9, UG

(x)-Fx (x)[(Fx::>

Gx)--Gy]

:>

(x)-Fx

2-10,

Simp.

C.P.

See.

4.5J 12.

13.

14.

(w){(x)[(Fx:> (x)[(Fx:> (x)-Fx

Gx)'-Gw)

Gx)'-Ga):>

:>

(x)-Fx

(x)-Fx)

Propositional

Functions

and

Quanlifiers

Sec.4.5] 7.

8.

(x)(Qx::> (x)(Sx .'. (x)(Rx (3x)Ux

Rx)

Tx)

:>

:> :>

(3x)Ux.(3x)Wx .'. 9.

(3x)(Ux.

(3x)Xx ..

:>

(3x)(Xx.

Sx)

(y)(Qy

:>

(y)[(Uy

v

Wx) (y)(Yy Yx)

:> :>

Vy)

Ty)

:> :>

Wy]

Zy)

(3y)(Xy.

Zy)

Propositional

Functions

and

Quantifiers

See.

4.6J

Logical

In

demonstrating

shall

have

forms

and

fication tion

the to

the rules

'(x)Fx

appeal strengthened as

:>

well.

(3x)Fx'

to

the

a

be

propositions list

original Conditional of

of

principle

Thus can

of

truth

logical only

not

demonstration set

down

as

of

Truths

quantifiers,

involving elementary

valid but

Proof, the

logical

Quantifiers

Involving

truth

argument quantiof the proposito

our

we

Propositional

Functions

and

Quantifiers

Sef'

4.6]

Logical

(x)Fx 

;

\Px

v

(x)Gx

Truths

Involving

Quantifiers

Functions

Propositional functions

fiers

'(x)',

we

whose

within

containing lent

any to

extend

over

wil

make

propositional let of

'Fx'

the

quantification of

another

'Fx'

this

and

'Fy' '(x)'

'(3y)',

Wherever

the

above,

writ en

expressions tional

and an

expression

scope free

lies

either

of in

clear.

the

'x'.

of

'Q',

'Fx' which

logical the

and

Q' is

more

entire

free

be

'Q'

containing equivalence conjunction briefly expressed

at

here

of

is

the as

equivap, does

example a

not

variable

least

one

free

is

between

a

or

'x', occurrence

the

two

Or

proposition

the

universal

/L not

logically on

either of

variable function

An

occurrence

them.

by the

on

4

proposiquanti-

the

affected

really quantifier or a propositional expression of the quantifier not

function. let

of

scopes

a

scope

propositional no

function first

Our

the

the

fol owing,

containing

propositional

any

p"

which or

In

are

the

the

and

'(z)Hz',

within

lying

containing proposition

a

occurrence

expression that proposition

variable with

'(z)',

have

and

'Ga'

propositions

although respectively,

'Gw',

function be

[Ch.

Quantifiers

and

universal

quantification

and

See.

4.6]

110

Sec.4-&]

The

Logic

of

Relations

See.

5.1J

so

of

Logic

The we

have

Relalions

the

relation

word

'taught'

common

to

the

propositions:

see.

5.IJ

6.

Lht: J Gt

7.

Gt

The

Logic

of

Relations

See.

5.1)

The

Logic A

similar

of

Relations

pair

of

inequivalent

propositions

may

be

writ en

as

5.1]

See.

3.

(x)[Vx::> (x)( Oxa (x)--Rxa

4.

Oza::>

5.

1. 2.

:>

(3y)Oyx] Rxa] /. .

--

Va

--Rza

2, UI 3, UI

6.

--Oza

4,5,

7. 8.

(y)--Oya --(3y)Oya

9.

Va::>

6, 7, 1, 9,

10.

--

Rza

(3y)Oya Va

M.T. UG

QN UI

8, M.T.

The

Logic

of

Relations

See.

50lJ

The

Logic

of

Relations

.

5.1)

tile

Logic *15. 16.

of

Relations

(x){[Wx'(y)[Py (x){[Px'(y)(-Vxy)]

-(3z)(Nz'

:>

:>

(3z)[(lz'Bzx)'-Dzy]])

Eyx))

:>

(y)[Xy

18. 19.

(x){Px:>

(3y)[Py'(3z)(Bxzy)]) (3y)[Py.(3z)(-Bxzy)])

*20. 21. 22.

23. 24.

25.

(x){Px:> (x){Px (x){Px (x)[(Nx'Dx) (x)[Px:>

:>

(y)[Py (y)[Py

:> :>

:>

Ix}

(z)(Wz

:>

Tgzx)}

(z)( -Bxzy)]}

(3z)( (y)(Lxy (3y)(Py'Xyx)).(3u)[Pu'(v)(Pv (x){[Qx,(y){[(py'Wyx)'(z)(-Kyz)) :>

:>

(z)(-Cx.:)}

:>

(x){Vx:> (x){[Lx'(3y)(Py'

17.

pxzy)]]

:>

-Bxzy)]} Myx)) :> :>

By}]

-Xuv)) :>

(u){[(Pu'Wux)'(v)(Kuv))

:>

Ou}}

See.

5.1]

The

Logic 4.

*5.

of

Relations

A

wise

He 6.

The

7.

Whoso

borrower

is

diggeth

The

have

(Ezekiel 9. 10.

a

fathers

The

foxes

have

hath

not

'"

hateth the

to

fal

pit shall (Proverbs eaten

(Proverbs his

10:1)

(Proverbs (Proverbs

son.

lender. therein:

13:24) 22:7)

and

he

the

children's

the

air

that

rolleth

a

stone,

are

set

it

sour

the

good (Romans

holes,

and

grapes,

and

where that

to

I would

7:19)

wil

26:27) teeth

edge.

on

18:2)

man

I do.

rod

servant

him.

upon

father.

glad

a

his

spareth

return

8.

maketh

son

that

the

lay

birds his

head.

I do

not;

of

have

(Matthew but

the

nests;

but

the

Son

of

8:20) evil

which

I would

not,

that

Sec.

5.2)

The

Logic

of

Relations

See.

5.2) 7.

(x)[Mx::>

(x)[Px .' 8.

.

(y)(Ny (y)(Oxy Px)

:>

(3x)(Mx'

-Sx)

(x)[(Rx'

:>

Oxy)]

Qy)] (y)(Ny (3y)(Txy' :>

:> :>

(3x)[Vx'Rx'(y)(Txy (x)(Vx .'.

9.

Vy))

-Sx)

(3x)(Vx'

Ux) Xx)

Xx)

:>

(x)(3y)(Yy' Ayx) Zy) (x)(y)[(AyX' '. (x)[(y)(Ayx (x){[BX'(3y)[Cy'DyX'(3z)(Ez'Fxz))) .

10.

Uy)]

:>

:>

(x)(Wx::>

(x)[(YX'

Qy)

:>

Zx] Zx]

:> :>

Wy)

:>

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

Zx] :>

(3w)Gxwx)

:>

:>

(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.

Finally,

5.4] 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'

uals

by

boldface

confused tus is

and

'GF'

good', complex

is

translate and to

into

propositions

is

the

symbolize can

a

stated

two

also

be

'C',

symbolized.

.

.

.

to

,

Here 'x

is

we

'F as

The

'FU'

propositions

is

a

and

appara-

'Tx',

'Ux',

use

truthful',

propositions

being

symbolic as 'Unpunctuality

propositions

quality'.

their

prevent

additional

this

such

good unpunctual', is

'B', With

notation

our

'x

'A',

individuals.

of

'Truthfulness abbreviate

and

let ers

capital

attributes

can

fault',

5

[Ch. italic

with we

a

Relations

fault', 'GT'.

'FF', and

'F More

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

[CI .

Calculus

Bl (A). (A) -(D)-(-((AI)'

BaA7-( )))( (

) (C

a

» )(

).(

)

7

See.

7.2]

A

Propositional

Calculus

See.

7.2]

A

Propositional

Calculus

See. in

7.2]

Primitive

R.S.

in

that

intended

its

the

normal

or

intended

P

That

RS.

is

-P

T

F

F

T

adequate

to

them

formulating when

P

Well

easily

is

of

RS.

as

and

when

P is true

P. The

function

f3(P)

expressible

in

therefore

is

and

case

-

-(-Pl!). There

be have

We

-P

formed

First

seen.

and

formulas

PQ

note

we

the

by

given

are

of These

thereby

P

Q

P

T

T

T

T

F

F

F

T

F

F

F

F

and

f2(l!),f3(F), f2(P) expressible when

true

RS.

as

expressed

in

RS.

as

and

that

all

and

true

truth

is

of

negation

P assumes, in

true

f3(l!),

every

that

is,

functions

truth

singulary

expressible

value

f4(l!)

false

function

The

is therefore

the

as

is

itself.

P

function

RS.

in

by actually

proved P

which

The

-PP.

shown

when

matter

nO

is

true

false,

P is

is false

fiP)

is

are

RS.

in are,

argument.

and

therefore

can

expressible

Ul!), function

The

therefore

is

and

and

express

RS.

in

false,

is

is false

fl (F)

as

and

tables

truth

in

interpretations interpretations

normal

or

Symbols

more

course,

defined

are

of

functions

truth

the

by

than

arguments

two

truth

fol owing

of

one

tables:

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

T

T

F

T

T

F

T

T

T

F

F

T

F

T

T

F

T

T

F

T

F

F

F

T

F

F

T

P

Q

f9(P.Q)

P

Q

T

T

T

T

T

F

F

F

T

F

F

fI(P.Q)

P

Q

fs(P.Q)

F

T

T

T

F

T

T

F

F

F

T

T

F

T

F

F

F

F

F

F

T

fIo(P.Q)

P

Q

T

T

T

T

T

F

T

T

T

F

T

F

F

F

F

F

f7(P.Q)

fu(P.Q)

fdP.Q)

P

Q

F

T

T

F

F

T

F

F

F

T

F

F

T

T

F

F

T

F

F

F

F

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

fI3(P.Q)

173

174

See.

7,2J

176

See.

7.2]

178

See.

7.3]

180

See.

7.3]

A

Propositional

Calculus

See.

7.4)

A

[Ch.7

Calculus

Propositional

Q)

:>

P

0

0

0

0

0

0

1

1

0

0

0

2

2

0

0

1

1

0

0

1

1

2

1

0

1

1

2

2

0

1

2

2

0

0

2

2

2

1

0

2

2

2

2

0

2

(P

[-

P)]

R)

:>

0

0

0

2

0

0

0

0

1

1

0

1

1

1

0

0

0

2

2

0

0

2

2

0

1

1

1

0

1

2

0

0

0

0

0

1

2

1

1

1

1

1

0

1

0

0

1

2

2

0

0

2

2

0

2

2

0

0

2

2

0

2

2

0

0

0

2

2

0

0

2

2

1

1

1

1

1

0

(Q

(P

:>

Q)

:>

0

0

0

0

2

0

0

0

0

0

1

0

0

0

0

0

1

1

0

0

1

1

0

1

0 0

(R

0

2

2

0

0

2

2

2

0

0

2

2

0

1

0

0

0

2

0

0

0

0

1

0

1

1

1

0

0

0

1

1

1

0

0

1

2

1

1

0

0

0

0

0

2

2

0

0

2

2

1

1

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

184

.0

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

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

The R.S.

characteristic

by

P and

P

consulting :> Q

have

belongs

teristic

them

by Finally

R

easily

is

to

one

seen

to

the

table

given

for

the

value

0,

or

more

Q wjJs

be

also it

has

also

with

hereditary ':>'.

In

the

the value

belongs

to

only O.

Hence

every

to

respect row

if

the

1 of

both charac-

from

deduced

wjJ

R

which

in

1. it

is

readily

seen

that

the

characteristic

in

question

does

not

belong

See.

7.4)

A

(P

0

0

0

1

0

1

0

1

2

0

2

2

2

[-

P) 0

P)]

:>

Q)

:>

R)

:>

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

0

0

2

I

0

2

1

0

0

0

2

0

0

0 0

(Q

(R

0

2

1

0

2

I

0

1

0

2

1

0

0

2

I

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

2

2

1

0

0

0

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

1

2

0

0

0

2

0

0

0

0

0

0

2

2

2

0

0

0

2

0

0

1

0

0

1

2

2

2

0

0

0

0

0

2

2

0

0

2

2

2

2

0

1

0

2

1

0

0

0

0

0

2

2

2

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

characteristic

by Q

:>

belongs by R Finally, to

:>

0

R.S. P

P 0

(P

The

186

[Ch.7

Calculus

Propositional

consulting have

independent.

or

0,

also

Q

wffs

more

':>'.

for it

In

the

has

belongs

the

characteristic

to

if

Hence

both

the

1

of

and

P

characteristic from

deduced

wff

every

R

to;

respect which

in

roW

O.

value

also

with

hereditary only

the

them

1. it

2.

Ax.

value

one

be

to

seen

table

value

the

to

easily

is

the

2

When rather

is

readily P

than

that

seen

and

Q 0

for

both

are

(I-I)

in

the

assigned :>

1

is

0

:>

value I

which

does

question 1, (Pis

2.

Q) Hence

belong

not :>

P

has Ax.

the 2

is

See.

To

model

which

of

Independence

7.4) prove and

fol ows,

the the

of

independence same

along

table with

for

Ax.

'-P'. the

3 of

The

derivative

RS.

dif erence table

we

the

use

same

lies

for

'P

in :>

the

Q'.

the

Axioms

for

'poQ',

three-element table

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 =

DR

16.

P =

DR

16,

COR.

QI--P Q, P =

-Q

=

R

=

Q,

5 I- PR R

=

=

5 I- P

Q5 v

R

=

Q

v

5

A

Propositional

Calculus

See.

7.6]

Deductive

TH.

26.

TH.

26,

COR.

TH.

27.

f-

--(PQ)

=

TH.

28.

f-

--(Pv

Q)

TH.

29.

f-

(P::>

30.

f-P(Qv

*TH.

f- P =

pp

(Tautology) f-p

TH.

30,

COR.

TH.

31.

f-

TH.

32.

f- Pv

(--Pv

=

= =

PR

Q)R

v

[PQ (Pv

= v

Q)(Pv

(De

Morg's

Theorem)

(De

Morgan's

Theorem)

(Material

Q)

PQv

=

(P

--Q)

(--P--Q)

=

Q) QR

(Tautology)

(--Pv

R)

=

pvP

=

Q) f-

(P

Completeness

(Distribution PR

v

R)

of

'.'

over

V)

over

,.')

QR (Material

--P--Q]

Implication)

(Distribution

Equivalence) of

V

A

Propositional

Calculus

See.

7.6]

204

See.

7.6]

206

7.6)

See.

by by

Qj:>

Deductive

DR

12

the

f3-case -So



Qj

: > -(Sl' assumption,

which

S2)' and

is

hence

I-

by

Qj

:>

DR

If

-5.

13



S2

to

Q;

:>

Completeness then

-(S1'



S2)'

Q;

:>

which

-52 is

A

Propositional

Calculus

See.

7.6]

Alternative Natations

Systems

and

See.

8.2)

212

Sec.

8.2]

Alternative

and

Systems P

Notations

-p

0

To of

2

0

0

PvQ

P

0

Q 0

1

0

1

0

1

0

0

2

0

2

1

0

0

0

1

1

0

0

independent

2

with

designated,

is

-p

1

2

1

1

2

0

0

0

2

1

1

0

2

2

2

0

we

the

the

use

model

three-element

P

Q

PvQ

PQ

1

0

0

0

1

0

0

1

0

1

2

2

0

2

0

1

1

0

0

0

1

1

1

0

1

2

1

0

2

0

0

0

2

1

1

1

2

2

1

1

3

{O, 1, 2}

tables:

0

Postulate

prove

Q

0

1

P

To

p

2

Postulate

prove

which

[Ch.8

independent

we

0

use

{O, 1, 2}

PvQ

with

and

designated

0

tables:

P

-P

P

Q

0

2

0

0

0

0

1

0

0

1

0

2

2

1

0

2

0

2

1

0

0

0

1

1

1

0

1

2

0

0

2

0

0

0

2

1

2

1

2

2

2

0

214

To

with

prove 0

Postulate

designated

4

and

independent tables:

we

use

the

PQ

four-element

model

{O, 1,2,

3}

Set:.

8.2]

The

Hilbert-Ackermann

System

p

Q

0

I

0

0

0

I

0

0

I

0

I

2

3

0

2

0

2

3

0

0

3

0

3

I

0

0

0

I

I

I

0

I

2

2

0

p

-p

p:J

PvQ

Q 0

I

3

3

0

2

0

0

0

2

I

2

3

2

2

2

0

2

3

0

3

3

0

0

0

3

I

3

0

3

2

0

0

3

3

3

0

Alternative

Notations

and

Systems

[Ch.8

P'Q

216

Q

0

0

0

0

1

0

0

2

0

PvQ

-(-Pv

-Pv-Q

-Q

-Q)

5

5

5

0

5

5

5

0

3

5

4

5

0

3

3

5

1

0

5

0

4

0

5

0

0

5

0

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

5

1

5

0

5

0

0

5

2

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

5

2

4

3

4

0

0

5

2

5

3

4

0

0

5

3

0

3

1

5

0

5

3

1

3

1

5

0

5

3

2

3

1

4

0

5

3

3

3

1

1

0

5

3

4

3

1

0

0

5

3

5

3

1

0

0

5

4

0

0

0

5

0

5

4

1

0

0

5

0

5

4

2

3

0

4

0

5

4

3

3

0

1

0

5

4

4

5

0

0

0

5

4

5

5

0

0

0

5

5

0

0

0

5

0

5

5

1

0

0

5

0

5

5

2

3

0

4

0

5

5

3

3

0

1

0

5

5

4

5

0

0

0

5

5

5

5

0

0

0

5

In

this

of

taking

P and -P

Pv

model

the

2See 65

-Q) designated =

Henry (1958),

2

three

elements

designated

only _(po only

take

vol.

-P

P

v

Hiz, pp.

-2

infer

to

values. 2

=

"A 613

the

Warning

f.;

Thomas

v

4

for

3

About W.

which

Translating

Scharle,

is

the not

Definitions

2

value a

Axioms," "Are

respect

formulations

H.A.

But =

with

three

The

designated.

are

hereditary

is

and

Q:

1, 2

0,

values

characteristic

to

the

rule:

of

the

R.S.

for

P,

American

Eliminable

we

have

2

value.

designated

From axioms

Mathematical in

Fonnal

Monthly, Systems"

Set:.

8.2]

Alternative

THEOREM

Systems

and

4.

Notations

mpv-p

Sec.

THEOREM

The

8.2J 9.

lux

[(Pv

Q)

v

R]

:>

[Pv

(Qv

R)]

Hilbert-Ackermann

System

Alternative

THE

and

Systems 0

REM

14.

Notations

1m: (PQ)

:>

P

The

8.2J

See.

THEOREM

Proof:

I.

P::>

2.

3.

(Pv [P: >

4.

P =

Proof:

10.

P)

IRA

v

P)

P]

:>

DR4 df.

(PP)

P =

Th.I6

(-Pv-F)

--p

3.

P =

--P

Th.l

4.

P =

DR9

5.

P =

-(-Pv-F) (PP)

Q

=

I.

lux

R

df.

(Pv

Q::>R (Pv Q) (Pv R) (Pv Q)

P::>

DR8

-(-Pv-P)

=

:>

4.

Q)

:>

(R

P)

v

premiss DR2

(Pv R) (R v P) (R vp)

:> :> :>

P3 DR

Iux(PvR)::>

8

Q,R: >

12.

premiss

(P

3.

P::>

4.

(8 (Pv

P =

Proof:

R) Q P) R)

v

v

Q,R

I.P

(8

v

P)

(Q (Q

v

8) S)

:>

:>

8

Q Q)( Q Q

=

(P::>

3.

P::>

4.

Q

5.

R

6.

(R

7.

R

8.

S :>

9.

(PvR)::>

10

DR

premiss :>

=

2.

:> =

v

Iux(PvR)

=

10 1

premiss :>

df.

P)

DR6 DR7

premiss

8)(S

:>

DR6 DR7

R

12.

(PvR)

221

df.

R)

S

S) (Q [(PvR)::>

(2, 4)

(Qv8)

S :> :>

DR DR

P

I.

10.

1

(Qv8)

I.R: >8 2.

5. DR

PI

P)][(P P)

v

-P

3.

Proof:

P2

P

:>

I.

2.

I.

P)

(Pv

=

System

2.

Proof:

DR

(Pv P) (P (Pv

17.

THEOREM

DR

luxp

16.

Hilbert-Ackermann

:>

v

=

(QvS) (Pv R) (QvS)][(QvS)::> (QvS)

(PvR)]

DR

11

DR

11

DR4

df.

(3, 5) (4, 8)

Afternative

Systems

and

Notations

8.2]

See.

The

Proof:

1.

(PvQ):> (QvP) [(Pv Q) (Pv Q)

2.

3. 4.

DR

13.

P:>

Proof:

=> =

R

Q)

(Qv

=

P3 P3

(QvP) (PvQ) (Q v P)][(Q (Q vp)

luxp

v

p)

=>

(Pv

DR4

Q)]

elf.

(QR)

=>

premiss

3.

P

4.

-R

5.

7.

(-Q -(Pv (PI')

:>

8.

P:>

(QR)

1.

=>

premiss

R

3. 4.

5.

14.

P

-P

v

-R) -p)

Proof:

15.

Proof:

(QR)] (QR)J

P

-Pv

3.

[-Pv(-QvR)] -Qv(-PvR) Q :> (P

P

1.

P

2.

-Pv

3.

(-Pv

4.

5.

=>

=>

v

=>

(Pv

(QR)]

=>

[(P

:>

(-Q -Q)

v

R)]

(Pv [(Pv

DR2

R)

R)]

Q)(Pv (P

=>

=>

DR

13

R)

R)

premiss df.

R)

[-Qv(-PvR)]

Th.7 R'I df.

R)

R)

:>

223

R

premiss

R)

v v

R

df. MT MT

R

17

Th.15

IRA (PQ)

(Q

Q)(P

Th.

DR2

Q)

=>

:>

v

I, Cor.,

Th.14

v

R)

--(-Pv-Q)vR (PQ) :>

[P

huQ

(Q

:>

MT

=>

(-Q

(Q

=>

elf.

=>

R)

1.

5.

DR5

-R)

R

=>

2. 4.

:>

Q

:>

(QR)]

=>

DRll

(-Pv-P) -(-Qv

=>

(QR)

(Q:>

=>

DR5

:>

hu

(QR) [Pv (QR) [Pv [Pv

2.

DR5

-P

-Q:>

19.

Proof

System

P)

I.P=>Q 2.

THEOREM

DR

=>

Q,P:>

6.

DR

\Hx(Pv

18.

THEOREM

Hilbert-Ackermann

df.

I, Cor., I, Cor.,

Th.

12

Th.

I

224

See.

8.2]

({3) P

2

The

the

Here ,

.

.

.

P

,

disjWlcts

of

Each We

.

MT

PI'

2

,.

.,

wffs

I, Cor.

Because

all

P

the

assumed

is

consider

Q Pn' Q

is

S

T

contains

that

PI

to

obtain

is

a

each

R

Sand

v

be

to

and T

and

R

of

T)

v

disjuncts

n

of

exactly

(>

n

PI' 1)

Y.

v

of

one

the

if

S, because

(S

=

<

k

any out

X

is

least

at

disjunct iHx Q

for

true

constructed

System

where

wffs

not

S

Pi (1 : :;

we

does

now

i

: :; n).

Th.

use

can

contain

18

PI

as

V

T),

disjunct.

fewer or

Metatheorem

Now

assume

can

and a

k

Hilbert-Ackermann

T

than

by the

the

contains n

{3-case

disjuncts

of

at

the

disjuncts

assumption of

S except

least

of

one

fIci

Pl'

P

S In

2

P

,

3

,

either

Hence

Pi'

(Sl

=

the

V

lat er

.

.

.

,

Pn

S is

S'),

as

S'

where case,

a

PI and

by

is

MT

disjunct, a

S contains

tHA

Q

wff

that

I,

Cor.

(PI

=

contains we

have

226

See.

8.3J

228

See.

8.3]

Afternative

Systems

and

Notations

See.

8.4)

The

Polish any

has

formula

for

8

[Ch.

Notations

notation

marks,

punctuation make

and

Systems

Altemative

obvious

the

the

unambiguous.

order

of

advantage in

which

its

symbols

with

dispensing are

writ en

all

suffices

special to

See.

8.5]

its

standard

true, the

The

truth

is

interpretation

which

is

table

the

same

as

to

affirming

that

deny that

of

either

they

Stroke

are

both

and

Dagger

formulas

the

false.

Operators P

It

is

defined

or

Q

is

by

234

See.

8.6J

Alternative

Systems

and

Notations

See.

THEOREM

The

8.6] 4.

!NPIP.I.P

Nicod

System

238

Sec. is

8.6]

The

Theorem

8

of

place P: to

I :Q I P.I.P and

of

S.

.P:I

:QIP.I.P: I applying

of

Line

15. 16.

Line

is

Theorem

19

: Q:.I the

Rule

Q:.I:.QIP.I.P:I:QIP.I.P: I: Q:.I:.QIP.I.P:I:QIP.I.P with Q in place 17

lines

20

and

and

21.

Line

22. to

lines

23

Line 23

and

18

is

22 the

is

Line

19.

4

of

place and

24.

P.

result

the

result

Theorem

is in

Rule

R.

Line 24

:.QIP.I.P:I:Q/P.I.P Nicod

lines

to

21 in

of

in

with

and

Nicod

System

with

P,

PIP.I.P:.I:.Q/P.I.P:I:QIP.I.P Q:.I:.Q/P.I.P:I:QIP.I.P: I: Q:.I:.QIP.I. of S. Line 16 is the of applying result is the Line 17 result of the applying 18 is Theorem 10 with QIP.I.P:I:QIP.I.P 3 with of S, and QI Q in place in place of P. Line I :QIP.I.P :.QIP.I.P:

and to

of

place

in

R,

place

and

12

14

Q of

in

lines

lines

with

and

Q

Nicod

of with Line

of

is

the

result

Nicod

Rule

Nicod

Rule in

with

Q:.I 20

result

the

is

8 with

of

place the

P

of

and

of

Nicod

Rule

Nicod

to

place :.QIP-I

Theorem

applying

the applying Q:.I:.QIP.I.P:I:QIP.I.P: j: Q:.\

25

is

the

applying

S,

Rule to

lines the

240

See.

The

B.6J 1.

DR

P,

Proof:

Line

of

P.

to

lines

of

applying

Line

IN

PIP.I.PIP:/:QIQ 2

1 is is

1 and

the 2.

the

P.

Nicod

4

1

(R'

9 with

Theorem

premiss Line

Q

is

Rule

the

P 3 is

Line

lines

3

in

place

the

result

and

System

H.A.)

PI P.I.PI

premiss to

of

Nicod

of P:

4.

S, and of

with

applying I:Q

I Q.

PIP

Line

in

5

is

place

Rule

Nicod

the

the

result

A

Calculus

First-Order

Function

Sec.

9.1] 3.

and

The

Infinitely without

many

subscripts,

let ers

capital

having

from

right-hand

the

first

superscripts

part

New

of

'1',

System

Logistic

with

alphabet,

the

'2',

'3'.

RS

.

.

.

I

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:

W6: 2.

W:

W6: 3.

W:

W6:

p.Q -Pv-Q (x)(Pv

Q) (3x)(-P._Q) (y)(3z)[Pv(-QvR.S)] (3y)(z)[-P.Q.(-R

v

-S)]

A

first-Order

Function

Calculus

Sec.

The

9.3] 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

the

Now R

with

an

(Qx2) II-

(3tH (3tH

.

normal

form

desired,

for

prenex

that

type

was

existential .

.

[D(t) [(Qxl)(Qx2)

G.

then

(Qxn)G', :>

.

of

(3t){

is

closed,

and

quantifier,

D(t)]} .

it

I- F

is

if

or .

(Qxn)G'J

-

-[D(t)-

[D(t)

G-

-D(t)]}

and

:>

in

D(t)]}

only

is

normal

prenex if

I- R.

the

Forms

formula

of

begins

form, Where

G

is

(Qxl)

A First-Grder

Function

Calculus

Sec.

9.5)

270

See.

9.6)

I- F

or

normal

Completeness I- -F.

interpretations,

This

kind

of

completeness

each

of

the

is

fol owing

not

desirable

wffs

either,

of

for

on

RS

their

I

272

See.

9.6]

274

Completeness

9.6]

Sec.

function in

it

a

by

is, is

as

calculus

RS

theorem.

To

not

valid.

In

defining

Seither to

only establish

to

the

that

say

if,

for

this

is

any

cwff

that

for

satisfiable.

by

'satisfiable'

satisfiable. And

S, if

any

and is S is

introducing

if every only cwffis provable S, if S cwff

and

valid

every

'valid' -S

or

-S

result

say

terms

valid

5 is

that

say

transposition,

if

complete

is

I

not

can

we

so

theorem

a a

characteristic

to

say then

cwff RS

in

is

not

that

say that

RS -S

I

is

as

a

then

that

for

S is

not

cp such

that

is

if We

both

S

wff

any valid

complete

satisfiable.

l

theorem

tlieorem

is

RS

provable

is 1

a

remarked

we

Hence

valid

of

and can

A First-Grder

Function

Calculus

Sec.

9.6]

278

See.

9.6]

280

See.

9.7)

A

First-Grder

Function

Calculus

APPENDIX

284

Normal

It

is

clear

that

by

invoking

the

defining

equivalences

Forms

and

Boolean

Expansions

286

Normal

Then

second

we

and

rearrange third

the

disjuncts,

by

terms

to

get

simply

commuting

Forms

and or

Boolean

interchanging

Expansions the

288

Forms

Normal

Expansion variables, at

least

one

n

variables

ing dif erent It

if

and

all

represent the

Since

2"

disjuncts

at

least of

of

one

2"

is

disjuncts

out

in

Chapter

possible

true,

2

that

in a

to

of

Section truth-functional

Expansions values

made

its

to

that

only

asserts

Expansion is

point

variables.

its

truth it

since

Boolean This

tautologous. again

Boolean

values

And

true.

disjunctive

any

and

truth

assignments

be

must

is

7.6

Section

in

pointed

of

assignments all

them

disjuncts

its

and terms

was

possible represent

and

containin

somewhat

8.2.

argument

is

valid

if its corresponding conditional statement antecedent is (whose only is the of the and whose the consequent argument's conjunction premisses is a tautology. number of disjuncts Since the conclusion) argument's counting us or a to decide whether not of its Boolean disjunctive Expansion permits an a of wi t h al t e rnat i v e met h od f o rm i s t h i s us given tautology, provides .'. q the Thus form the of arguments. p v q, -p validity argument deciding of its is proved valid t h e Bool e an disjunctive by constructing Expansion : > v and t h at t h e number of condi t i o nal observing [(p q)'p] q, corresponding 2 i s 2 its disjuncts an is valid Since of a tautology is a contradiction, the negation argument if and i t s i s of condi t i o nal a contradiction. i f t h e only negation corresponding is to another of an Hence method of deciding the form validity argument of the Bool e an of i t s t h e conjunctive Expansion negation corresponding conditional of its conjuncts. H it contains n and count the number distinct and 2" i s it is variables has ot h erwi s e t h en the valid; conjuncts, argument .

invalid.

290

The

Algebra

of

Classes

292

The a

and

{3:

I

products which is n

a

dif erent

of

2

does

make

not

provide designates simple A

it

Expansion if

it

whether what The

classes

E

subject-predicate

the

classes

product

a

is

as

Since

empty.

and

sum

terms

thus

and

count

the

expression by

designated

are

expression,

2

distinct

11

need

we

of

number

by thus

propositions. {3 have The

E

the far

the

only

products

no

class

E is

the common,

therefore

a

a

into

Boolean

for

of

symbolized

is

of

contains.

it

symbolization which

deciding

regardless

which a

class

empty

method class

No

a

terms

Expansion the

empty terms

com-

class

conjunctive

designates the

class

0,

De

By the

Boolean

have

we

proposition: in

1 is

simple occurs

simple

same

terms

designates simple permit

every

sum.

disjunctive of

members

proposition

the

which

transformed

be

can

the

of

sums

term

mentioned,

already

as

The

in

once

Hence

sums.

distinct class

simple

involves

complement simple

expression

designated

of

exactly

sums

many the

dif erent

introduced

notations

classes

any

which

of

class

any are

the

is

their

class

any

class

equivalences Expansion

Boolean

n

of

not

it

of

Expansions

not

any

product

a

occur

other

Expansion

product or

wil

disjunctive

containing the

is

is

where

the

and

product products.

of

sum

if

Boolean or

what

of

Given

Expansion

is the

which

whether

Expansion

expression

any Boolean

conjunctive

Disjunctive

observed,

containing

class order

the

in

deciding regardless

expression be

Expansion

universal

distinct).

Boolean

Theorem of

the

universal

class

should

it

Boolean

dif erence

mere

contains.

complements,

the

in

the

it

class,

four

those

the

The

exhaustive.

disjunctive

of

divide

Classes

sum.

their

or

anywhere Morgan's plement is

class

Boolean

terms

and

the

is

conjunctive

class

for

which

disjunctive

its

which

a

products

universal terms

and

01

two

any

wil

classes

universal

designates

terms

(where method

a

the class

construct

of

class

two

with

us

A

of

product n

any

Expansion.

simple products

distinct

11

the

and

exclusive of the

are

division

a

Boolean

ap,

U

Similarly,

which such

symbolizes disjunctive

af3

U

class.

empty

subclasses

11

2

into

ap

U

the

is

class

a{3

=

Algebra

{3,

the asserts

that

means as

A

and

that their

B

Appendix 0

The

of

empty.

a

Some

proposition: which

is

In

symbols,

not

a

the

is

not

of

member

a

0

proposition

fJ, asserts fJ, i.e.,

there

that

that is

expressed

the

is

product as

at

least of

a

one

member

and

P

is

not

The not

only

but

is

validate

to

The

'

C

of

The

classes. also members

All

introduced

in

reflexive that

if

a

and

transitive

C

p then

is

rewrit en

as

'aa

of

proof

and

p.

It

(see

pages The when

'p& and

0',

validity

0'.

=

transitivity

for

categorical

obviously 131-132)

lat er

is

'a

a

=

and

The

has

the

is

is

has

already syllogisms

a

'ap

as

obvious

when

been

established

'a

=

0'

C

a' in

only

containing

is

C

i'

double

and

'P

is

rewrit en

algebraic proposi-

our

W1iversal

tions.

The a

sets as

of

algebra is called

system for

Boolean

classes a

can

Boolean

set

been

fol ows.

Special

W1defined

primitive

symbols:

as

up and

Algebra, have

Algebra

be

a

proposed.

a

vast

formal

deductive of

number One

of

Such

system. alternative

them

p

=

C

property of

consequence

rewrit en

p

U

relation

(transposition)

immediate

an

already

as

or

A

the

symbols

the

ap

as

or

are

any,

of

equivalent.

p'

C

of

terms

0

=

reflexiveness

Its

its

ap

as are

in

algebra

if

a,

symbolization

defined

either

the

with of

members

alternative

an

be

can

ways: of which

pea.

as

all

Classes

propositions,

working

in

that

asserts

used

categorical also.

used

is often

p'

C

is

commutation as

=

inclusion 'a

all

1,

involving syllogisms

categorical class

is

a

=

and

negation

for

various

p

U

a

as

inferences

validating

expression of p,

proposition: or

immediate of

capable symbol'

of

Algebra

can

be

postulate set

forth

296

The

Class

inclusion,

equality,

and

inequality

may

be

defined

as

Algebra fol ows:

of

Classes

298

The

(A

f)

U

that

true

not

a

And

1.

#: n

of

effective

class The

connection

logically

is

which R.S.

theorems

for

fol ows Since and

should the

algebra

that

n

that

the

classes

of

R.S.,

the the

Classes

logically

not

IT

propositional

therein

have

we

intimacy

is

for

criterion and

equations indicate

and

is

of

designates

effective

an

true

to

it

which

wff

nontheorems

suffice

of

then

I,

#:

have

we

logically

recognizing

discussion

between

true

it in

between criterion

algebra. preceding

it

theorem

provable

distinguishing an

if from

I,

=

Algebra

inequalities of

calculus.

the

APPENDIX

The

relation, definition

so

that's

desigflates

tf>'

is

symbolized

as

'sDes :>

-[(3x)Fx (3x)(Fx

v

v

(3x)Gx] Gx)

M.

De

C.P. 1-11, 12, Trans.

16,QN 17, UI 15, 18, 19,EG

-(3x)Fx::>

(3x)Gx v

23.

(3x)Fx

v

24.

(3x)(Fx {13}'{24} [(3x)Fx

(3x)Gx v

Gx)

v

(3x)Gx]

14, :>

D.S.

C.P. 16-20, D.N. 21, Impl.,

(3x)Gx

[(3x)Fx =

v

(3x)(Fx

(3x)Gx] v

Gx)

Simp.

4,

7, 8, Conj.

-(3x)Gx v (3x)Gx]

(3'.\Fx

26.

6,QN

5,UG 6,UG

22.

25.

5,UG

2, UI 3,DeM.

Gx)

v

2, UI 3, DeM. 4, Simp.

Q)

v

I,QN

Gx)

1,QN

Q)

Gx)

v v

Q)

v v

15-22,

C.P. 14-23, 13, 24, Conj. 25, Equiv.

EI

7,8,

Conj.

9,DeM. :>

C.P.

1-10, :>

11,

Trans.

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 Exercises 1.

(3x){Px.Sx.(y)((Py.Sy)

2.

Pz.

3. 4.

Px.Sx.(y)((Py.Sy) (y)((Py.Sy)

5.

(Pz.

6.

x

7.

Lx

8.

Lz.

9.

Lz

(x)[(Px.

:>

Sz)

:>

x

:>

L::;

Sx)

1.

(3x){Px.(y)[(Py.x

2.

Py.-Sy

3.

Px.(y)((Py.x

4.

Sx

5.

-Sy x¥'y

8. 9.

(y)[(Py.x

x

¥'

x

3,

14.

Px.Fxy (3x)(Px-

Fxy)

15.

(3x)(Px.

Fxy)

16.

(Py.-Sy)::> (y)((Py.

y)

:>

Fxy].Sx}

:>

y)

/:.

y)

¥'

y)

:>

:>

Fxy]

1,3-14,

(3x)(Px-Fxy) :> (3x)(Px.

-Sy)

Fxy)]

(x){Fx::>

(y)[(Fy.x

6.

Fx

15.

¥'

y)

:>

Lxy]

: > Sxy] :> (y)[(Fy. Lxy) :> Sxy] (y)((Fy.Lxy) :> (Fy.Lxy) Sxy ¥' y) :> Lxy (Fy.x :> (Fy : > Sxy) Lxy ¥' y) : > (Fy : > Sxy) (Fy.x : > Sxy ¥' y.Fy) (Fy.x ¥' y) :> Sxy (Fy.x ¥' y) : > Sxy] (y)((Fy.x ¥' y) : > Sxy] Fx-(y)((Fy.x

17.

18.

2

:>

EI

:>

(3x){Fx.(y)((Fy.x (3x){Fx.(y)((Fy.x

16.

Fxy)]

C.P. 2-15, 16, UG

Fx

13.

(3x)(Px.

:>

13,EG

5.

14.

-Sy)

9, UI M.P. 10,8, 3, Simp. 12, 11, Conj.

4.

12.

(y)((Py.

Fxy

3.

11.

EI C.P.

4,5,Id. 2, Simp. 7, 6, Conj. 3, Simp.

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

10.

Lx]

:>

Fxy].Sx

/:.

8.

x)

Simp.

2-9, 1O,UG

Lx]

¥'

Px

9.

(x)((Px.

y].Lx

=

y

13.

7.

.

4, UI 5, 2, M.P. 3, Simp. 6, 7, Id.

¥'

12.

2.

.

y]

=

¥'

11.

1.

/.

Z

=

:>

(Py.x Fxy

6.

y].Lx}

3, Simp. 2, Simp.

Py Py.x

17.

=

1,3-8,

(Pz.

10.

x

Z

11.

7.

:>

:>

Sz) =

6.

145

Sz

10.

4.

Page

on

145:

page

on

2.

Exercises

Selected

to

17

¥'

y)

:>

SxyJ}

¥'

y)

:>

SxyJ}

:>

(3x){Fx-(y)[(Fy.x

3, Simp. 3, Simp. 1, UI M.P. 6,4, 7,UI 5,UI 8, Com., Exp. 9, 10, H.S. 11, Exp. Taut. 12, Com., 13, UG 4,14, Conj. 15,EG EI

2,3-16, 2-17,

C.P.

¥'

y)

:>

SxyJ}

327

Solutions

Exercises

on

I.

I.

Exercises

Selected

to

(x)(y)[x

6.

(3x){Fxd'(y)(Fyd

9.

(x){[Mx-(F)(VF:J

¥'

2.

150-170

150-151:

pages

3.

Pages

on

(3F)(Fx'-Fy)]

:J

y

:J

x

y)'(G)[(Gx-FG) Vx}'(3x)[Mx-Vx-(3F)(VF'-Fx)]

=

Fx)]

:J

:J

Gd]'(H)[(Hx-VH)

(3x)(3F)Fx

1.

3.

Fx

4.

5.

(3x)Fx (3F)(3x)Fx

3, EG 4, EG

6.

(3F)(3x)Fx

2, 3-5,

7.

(3F)(3x)Fx

8.

1 :J

EI

1, 2-6, 7

EI

1-7,

C.P.

Rxz]'(x)-Rxx Rxz]

3.

Fx

4.

5.

(3F)Fx (3x)(3F)Fx

3, EG 4, EG

6.

(3x)(3F)Fx

2, 3-5,

7.

(3x)(3F)Fx

8.

1 :J

1.

(x)(y)(z)[(Rxy'Ryz)

5.

(x)(y)(z)[(Rxy' (y)(z)[(Rxy' (z)[(Rxy' (Rxy'Ryx)

6.

(x)-Rxx

1, Simp.

6, UI 5, 7, M.T. 8, De M.

3. 4.

7.

-Rxx

8.

-(Rxy'

9.

-Rxy Rxy

10. 11.

13. 14.

10.

1.

(x)(y)[(x Fx

3.

--Fx

4.

Fx

5.

Fx

6.

(Fx

7.

Fx

8.

(F)(Fx (y)[(x (x [(x [(F)(Fx (F)(Fx

10. 11. 12.

Ryz)

-Ryx

-Ryx)

:J

-Ryx)

:J

12

:J

Simp. UI UI UI

9, 10,

Impl.

11,

UG

UG

C.P. 1-12, 13, UG

y)

=

(F)(Fx

=

=

Fy)]

/.'.

(x)(x

=

2, D.N. 3, D.N. C.P. 2-4, 5, Taut. 6, Equiv. 7, UG

Fx

:J

Fx)'(Fx

:J

:J

Fx)

Fx

=

Fx)

=

y)

= = (F)(Fx Fy)] = (F)(Fx Fx) = Fx)]' x) :J (F)(Fx = Fx) :J (x x)] Fx) :J (x x)

=

x)

=

(x)(x

Rxx

Ryx) -Ryx

=

=

=

-

=

nx=x 14.

1, 2, 3, 4,

Rxz]

Rxz]

:J :J

:J :J

Ryz)

:>

2.

9.

:J

Ryz)

v

(y)(Rxy (x)(y)(Rxy {I} (R){13}

12.

328



2.

6.

-HdJ}

(3F)(3x)f.x

1.



:J

=

x)

1, UI 9, UI

10, Equiv. 11, Simp. M 13, UG

x)

EI EI

1, 2-6, 7

1-7,

C.P.

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 THEOREM

Proof:

Selected

to

Exercises

IP;

1.

1.

[Q:>

2.

Q:>

3.

(Pv [Pv (Pv [(PvP):> (P v

4.

5. 6. 7.

(P

Q)

v

(Q

:>

v

:>

P)

:>

(Pv

P)] P]:>

8.

[Qv(PvP)]:>(Qvp)

9.

(PvQ):>

(Q

P

(QvP)

v

[Qv

:>

[Q

:>

227-257

P)

{(PvQ):>

(QvP)]:> (Q v P) Q) (Qv Q)

Pages

on

v

(Pv

P)] (Pv P)]

{(Qv(PvP)]:>

[Pv(QvP)]} P)]

(QvP)}

Solutions

to

Selected

Exercises

on

Pages

258-296

Solutions

to

Selected

Exercises

on

Page

296

Solutions 1.

20.

If

a

n

/3 -:;6

0

and

/3

n

-

y

=

0,

then

to

any

Selected

-:;6 0

Exercises

on

Page

296

SPECIAL

SYMBOLS

INDEX

Index

Completeness deductive, 203-207, expressive, functional, the

of

160-161, 215,

201,

182,

270-280

225-227, 170

159, 170,

172-175,232-233 deduction,

method

of

'natural

deduction'

56,

208-209 the

of

apparatus,

280

Compound

3,

71,

8-9,

statement,

Conclusion,

170

179

5,

Condition necessary,

16

sufficient,

16

Conditional

28,

corresponding,

proof,

55,

50-53,

51, 56-57,

289

28,

50-52,

58-61,

201,

259

250,

14-16,

statement,

68,

56-57,

289

170-171, 32

Conj., Conjunct, Conjunction,

8 297 70, 112, 101, 195, 192,

8-9,

principle

of,

32,

198,

200

Conjunctive

normal

form,

Boolean, Connective,

288-289, 9, 13

293

Consequent, Consistency,

62-63,

225-227,

286

159-161,

182,

14

79n.,

275-279

247-248, Constant

individual,

Constructive

Contingent, Contradiction, 148-149,

243,

64,

predicate, propositional,

278

276,

243 242

dilemma, 32, 49, 50, 200 26, 62-63 57, 61, 63, 26, 53-55, see also 159, 166-167,

Appendix Contradictory,

C

9-10,

26,

62-63,

67,

294

69

67,

cotraries, Convention

of

association

to

the

left,

171-172,

245

governing governing

'-', '.',

11, 171,244

171,

244

69,

342

Fallacy of

the

affirming denying

of

Feigl,

22

consequent, antecedent,

the

22-23

309n.

H.,

First-order

function,

305

302, calculus,

function

9

Chapter

see

303-304

proposition, Form

28, 289 287-289,

18-23, normal,

argument, Boolean

valid

elementary normal,

263-270, normal, normal,

prenex Skolem

31-32

263-265 267-270

specific-of specific-of

19,

argument,

an

25,

32

23,

25

statement, 27-28

a

statement, of valid

292-293

argument, 287 286,

syllogism,

categorical

294-295 Formal 163

criterion, deductive

295,

157-161,

system,

297

definition,

147

equivalence,

305

of

nature

of

proof

validity, validity,

18

37-40,

30-32,

61,

89-90

truth, Formula,

25-26

of

267

R,

type formed,

Free

247-248

propositional,

well

245

167,

163,

158-159,

associated

of

245

168-169,

162-164,

occurrence

84,

variable,

a

108,

246

of

Freeing G.,

Frege,

bound

variables,

142n.,

149n.,

90;

also

see

UI

EI,

282n.

188,

Function

binary,

174

calculus, ,

210,

dyadic, monadic, order

also

65,

83-84,

Chapter

9

172

of,

302-303

propositional, singulary, ternary, triadic,

see

174

172 174 174

89-93,

114

Index

188

L.S., Lambtla,

274

Langford,

C.

H., 23On. 165-167 5-6, 163-164, levels 306-309 of, 165-167, 306-309 165-167, object, 100 165, Syntax, 305 Least bound, upper ix Lee, Karen,

Language,

Leibniz,

137

Levels

of

language,

Lewis,

C.

I.,

Liar

165-167,306-309

230n.

paradox,

166;

also

see

C

Appendix

Limited 120

generality, of

scope

60-61,

assumption,

an

75,

90,96-97

Lincoln,

112, Robert

Loftin,

136,

122, 156, W.,

Lobachevsky,

141

210 ix

Logic definition,

1

of,

science

of,

study symbolic,

152 1

5-7

task

deductive,

of

Logical analogy, equivalence, proof,

3

18-19

27-28,

37

162

290 202, 203, 79n., 104-111, 116, 271, also see 148-149; Appendix 161-164, 295; Logistic system, 7, 8, 9 passim. Chapters sum,

truth, types,

function

Lower

Lukasiewicz, Luke,

calculus,

J., lOOn., 71

273

C also

see

242n.

188,

210,

231

Index

Proof

(cont.) of

functional

completeness,

172-175,

232-233

of

functional

incompleteness,

176-177

of

of

incompleteness independence

of

47-50

rules,

of

160,

axioms,

182-187

indirect,

53-56,

of

invalidity,

in

R

57,

61,

201

78-81

45-46, 190-192

S.,

ad

reductio

absurdum,

53-56,57,61,

62-63

shorter, of

101 56-57

tautology, validity,

of

30-32,

Proposition, categorical,

190-192

also

see

64-70,

negative, numerical,

65,

Statement

293-294

290,

general, multiply

78 83-87

general, 67 140-141

orders

of,

303-304

particular,

67

relational,

112-122

singly singular,

37-40

demonstration, 2, 5;

versus

83

general,

84

64-65,

subject-predicate,

67-70,

universal,

290

67

Propositional calculus, and

164;

also

89-93, 65, 83-84, 272 167, 242, 242 168, 126 14, 71,

symbol, variable, Proverbs, 14

Protasis,

Psalms,

14, 69,

Psychology, Punctuation,

Pythagoras,

7

Chapters

242

constant,

function,

Psi,

see

8

125

71,

90-91 1-2

11-12, 153-154

227-232

114

Index

Strong 170

10, 176-177

disjunction, induction, Subclass, Subcontraries,

292-293

Subject

69

67, 64,

term,

84,290

Subject-predicate Subset,

67-70,

propositions, 290

118, 274

Substitution

18,

instance, rule

Substitution

functional

for 269n.

variables,

logical,

'Sum,

65,

31-32,

25,

90

84-85,

68,

66,

290

203,

202, 203

Summand,

143

Superlative, Suppe,

Frederick,

Swartz,

Norman,

ix ix

Syllogism

categorical,

294-295

disjunctive,

10-11, 21,

hypothetical,

19-20,

32,

32,

22,

48-49,

200

relations, 163-164,

Symmetrical Syntactical,

244

variable,

Syntax language, System deductive,

logistic, Chapters

130

165

165, see

178

166, 6

Chapter

161-164,

295; 7, 8,

9

passim

see

also

200

194,

Index