alan ross anderson and nuel d.belnap, jr. entailment the ...entailment the logic of relevance and...

, ,l j Ii l i J

EN

TA

ILM

EN

T

TH

E L

OG

IC O

F RE

LE

VA

NC

E

AN

D N

EC

ESSIT

Y

by

ALA

N RO

SS AN

DER

SON

and I i--'

NU

EL D. B

ELNA

P, JR.

wUh contributions by

J. M

ICH

AE

L

DU

NN

R

OB

ERT K

. M

EYER

and further contributions by

JOH

N R

. C

HID

GE

Y

STOR

RS M

CC

AL

L

J. A

LBER

TO C

OPPA

Z

AN

E PA

RK

S D

OR

OT

HY

L. G

RO

YE

R

GA

RR

EL

POTTIN

GER

B

AS Y

AN

FRA

ASSEN

R

ICH

AR

D

RO

UTLEY

H

UG

UE

S LEB

LAN

C

ALA

SDA

IR U

RQ

UH

AR

T

RO

BER

T G

. WO

LF

VO

LUM

E I

PR

INC

ET

ON

UN

IVE

RS

ITY

PRE

SS

(",

CO

PYR

IGH

T ©

1975 BY

PRIN

CE

TO

N U

NIV

ER

SITY

PRESS

Published by Princeton University Press

Princeton and London

All Rights R

eserved

LCC

: 72-14016 ISB

N: G

-691-07192-6

Library of Congress cataloging in Publication D

ata will be found on the last printed page of this book

Printed in the United States of A

merica by

Princeton University Press

Princeton, New

Jersey

Dedicated to the m

emory of

WILH

ELM A

CK

ER

MA

NN

(1896-1962)

whose insights in

BegrUndung einer strengen Im

plikatiol1 (Journal of sym

bolic logic, 1956) provided the im

petus for this enterprise

CO

NT

EN

TS

VO

LUM

E I A

nalytical Table of Contents

Preface A

cknowledgm

ents I. T

HE

PUR

E C

ALC

ULU

S OF

ENTA

ILMEN

T II.

ENTA

ILMEN

T AN

D

NEG

ATIO

N

III. EN

TAILM

ENT B

ETWEEN

TR

UT

H FU

NC

TION

S IV

. TH

E C

ALC

ULU

S E O

F ENTA

ILMEN

T V

. N

EIGH

BO

RS O

F E A

ppendix: Gram

matical propaedeutic

Bibliography for V

olume I

Indices to Volum

e I

VO

LUM

E II (tentative) V

I. TH

E TH

EOR

Y O

F EN

TAILM

ENT

VII.

IND

IVID

UA

L Q

UA

NTIFIC

ATIO

N

VIII.

AC

KER

MA

NN

'S Strengen Implikation

IX.

SEMA

NTIC

AN

ALY

SIS OF R

ELEVA

NC

E LOG

ICS

X.

ASSO

RTED

TO

PICS

Com

prehensive Bibliography (by R

obert G. W

olf) C

ombined Indices

vii

IX xxi

xxix 3 107 150 231 339 473 493 517

i , I I :'[ {

AN

AL

YT

ICA

L T

AB

LE

OF C

ON

TE

NT

S

VO

LUM

E I

I. TH

E PU

RE

CA

LCU

LUS O

F EN

TAILM

ENT

§1. The heart of logic 3

§ 1.1. "If ... then -" and the paradoxes

3 § 1.2. Program

5

§1.3. Natural deduction

6 §IA

. Intuitionistic implication

10 §2. N

ecessity: strict implication

14 §3. Relevance: relevant im

plication 17

§4. Necessity and relevance: entailm

ent 23

§4.1. The pure calculus of entailment: natural deduction

formulation

23 §4.2. A

strong and natural list of valid entailments

26 §4.3. T

hat A is necessary P .A-+

A-+A

27 §5. Fallacies

30 §5.1. Fallacies of relevance

30 §5.1.1. Subscripting (in §5.1.2. V

ariable-sharing (in §5.2. Fallacies of m

odality 35

30 32

§5.2.1. Propositional variables entailing entailments

(in 37

§5.2.2. Use of propositional variables in establishing entailm

ents (in

40 §6. Ticket entailm

ent 41

§7. Gentzen consecution calculuses

50 § 7.1. Perspectives in the philosophy of logic § 7 .2. C

onsecution, elimination, and m

erge §7.3. M

erge formulations

57 §7A. Elim

ination theorem

62 §7.5. Equivalence

67 §8. M

iscellany 69

§8.1. An analysis of subordinate proofs

70

50 51

§8.2. Ackerm

ann's "strengen Implikation" and the rule (0)

§8.3. Axiom

-chopping 75

§8.3.1. Terminology for derived rules of inference

§8.3.2. Alternative form

ulations of 76

ix

72

75

x A

nalytical table of contents A

nalytical table of contents xi


ulations of 77


ulations of 139


ulations of 79


ulations of E.. 142

§8.4. Independence 80


ulations of 142

§8.4.1. Matrices

84 §14.2. Independence (by John R. Chidgey)

143 §8.4.2. Independent axiom

s for 87

§14.2.1. Matrices


s for 87

§14.2.2. Independent axioms for T..


s for 88

§14.2.3. Independent axioms for E..

144 §8.5. Single-axiom

formulations


s for R"

144 §8.5.1. Problem

89

§14.3. Negation w

ith das Fa/sche 145

§8.5.2. Solution for L (by Zane Parks)

89 §14.4. C

onservative extensions 145

§8.6. Transitivity

90 §14.5.

and R" w

ith co-entailment

147 §8.7.

Co-entailm

ent 91

§14.6. Paradox regained 147

§8.S. A

ntecedent and consequent parts 93

§14.7. Mingle again

148 §S.9.

Replacem

ent theorem

93 §S.IO

. is not the intersection of

and 94

§8.11. Minim

al logic 94

III. ENTAILMENT BETW

EEN TRUTH FUNCTIONS ISO

§S.12. Converse A

ckermann property

95 §15. Tautological entailm

ents 150

§S.13. Converse of contraction

96 §15.1. 'Tautological entailm

ents 151

§S.14. Weakest and strongest form

ulas 96

§15.2. A form

alization of tautological entailments (E

fde ) ISS

§S.15. Mingle

97 §15.3. C

haracteristic matrix

161 §8.16.

without subscripts

99 §16. Fallacies

162 §S.17. N

o finite characteristic matrix

99 §16.1. The Lew

is argument

163 §8.IS. Indefinability of necessity in

(by Zane Parks) 99

§16.2. Distinguished and undistinguished norm

al forms

167 §S.19. N

ecessity in 100

§16.2.1. Set-ups 169

§8.20. The Cr system

s: an irenic theory of implications (by G

arrel §16.2.2. Facts, and som

e philosophical animadversions

171 Pottinger)

101 §16.2.3. A

special case of the disjunctive syllogism

174 §8.20.1. The system

s FCr and Cr

101 §16.3. A

remark on intensional disjunction and subjunctive con-

§8.20.2. Some theorem

s 103

ditionals 176

§S.21. Fogelin's restriction 106

§17. Gentzen consecution calculuses

177 "

§18. Intensional algebras (Efd ,) (by J. M

ichael Dunn)

ISO

II. ENTAILMENT AND NEGATION

107 § 18.1. Prelim

inary definitions 190

§18.2. Intensional lattices 193

§9. Preliminaries

107 §IS.3 The existence of truth filters

194 • §1O. M

odalities 110

§IS.4. Hom

omorphism

s of intensional lattices 197

§II. Necessity: historical rem

arks lIS

§IS.5. An em

bedding theorem

200 §12. Fallacies

119 §18.6. Intensional lattices as m

odels 202

§13. Gentzen consecution calculuses: decision procedure

124 §IS.7. The Lindenbaum

algebra of Efd ,

202 § 13.1. C

alculuses 124

§IS.S. An algebraic com

pleteness theorem for E

fd , 204

§13.2. Com

pleting the circle 126

§19 First degree formulas E

fdf 206

§13.3. Decision procedure

136 §19.1. Sem

antics 206

§14. Miscellany

139 §19.2. A

xiomatization

207 §14.1. A

xiom-chopping

139 §19.3. C

onsistency 209

xii A

nalytical table of contents A

nalytical table of contents xiii

§19.4. Facts 209

§19.5. Com

pleteness 212

§24.1.2. Two valued logic is a fragm

ent of E 283

§24.2. E and first degree entailments

285 §20. M

iscellany 215

§24.3. E and first degree formulas

285 §20.1. The von W

right-Geach-Sm

iley criterion for entailment

215 §20.1.1. The intensional W

GS criterion

217 §20.1.2. The extensional W

GS criterion

218 §20.2. A

howler

220

§24.4. E and its positive fragment

286 §24.4.1. E+: the positive fragm

ent of E 287

§24.4.2. On conserving positive logics I (by R

obert K.

Meyer)

288 §20.3. Facts and tautological entailm

ents (by Bas van §24.5. E and its pure entailm

ent fragment

296 F raassen)

221 §25. The disjunctive syllogism

296

§20.3.1. Facts 221

§25.1. The Dog

296 §20.3.2. A

nd tautological entailments

226 §25.2. The adm

issibility of (,,) in E; first proof (by Robert K

. Meyer

and J. Michael D

unn) 300

IV.

TH

E C

ALC

ULU

S E O

F E

NTA

ILME

NT

231 §25.2.1. E-theories

300 §25.2.2. Sem

antics 303

§21. E E"+E/d,,

231 §2S.2.3. G

eneralizations 311

§21.1. Axiom

atic formulation of E

231 §21.2. Choice of axiom

s 232

§25.3. Meyer-D

unn theorem; second proof

314 §25.3.1. D

efinitions 315

§21.2.1. Conjunction

233 §21.2.2. N

ecessity 235

§25.3.2. Abstract properties

316 §25.3.3. Facts

318 §22. Fallacies

236 §25.3.4. Punch line

319 §22.1. Form

al fallacies 237

§22.1.1. Ackerm

ann-Maksim

ova modal fallacies

237 §22.1.2. Fallacies of m

odality (by J. Alberto Coffa)

244 §22.1.3. Fallacies of relevance

252 §22.2. M

aterial fallacies 255

§22.2.1. The Official deduction theorem

256

§22.2.2. Fallacies of exportation 261

§22.2.3. Christine Ladd-Franklin

262

§26. Miscellany

321 §26.1. A

xiom-chopping

321 §26.2. Independence (by John R

. Chidgey) 322

§26.3. Intensional conjunctive and disjunctive normal form

s 323

§26.4. Negative form

ulas; decision procedure 325

§26.5. Negative im

plication formulas

326 §26.6. Further philosophical rum

inations on implications

328 §26.6.1. Facetious

329 •

§22.3. On coherence in m

odal logics (by Robert K

. Meyer)

263 §22.3.1. C

oherence 264

§22.3.2. Regular m

odal logics 265

§22.3.3. Regularity and relevance

268 §23. N

atural deduction 271

§26.6.2. Serious 330

§26. 7. A --. B, C --. D

, and A

"'''':--.--C;;B---.-.''''CC;---.-;C;:D 3 3 3

§26.8. Material "im

plication" is sometim

es implication

334 §26.9. Sugihara's characterization of paradox, his system

, and his m

atrix. 334

§23.1. Conjunction

271 §23.2. D

isjunction 272

§23.3. Distribution of conjunction over disjunction

273 V.

NEIG

HB

OR

S OF E

339 §23.4. N

ecessity and conjunction 274

§23.5. Equivalence of FE and E

276 §23.6. The Entailm

ent theorem

277 §24. Fragm

ents of E 279

§27. A survey of neighbors of E

339 §27.1. A

xiomatic survey

339 §27.1.1. N

eighbors with sam

e vocabulary: T, E

, R, E

M, and

RM

339

§24.1. E and zero degree formulas

280 §24.1.1. The tw

o valued calculus (TV)

280 §27.1.2. N

eighbors with propositional constants: R

and E with

t,j, w, w

', T, and F 342

xiv A

nalytical table of contents

§27.1.3. Neighbors

with

necessity as

primitive:

RD and

ED 343

§27.1.4. R with intensional disjunction and co-tenability as

primitive

344 §27.2. N

atural deduction

survey: FR

, FE

, FT

, FR

M,

and FE

M

346 §27.3. M

ore distant neighbors 348

§28. Relevant im

plication: R 349

§28.1. Why R is interesting

349 §28.2. The algebra of R (by J. M

ichael Dunn)

352 §28.2.1. Prelim

inaries on lattice-ordered semi-groups

353 §28.2.2. R and D

e Morgan sem

i-groups 360

§28.2.3. R' and D

e Morgan m

onoids 363

§28.2.4. An

algebraic analogue

to the

admissibility

of (oy)

366 §28.2.5. The

algebra of E

and RD

t: closure

De M

organ m

onoids 369

§28.3. Conservative extensions in R (by R

obert K. M

eyer) 371

§28.3.1. On conserving positive logics II

371 §28.3.2. R is w

ell-axiomatized

374 §28.4. O

n relevantly derivable disjunctions (by Robert K

. M

eyer) 378

§28.5. Consecution form

ulation of positive R with co-tenability and t

(by J. Michael D

unn) 381

§28.5.1. The consecution calculus LR+ 382

§28.5.2. Translation 385

§28.5.3. Regularity

386 §28.5.4. Elim

ination theorem

387 §29. M

iscellany 391

§29. I. Goble's m

odal extension of R 391

§29.1.1. The system G

391

§29.1.2. Dunn's translation of G

into RD 391

§29.2. The bounds of finitude 392

§29.3. Sugihara is a characteristic matrix for R

M (by R

obert K.

Meyer)

393 §29.3.1. D

evelopment and com

parison of RM

and R 394

§29.3.2. Syntactic and semantic com

pleteness of RM

400

§29.3.3. Glim

pses about 415

§29.4. Extensions of RM

(by J. Michael D

unn) 420

§29.5. Why we don't like m

ingle 429

§29.6. " ... the connection of the predicate with the subject is thought

through identity .... " 429

Analytical table of contents

xv

§29.6.1. Parry's analytic implication

430 §29.6.2. D

unn's analytic deduction and completeness

theorems

432 §29.7. C

o-entailment again

434 §29.8. C

onnexive implication (by Storrs M

cCall)

434 §29.8.1. C

onnexive logic and connexive models

435 §29.8.2. A

xiomatization of the fam

ily of connexive m

odels 441

§29.8.3. Scroggs property 447

§29.8.4. Whither connexive im

plication? 450

§29.9. Independence (by John R. Chidgey)

452 §29.1O

. Consecution form

ulation of 460

§29.11. Inconsistent extensions of R 461

§29.12. Relevance is not reducible to m

odality (by Robert K

. M

eyer) 462

APPEN

DIX

(to Volum

e I). Gram

matical propaedeutic.

473

A!. Logical gram

mar

473 A

2. The table 480

A3. Eight theses

481 A

3.1. Logical gramm

ar and logical concepts 481

A3.2. A

questiou of fit 482

A3.2.l. Sim

plest functors 482

A3.2.2. M

ore complex functors

482 A

3.3. Parsing logical concepts 484

A3.4. R

eading formal constructions into English: the roles of "true"

and "that" 486

A4. A

word about quantifiers

489 A

5. Conditional and entailm

ent 490

VO

LUM

E II (tentative)

VI. T

HE

THEO

RY

OF EN

TAILM

ENT

§30. Propositional quantifiers §30.l. M

otivation §30.2. N

otation §31. N

atural deduction: FE"P §31.l. U

niversal quantification §31.2. Existential quantification

xvi A


§31.3. Distribution of universality over disjunction

§31.4. Necessity

§31.5. FE"P and its neighbors; summ

ary §32. E"P and its neighbors: sum

mary and equivalence

§33. Truth values §33.1. TV

'P §33.2. For every individual x, x is president of the U

nited States betw

een 1850 and 1857 §33.3. E

rdc and truth values §33.4. Truth value quantifiers §33.5. R

"P and TV

§34. First degree entailm

ents in E"P (by Dorothy L. G

rover) §34.1. The algebra of first degree entailm

ents of EV

3p

§34.2. A consistency theorem

§34.3. Provability theorem

s §34.4. C

ompleteness and decidability

§35. Enthymem

es §35.1. Intuitionistic enthym

emes

§3S.2. Strict enthymem

es §35.3. Enthym

ematic im

plications in E §35.4. Sum

mary

§36. Enthymem

atic implications:

representations of irrelevant logics In

relevant logics §36.1. H

in E'I'P §36.1.1. U

nder translation, E'I'P contains at least H

§36.1.2. Under translation, E'I'P contains no m

ore than H

§36.2. A logic is contained in one of the relevance logics if and only if

it ought to be (by Robert K

. Meyer)

§36.2.1. D (but not exactly H

) in R §36.2.2. TV

in R §36.2.3. D

and TV in R

"P

§36.2.4. S4+ and S4 in E, and S4+, S4, D, and TV

in E V3p

§36.2.5. H in E"P

§37. Miscellany

§37.1. Prenex normal form

s (in T V3P) §37.2. The w

eak falsehood of VpVq(p->,q->

p) §37.3.

RV

3p is not a conservative extension of

VII.

IND

IVID

UA

L QU

AN

TIFICA

TION

§38. R

V3X

, EV3X

, and TV3x

§38.1. Natural deduction form

ulations


§38.2. Axiom

atic formulations and equivalence

§39. Classical results in first order quantification theory

§39.1. Godel com

pleteness theorem

§39.2. Lowenheim

-Skolem theorem

xvii

§40. Algebra and sem

antics for first degree form

ulas w

ith quantifiers §40.1. C

omplete intensional lattices (w

ith J,. Michael D

unn) §40.2. Som

e special facts about complete intensional lattices

§40.3. The theory of propositions §40.4. Intensional m

odels §40.5. B

ranches and trees §40.6. C

ritical models

§40.7. Main theorem

s §41. U

ndecidability of monadic R

"x and EV3x (by R

obert K. M

eyer) §42. Extension of (,,) to RV3x (by R

obert K. M

eyer, J. Michael D

unn, and H

ugues Leblanc) §42.1. G

ramm

ar, axiomatics, and theories

§42.2. Norm

al De M

organ monoids and R: prim

ing and splitting §42.3. (,,) holds for R

V3x §42.3.1. N

ormal RV3x-validity; consistency

§42.3.2. Deduction and confinem

ent §42.3.3. Prim

e and rich extensions: relevant Henkinning

§42.3.4. Splitting to normalize

§42.3.5. Yes, V

irginia §43. M

iscellany

VIII.

AC

KER

MA

NN

'S strengen Implikation

§44. Ackerm

ann's 1: systems

§44.1. Motivation

§44.2. 1:E §44.3. 1:E contains E §44.4. E contains 1:E

§45. :1;', n', n", and E (historical) §45.1. f goes §45.2. (0) goes §45.3. (,,) goes

§46. Miscellany

§46. I. Ackerm

ann on strict "implication"

§46.2. E and S4 §46.2.1. R

esults §46.2.2. D

iscussion (by Robert K

. Meyer)

xviii A


IX. SEM

ANTIC ANALYSIS OF

RELEVANCE LOGICS ·(w

ith §47 by Alasdair

Urquhart and §§48-60 by R

obert K. M

eyer and Richard R

outley)

§47. Semilattice sem

antics for relevance logics §47.1. Sem

antics for R_

§47.2. Semantics for E_

§47.3. Semantics for L

§47.4. Variations on a them

e §48. R

elational semantics for relevant logics

§48.1. Bringing it all back hom

e §48.2. Preview

§49. Relevance: relational sem

antics for R §49.1. M

otivation §49.2. Syntactic prelim

inaries §49.3. R

elevant model structures (R

ms)

§49.4. Examples of R

ms

§49.5. Relevant m

odels (Rm

odels) §49.6. The valuation lem

ma

§49.7. The semantic entailm

ent lemm

a §49.8. A

pplications: relevance, Urquhart, (y), H

allden §49.9. The first-order theories R

MO

DE

L and R

+MO

DEL

§49.1O. Sem

antic consistency of R+ and R §50. Im

plication, conjunction, disjunction: relational semantics for posi-

tive relevant logics §50.1. The basic positive logic B+; +m

s; +models

§50.2. Ringing the changes I: E+, T+, R+, and their kin

§50.3. Paradoxical postlude: H+, S4+, TV

+ §51. N

egation §51.1. The m

inimal basic logic M

B; m

odel structures (ms); m

odels §51.2. R

inging the changes II: T, R, and their kin §51.3. The basic logic B §51.4. E cops out §51.5. Postulate-chopping and independence

§52. Entailment: relational sem

antics for E §52.1. Entailm

ent model structures (Em

s); Emodels

§52.2. Semantic consistency of E

§53. Modality: relational sem

antics for RD §53.1. M

odality means a new

semantical view

point §53.2. R ° m

odel structures (Rom

s); R Dm

odels; sem

antic consis-tency of RD

§53.3. Minim

al and other modal relevant logics

§53.4. Improving (?) E


§53.5. Modal fallacies

§54. Paradoxical logics: RM

, Lewis system

s, TV

§54.1. Relational sem

antics for RM

§54.2. The sem

antics of RM

3 §54.3. Lew

is systems of relevant logics

§54.4. TV as a relevant logic

§55. Com

pleteness theorems

§56. Classical relevant logics

§57. Individual quantification

xix

§58. Propositions and propositional quantifiers; higher-order relevant logics §59. A

lgebras of relevant logics §60. M

iscellany §60.1. H

istory §60.2. First degree sem

antics §60.3. O

perational semantics (Fine, R

outley, Urquhart)

§60A. Conservative extension results

§60.5. (y), Hallden, etc.

§60.6. Decidable relevant logics

§60.7. Word problem

s §60.8. U

ndecidable relevant logics

X. ASSORTED TOPICS

§61. Relevant logic without m

etaphysics (by Robert K

. Meyer)

§61.1. Beyond Frege and Tarski

§61.2. Truth conditions §61.3. H

enkin's lemm

a §61.4. The converse Lindenbaum

lemm

a §61.5. G

entzen, Takeuti, und Schnitt §62. O

n Brouw

er and other formalists (by R

obert K. M

eyer) §62.1. N

egation disarmed

§62.2. Coherence revisited

§62.3. Metacanonical m

odels §62A. Prim

eness theorems

§62.5. Applications

PRE

FAC

E

THIS BO

OK

is intended as an introduction to what we conceive of, rightly or

wrongly, as a new

branch of mathem

atical logic, initiated by a seminal

paper of 1956 by Wilhelm

Ackerm

ann, to whose m

emory the book is

dedicated. It is also intended as a summ

ary, seventeen years later, of the current state of know

ledge concerning systems akin to those of A

ckermann's

original paper, together with philosophical com

mentary on their signifi-

cance. W

e argue below that one of the principal m

erits of his system of strengen

Implikation is that it, and its neighbors, give us for the first tim

e a mathe-

matically satisfactory w

ay of grasping the elusive notion of relevance of antecedent to consequent in "if ...

propositions; such is the topic of this book.

As is w

ell-known, this notion of relevance w

as central to logic from the

time of A

ristotle until, beginning in the nineteenth century, logic fell in-creasingly into the hands of those w

ith mathem

atical inclinations. The m

odern classical tradition, however, stem

ming from

Frege and Whitehead-

Russell, gave no consideration w

hatever to the classical notion of relevance, and, in spite of com

plaints from certain quarters that relevance of antecedent

to consequent was im

portant, this tradition rolled on like a juggernaut, recording m

ore and more im

pressive and profound results in metam

athe-m

atics, set theory, recursive function theory, modal logic, extensional1ogic

tout pur, etc., without seem

ing to require the traditional notiol1 of relevance at all.

To be sure, even in

the modern m

athematical tradition,

textbooks frequently give som

e space in earlier pages to the notion of relevance, or logical dependence of one proposition on another, but the m

athematical

developments in later chapters explicitly give the lie to the earlier dem

and for relevance by presenting a theory of "if ...

(the classical two

valued theory) in which relevance of antecedent to consequent plays no

part whatever.

Indeed the difficulty of treating relevance with the sam

e degree of mathe-

matical sophistication and exactness characteristic of treatm

ents of ex-tensional logic led m

any influential philosopher-logicians to believe that it w

as impossible to find a satisfactory treatm

ent of the topic. And in con-

sequence, many of the m

ost acute logicians in the past thirty years have xxi

xxii Preface

marched under a philosophical banner reading·" D

own with relevance,

meanings, and intensions generally!" That m

etaphor is perhaps implausible, but it serves us in pointing out

that, in addition to the mathem

atical bits to follow, there are philosophical

battles to be fought. Am

ong them are the principal issues touched on in

contemporary philosophical discussions of logic: controversies about ex-

tensions and intensions, alethic modalities, and the like. W

hat we have tried to do is to jum

p into the skirmishes am

ong neo-Platonists, neo-concep-tualists, neo-nom

inalists, and generally exponents of neo-what-have-you

in logic, and to hit everyone over the head with a theorem

or two.

Such a program seem

ed to us to require, from an expository view

, that the m

athematical and philosophical tones of voice be intertw

ined. Not

inextricably, of course: we expect the reader to be able to tell when we are

(a) offering serious m

athematical argum

ents, (b)

propounding serious philosophical m

orals, and (c) making ad hom

inem jokes at the expense of

the opposition. No doubt som

e readers will find item

s under (c) undignified, or in som

e other way offensive. T

o such readers we apologize. We suggest

that they simply strike from

the book those passages in which they find an

unseemly lack of solem

nity -nothing m

uch in the argument hinges on

them anyw

ay, though ad hominem

arguments are som

etimes persuasive,

especially if the opposition can be made to look"ludicrous enough. For

the classical tradition we are attacking, this task is not difficult, but of course we certainly do not dem

and of our readers that they be entertained by side-com

ments. W

hich observation leads us to make another rem

ark about how

the book may be read.

We share w

ith many the conviction that the grow

th of (western) logic

from

Aristotle to the present day,

despite temporal discontinuities in

historical development, represents a progressively developing tradition:

the more m

athematical character of contem

porary work in logic does not

represent a sharp break with tradition but rather a natural evolution in

which m

ore sharp and subtle tools are used in the analysis of the "same

subject" -logic. (M

athematical treatm

ent of logic was initiated, so far as

we know, by Leibniz in 1679, though his insights did not catch on -

which

is hardly surprising: they weren't published until 1903; see .Lukasiew

icz 195 L) W

e believe in consequence that our hope of supplementing the

modern classical m

athematical analysis of logic w

ith a sharper, subtler, and m

ore comprehensive analysis of the sam

e topic, whatever its m

erits, will be thoroughly understood only by those prepared to study both the philosoph-ical and m

athematical argum

ents offered below.

Nevertheless, it is not necessary in m

any cases to check through the m

athematical argum

ents in detail, provided the sense of the theorems is

understood. Proofs frequently, in some m

ysterious way, illum

inate the

Preface xxiii

theorems they prove, and we hope that som

e readers will read proofs care-

fully enough to find such errors as are no doubt to be found. But the philo-

sophical thrust of arguments under (b) above can be gathered independently

of the compulsive checking of all the m

athematical details. Equally, of

course, the mathem

atical arguments under (a) are independent of the

philosophical polemics. A

nd we would be delighted if som

eone were to read

the book just for the jokes (c). W

e have used earlier versions of large parts of this book in advanced undergraduate and graduate courses at Y

ale and the Universities of M

an-chester and Pittsburgh. Students w

ith one year of mathem

atical logic have been able to grasp the m

aterial without too m

uch difficulty, though, as one is alw

ays supposed to say, there is more here than we have been able to

cover in a two-sem

ester course. Enough theorems, lem

mas, and the like have

been left unproved in the text to provide an ample source of exercises.

As is explained at the outset of §8, the m

iscellany sections may all be

skipped without loss of m

omentum

. A one-sem

ester course designed to touch the m

ost pervasive philosophical and mathem

atical points in the book m

ight include §§1-5 of Chapter I, §§9-12 of C

hapter II, §§15-16 of Chapter

III, and §§21-23 of C

hapter IV.

A

second-term continuation should

probably include the Gentzen form

ulations of §§7 and 13, and the algebraic sem

antics of §§18-19 and 25. The deepest insights into the semantics of

these systems w

ill be found in Chapter IX

(by Urquhart, M

eyer, and R

outley), which brings us to the edge of current research in this aspect of

the topic. Sections of the book not m

entioned above are designed to bring the reader to the edge of current research in other aspects of the topic. The book is intended to be "encyclopedic," in the m

odest sense that we have tried to tell the reader everything that is know

n (at present writing) about the

family of system

s of logic that grew out of A

ckermann's 1956 paper. B

ut there are still m

any entertaining open questions, chief of which are the

decision problems for R

and E, (and perhaps T), which have proved to be

especially recalcitrant. O

ld friends of our project will be surprised to find that we w

ere forced to split the book into tw

o volumes -

in order, of course, to avoid weighing

the reader down either literally or financially -

when we finally realized

that the universe of relevance logics had expanded unnoticed overnight. The second volum

e should appear about a year after this one; we include as part of the A

nalytical table of contents a tentative listing for Volum

e II.

Gram

matical propaedeutic. A

word should be said about the G

ramm

atical propaedeutic, w

hich is placed at the end of the first volume. There has been

abroad for a long time the view

that one cannot discuss the topic of this book

xxiv Preface

without m

aking certain essential mistakes. W

e complain about this from

tim

e to time in an incidental w

ay in the first few chapters, w

hich are devoted m

ainly to logical discussions near to our hearts. But we can already hear

unsympathetic readers w

hispering as they read that our project has no m

erit. Rather than try to sprinkle passim

remarks about our view

of the canard that there is no w

ay of talking about "entailment" w

ithout making

object-language meta-language betises, we gather our gram

matical view

s in one section, w

hich such an unsympathetic reader should indeed read

first, as a propaedeutic. But the reader w

ho has not been moved by listening

to a priori rejections of our entire topic on gramm

atical grounds is advised to postpone reading the G

ramm

atical propaedeutic for a long time-

maybe indefinitely; and w

ith such a reader in mind, we have relegated the

propaedeutic to an appendix, where it is less likely to constitute an obstacle

to beginning this book where it should be begun, §1.

Cross-references. The amount of cross-referencing in this book w

ill annoy those readers w

ho feel obliged to look, say, at §27.S whenever that section

(or any other) is mentioned in the text. W

e have attempted to w

rite and edit in such a w

ay that the reader seldom is forced to stick his finger in the

book at one page while he refers to another; the aim

of the cross-references has sim

ply been to assist those who w

ish to find other places where the sam

e or sim

ilar topics are treated.

Citations. W

ith respect to referencing the literature, we have tried to be liberal in V

olume I, except in one respect: our ow

n work has been cited in

the text only in the case of joint authorship with another, or occasionally in

sections by one of our co-authors. The method of citation, explained at the

beginning of the bibliography at the end of this volume, w

as to the best of our know

ledge invented by Kleene 1952, and is the best and m

ost economi-

cal we know, sim

ultaneously avoiding footnotes, cross-references among

footnotes, and willy-nilly giving (perhaps even forcing on) the reader som

e sense of history as he reads.

Notation. This has been a headache; as G

rover has pointed out to us, the R

oman alphabet w

as not designed to assist logicians, nor were the type-

setter's fonts. Those in the market w

ill find a remarkable exam

ple on page 206 of K

leene 1952, where Lem

ma 21 presents us (w

ithin two lines) w

ith the first letter of the R

oman alphabet in six different typefaces, of different

significance; it is an extraordinary tribute to that author's ingenuity, to the typesetters, and to the proofreaders, to have got it to com

e out right. W

e have in fact been sparing in our use of fonts. Our policy is not alto-

gether rational, but it is anyhow straightforw

ard.

Preface xxv

Bold face roman only for system

names, and variables over system

s, as explained below

.

Bold face italic for propositional and algebraic (or m

atrix) constants.

Italic for

expression-variables (variables over form

ulas, individual

variables, propositional variables, predicate constants, etc.), and italic also for num

erical variables.

rpLK for sequences of formulas, w

ith occasional other uses.

Plain roman for all other variables, tem

porary constants, etc.

There is of course lots of special notation, explained as it is introduced. W

e assume, how

ever, familiarity w

ith the standard set-theoretical concepts: !O (the em

pty set), n (intersection of sets), U (union of sets),

(comple-

mentation of sets -

relative to some indicated dom

ain), {ai, ... , a,} (the set containing just al, ... , a,), {x:A

x} (the set ofx's such that Ax).

For further information, consult the Index under "notation for."

System nom

enclature. This book mentions or discusses so m

anv different system

s (Meyer claim

s the count exceeds that of the number ;f ships in

Iliad II) that we have been driven -m

ostly at Bacon's gentle urging -

to try to devise a reasonably rational nom

enclature. In sum it goes like this:

1. B

old face: principal system (of propositional logic)

2. Subscripts: fragm

ent (involving subscripted connectives) 3.

Superscripts: extension (by adding superscripted connectives) 4.

Prefixes: formulations (axiom

atic, Fitch, or Gentzen)

Details, exam

ples, additions, and exceptions follow.

1. Bold face characters are used to designate principal system

s of propo-sitional logic, usually involving the connectives

v, &, and -

. Thus: E

, R, R

M, T

, S4, TV, etc.

2. Subscripts indicate fragm

ents of principal systems (of w

hich the principal system

s are presumably conservative extensions). W

e subscript w

ith the connective(s) in the fragment w

hen this does not get out of hand; otherw

Ise, we use som

e other (hopefully) mnem

onic device. The m

ost im

portant subscripts we illustrate with respect to E:

K,

the pure arrow fragm

ent of E entailm

ent with negation fragm

ent of E E+

positive (negation-free) fragment of E

Eldl first degree form

ula fragment of E

xxvi Preface

3. Superscripts indicate (presum

ably conservative) extensions of principal system

s by the addition of new pieces of notation, w

ith axioms governing

them. W

e superscript with the actual notation added, to tbe extent to w

hich this is feasible. W

e illustrate with respect to R

:

R D

a system

which adds a necessity m

odality to R

R m

adds both 0 and the propositional constant t to R

R'P

adds universal quantification (V) for propositional variables

(p) to R

R

"· adds universal (V

) and existential (3) quantification for in-

dividual variables (x) to R

4. Prefixes (not in boldface) distinguish startlingly differentJorm

ulations of the "sam

e" system. Since w

e use the null prefix for Hilbert (axiom

atic) form

ulations, we obtain as illustrations

Hilbert form

ulation of Fitch (natural deduction) form

ulation of C

onsecution (Gentzen) form

ulation of the "L

" derives from

Gentzen 1934

Merge-style consecution form

ulation of

5. N

umerical subscripts are occasionally used to indicate m

ildly different form

ulations; e.g.

are different axiomatic form

ulations of

6. V

arious combinations should be self-explanatory; e.g.,

is a consecution form

ulation of the positive fragment of the {D

, t l extension of R.

7. The principal exceptions to our policies occur in C

hapter VIII,

where we use nam

es used or suggested by Ackerm

ann's original notation for the fam

ily of systems w

ith which this book concerns itself. Thus 1;, II',

II" are all due to him (w

ith more thanks than w

e can, at this late date, m

uster); principal changes from the policies of 1-5 above are confined to

Chapter V

III. But the reader w

ill doubtless recognize cases in which the

price of systematic nom

enclature was deem

ed by us to be too high.

8. O

ne of the prices we have already paid -

with, let it be said, no

cheer -is to change the nam

es of a variety of systems w

hich have been

,

J

Preface xxvii

already used in the literature. We report here the m

ost significant cor-relations.

EQ, R

Q, etc.

= E'v':lX, R'v'3X, etc.

Et, R'i', etc.

= EV:l P, RV:l P, etc.

El,

R1 ,

etc. = E .... , R

.... , etc. E+,

R+, etc.

= Rt, R+, etc. N

R

RD

El *, etc. = FE

.... , etc.

See the Index of systems for a guide to w

here each system is defined, etc.

Pronouns. We have used "w

e" because of our essential multiplicity. Then

in editing papers by co-authors We have changed "I" to "w

e" for uniform-

ity; but in these contexts the "we" is editorial.

For example, w

hen a co-author says "we are going to prove X

," he or she doesn't m

ean that we are going to prove X, but that the co-author is.

In a few cases, how

ever, in which m

atters of history or opinion have arisen in a particularly delicate w

ay, we have let the first-person singular stand in order to avoid any tinge of am

biguity. (See Meyer's §28.3.2 for highly

refined sensitivity to these issues.)

AC

KN

OW

LEDG

MEN

TS

THIS BO

OK

has been in preparation since 1959. In the ensuing fourteen (see end of § II) years, so m

any persons have helped us in so many w

ays that we first considered the possibility of m

aking a list of those who had not helped,

on the grounds that such a list would be shorter. That policy w

ould however

deprive us of the pleasure of expressing our gratitude to those who have

contributed so much in producing w

hatever of value may be found here.

We should m

ention first our principal teachers, Frederic Brenton Fitch

(AR

A and N

DB

), George H

enrik von Wright (A

RA), and the late C

anon R

obert Feys (ND

B). The help they gave us, both directly, in providing us

information about the field, and indirectly, by providing fascination and

enthusiasm for techniques in logic, has been im

measurable.

We both profited from

Fulbright Fellowships (A

RA

with von W

right in C

ambridge, 1950-52, and N

DB

with Feys in Louvain, 1957-58), and we

wish to express our gratitude to the Fulbright-H

ays program for its assis-

tance in helping us learn much from

many European friends and colleagues.

Our initial collaboration at Y

ale in 1958 was sponsored in part by O

ffice of N

aval Research (Group Psychology Branch) C

ontract SAR

/Nonr-609

(16) (concerned with Sm

all Groups Sociology), under the direction of O

mar

Khayyam

Moore as Principal Investigator, to w

hom we are very grateful

for initial and continued support and encouragement. W

ith his help, and partially w

ith the assistance of summ

er programs sponsored by the N

ational Science Foundation through G

rants G 11848, G

21871, and GE

2929, we m

anaged to engage the interest of a number of extraordinarily able students

at or near Yale, am

ong them John Bacon, Jon Barw

ise, Neil G

allagher, Saul K

ripke (Harvard), D

avid Levin, William

Snavely, Joel Spencer (MIT),

Richm

ond Thomason, and John W

allace. We thank each of these for his

contributions during those salad days, when alm

ost every week saw

the solution of a problem

, or the genesis of a fruitful conjecture. W

e are also grateful to Yale U

niversity for Morse Fellow

ships (AR

A

1960-61, ND

B 1962-63), w

hich gave us both time off for research, and to

the National Science Foundation, w

hich partially supported both our work

and that of many of the students m

entioned above and below through

Grants G

S 190, GS 689, and G

S 28478; we further join Robert K

. Meyer

and J. Michael D

unn in thanking the National Science Foundation for

support to them through G

rant GS 2648.

xxix

xxx A

cknowledgm

ents

The years 1963-65 separated us, ND

B being in Pittsburgb (then and

since), and AR

A at Y

ale 1963-64, and Manchester (under the auspices of

the Fulbright-Hays Program

) in 1964-65, a separation which understand-

ably slowed progress. B

ut while separated, both of us w

ere fortunate in finding colleagues and students w

ho combined helpful collaboration w

ith good-natured, abrasive criticism

. In addition to those mentioned at Y

ale, others contributed clarifying insights (both as to agreem

ents and differ-ences) in M

anchester, among them

E. E.

Daw

son, Czeslaw

Lejewski,

David M

akinson, John R. C

hidgey (who later spent a helpful year w

ith us in Pittsburgh), and in particular the late A

rthur Norm

an Prior, a close friend w

ith whom

we had valuable correspondence on m

any topics for m

any years. M

eanwhile N

DB

had the good fortune to find able allies among students

at Pittsburgh, especially J. Michael D

unn, Robert K

. Meyer, Bas van

Fraassen, and Peter Woodruff.

AR

A and N

DB

rejoined forces in Pittsburgh in 1965, and since then we

have enjoyed not only sabbatical leaves, AR

A 1971, N

DB

1970, but also stim

ulating discussions with our colleagues Steven D

avis, Storrs McC

all, N

icholas Rescher, Sally Thom

ason, and Richm

ond Thomason, and w

ith a num

ber of delightful and dedicated colleagues-in-students' -clothing. Am

ong the latter w

e have depended for assistance upon Kenneth C

ollier, Jose A

lberto Coffa, Louis G

oble, R

ichard Goodw

in, D

orothy L. G

rover, Sandy K

err, Virginia K

lenk, Myra N

assau, Zane Parks, Garrel Pottinger,

and Alasdair U

rquhart; and more recently Pedro A

maral, Jonathan B

roido, A

nil Gupta, M

artha Hall, G

len Helm

an, and Carol H

oey. W

ord spreads, in this case through Kenneth C

ollier, who interested his

colleague Robert G

. Wolf in the enterprise. W

olf has not only lent us his expertise in the later stages of the preparation of the bibliography for this volum

e, but has generously assumed total responsibility for the C

ompre-

hensive bibliography promised for V

olume II.

One of our principal debts of course is to those w

ho have consented to stand as co-authors of this book. Tw

o options were open to us: (a) to

paraphrase and re-prove their results; (b) to ask their permission sim

ply to include their results verbatim

(or very nearly). Both of us w

ere convinced that course (a) w

ould not improve the total result at all, so w

e resolved on course (b), and our gratitude to those w

ho have allowed their nam

es to appear on the title page w

ith ours is again imm

easurably large. Som

e of the sections by ourselves and our co-authors were w

ritten es-pecially for this w

ork, while others w

ere edited by us -w

ith an eye to cross-referencing, reduction of repetition, reasonable uniform

ity of nota-tion, and the like -

from pieces and parts of pieces w

hich have appeared elsew

here; with respect to the latter w

e wish to thank not only those w

ho

AcknO

Wledgm

ents xxxi

allowed us to include their w

ritings, but also the following journals and

publishers. Detailed inform

ation appears in the bibliography at the end of this volum

e under the heading given below; section num

bers in parentheses indicate w

here in this volume (a portion of) the cited m

aterial appears. The journal of philosophy:

van Fraassen 1969

(§20.3). The

journal of sym

bolic logic: A

nderson and

Belnap

1962a (§§1-5,

8.1), B

elnap 1960b (§§5.1.2, 22.1.3), 1967 (§19); M

eyer and Dunn 1969 (§25.2); D

unn 1970 (§29.4).

Australasian journal of philosophy:

Meyer

1974 (§29.12). N

otre Dam

e journal of formal logic: M

eyer 1972 (§28.4), 1973b (§§24.4.2, 28.3.1).

Philosophical studies: Anderson and B

elnap 1962 (§§15,

16.1). Zeitschrift

fiir m

athematische

logik und

Grundlagen

der M

athematik:

Anderson 1959 (§23); A

nderson, Belnap and W

allace 1960 (§8.4.3); Belnap

and Wallace 1961 (§13). Logique et analyse: M

eyer 1971 (§22.3). Interm

ittently, many others have m

ade helpful suggestions and offhand illum

inating side-comm

ents; to those whose inform

al conversational ideas and perhaps w

ords, may bave found their w

ay into this book, and unw

ittingly without explicit acknow

ledgment, we offer apO

logies -and

gratitude. W

e have also had the good fortune to secure the services of a large num-

ber of superlatively good secretaries, about one of whom

we would like to

recount the following paradigm

atic tale. On her first day, she w

as given a m

essy manuscript to turn into typescript, involving m

any special logical sym

bols, various under linings for different typefonts, etc. The typescript cam

e back with lots of errors, w

hich we explained to her, finally getting the

reply, "Oh ... I see .... I w

asn't being careful enough." And for the re-

mainder of the year we had the pleasure of w

orking with her, w

e could find (w

ith great difficulty) only a handful of typographical errors. Our luck in

this respect continued to follow us through the follow

ing list: Phyllis Buck,

Berry Coy, B

onnie Towner, B

arbara Care, C

atherine Berrett, M

argaret Ross, M

ary Bender, R

ita Levine, Rita Perstac, and (lastly) R

ita DeM

a-jistre, w

ho is responsible for the superlatively elegant completion of the

whole of the final typescript. Y

ou can well im

agine the sinking sensation you would experience w

ere you handed a body of typescript w

ell sprinkled with henscratches and then

asked to undertake the horrendous task of faithfully translating it onto the printed page. W

e are indebted to Trade Com

position, and in particular to their rem

arkable crew of craftsm

en, for carrying out this task with a degree

of fidelity and sensitivity we would not have believed possible.

Finally, we w

ould like to express our gratitude to two m

embers of the

staff of the Princeton University Press: both helped us enorm

ously. Sanford Thatcher opened our neophyte eyes to the possibility of publishing our w

ork with his press, for w

hich we are extrem

ely grateful, and Gail Filion

xxxii Acknowledgm

ents

addressed herself beautifully to the task of careful word-by-w

ord editing, w

hich saved us from m

any infelicities. The problems of editing and proof-

reading a printed volume of this kind are form

idable, as we know w

ell sim

ply from trying to get the typescript right. W

e will both rem

ember the

unfailing helpfulness, kindness, and patience of those we have dealt with at

the Princeton University Press w

ith great pleasure. Tw

o concluding notes: We are of course dissatisfied w

ith the customary,

and also apparently inevitable, practice of making the" A

cknowledgm

ents" section of a book of this kind appear as a m

ere catalogue of mentors,

colleagues, friends, students, publishers, and the like, who have contributed

to the enterprise. The reason is sadly obvious; it leaves out of account entirely the exciting sense of adventure involved in the actual w

ork: sUdden

unexpected insights and flashes of illumination, as w

ell as disappointments

at being unable to prove a true conjecture or disprove a false one. The sense of joy in creation or discovery is lost in a catalogue of nam

es; it is really the close personal interaction w

ith those friends listed above which accounts

for the euphoric sense of the enterprise. Secondly, w

hile it is not comm

on for co-authors to include each other am

ong acknowledgm

ents, we see nothing in principle that m

akes the practice im

proper or reprehensible. It is, however, difficult in our case

because our respective contributions are intertwined like the "tw

o parts" of a double helix. The closeness of our collaboration is indicated, as an exam

ple, by a scribbled manuscript page in w

hich one of us wrote "re-

duncies," corrected by the other to "rednndacies," and finally corrected by the first again to "redundancies." W

hile it is not true, in general, that we

have together gone over each word in this w

ork in letter-by-letter fashion, we have certainly looked together at every sentence w

ord-by-word, each

paragraph sentence-by-sentence, etc. So we will stand equally convicted

for the errors, inaccuracies, and infelicities which are inevitably to be discov-

ered, and it is unlikely, on points of authorship, that either of us will point

an accusing finger at the other -unless, perhaps, the book com

es under especially brutal or vitriolic attack by the profession at large.

[Note by N

DB.J A

RA

died Decem

ber 5, 1973. He took pleasnre in the

fact that he lived to see the completion of the preparation of the m

anuscript for the printer; each w

ord, save this note, bears his irreplaceable stamp.

EN

TA

ILM

EN

T

CH

APT

ER

I

TH

E PU

RE C

ALC

ULU

S OF EN

TAILM

ENT

§1. The

heart of logic. A

lthough there are

many candidates for

Hlogical connectives," such as conjunction, disjunction, negation, quanti-

fiers, and for some w

riters even identity of individuals, we take the heart of logic to lie in the notion "if ... then -"

; we therefore devote the first chapter to this topic, com

mencing w

ith some rem

arks about the familiar

paradoxes of material and strict "im

plication."

§1.1. "if ... then -

" and the paradoxes.

The "implicational" para-

doxes are treated by most contem

porary logicians somew

hat as follows.

The two-valued propositional calculus sanctions as valid m

any of the obvious and satisfactory principles w

hich we recognize intuitively as valid, such as

(A ---7( B---7 C) )---7( ( A ---7 B)---7( A ---7 C»

and (A ---7 B)---7( (B---7 C)---7( A ---7 C);

it consequently suggests itself as a candidate for a formal analysis of

"if ... then -." To be sure, there are certain odd theorem

s such as

A---7(B---7 A)

and A---7(B---7 B)

which m

ight offend the naive, and indeed these have been referred to in the literature as "paradoxes of im

plication." But this term

inology reflects a m

isunderstanding. "If A, then if B then A" really m

eans no more than

"Either not-A, or else not-B or A," and the latter is clearly a logical

truth; hence so is the former. Properly understood there are no "para-

doxes" of implication.

Of course this is a rather w

eak sense of "implication," and one m

ay for certain purposes be interested in a stronger sense of the w

ord. We

find a formalization of a stronger sense in sem

antics, where" A im

plies B

" means that there is no assignm

ent of values to variables which m

akes A true and B faIse, or in m

odal logics, where we consider strict im

plica-3

4 The heart of logic

Ch. I

§1

tion, taking "if A then B" to m

ean "It is impossible that (A and not-B

)." A

nd, mutatis m

utandis, some rather odd things happen here too. But

again nothing "paradoxical" is going on; the matter just needs to be

understood properly -that's all.

And the w

eak sense of "if ... then -" can be given form

al clothing, after Tarski-Bernays,

as in Lukasiew

icz 1929 (see bibliography), as

follows:

A->(B->

A), (A

-tB)-t((B

->C

)->(A

-tC»,

((A->B)->

A)->A,

with a rule of m

odus ponens. (For reference let this system be T

V

The position just outlined will be found stated in m

any places and by m

any people; we shall refer to it as the Official view

. We agree w

ith the O

fficial view that there are no paradoxes of im

plication, but for reasons w

hich are quite different from those ordinarily given. To be sure, there is a

misunderstanding involved, but it does not consist in the fact that the

strict and material "im

plication" connectives are "odd kinds" of implica-

tion, but rather in the claim that m

aterial and strict "implication" are

"kinds" of implication at all. In w

hat follows, we w

ill defend in detail the view

that material "im

plication" isnot an implication connective. Since our

reasons for this view are logical (and not the usual gram

matical petti-

foggery examined in the G

ramm

atical propaeduetic appearing as an appen-dix to this volum

e), it might help at the outset to give an exam

ple which

will indicate the sort of criticism

we plan to lodge. Let us im

agine a logician who offers the follow

ing formalization as an

explication or reconstruction of implication in form

al terms. In addition to

the rule of modus ponens he takes as prim

itive the following three axiom

s:

A->A

(A->B)->

((B->C

)->(A->

C», and

(A->B)->

(B-tA).

One m

ight find those who w

ould object that "if ... then -" doesn't seem

to be sym

metrical, and that the third axiom

is objectionable. But our logi-

cian has an answer to that.

There is nothing paradoxical about the third axiom; it is just a m

atter of understanding the form

ulas properly. "If A

then B" m

eans simply

"Either A and B are both true, or else they are both false," and if we understand the arrow

in that way, then our rule w

ill never allow us to

infer a false proposition from a true one, and m

oreover all the axioms

are evidently logical truths. The implication connective of this system

§1.2 Program

5

may not exactly coincide w

ith the intuitions of naive, untutored folk, but it is quite adequate for m

y needs, and for the rest of us who are reasonably

sophisticated. And it has the im

portant property, comm

on to all kinds of im

plication, of never leading from truth to falsehood.

There are of course some differences betw

een the situation just sketched and the O

fficial view outlined above, but in point of perversity, m

uddle-headedness, and dow

nright error, they seem to us entirely on a par. O

f course proponents of the view

that material and strict "im

plication" have som

ething to do with im

plication have frequently apologized by saying that the nam

e "material im

plication" is "somew

hat misleading," since it suggests

a closer relation with im

plication than actually obtains. But we can think of lots of no m

ore "misleading" nam

es for the relation: "material conjunc-

tion," for example, or "m

aterial disjunction," or "imm

aterial negation." M

aterial implication is not a "kInd" of im

plication, or so we hold; it is no m

ore a kind of implication than a blunderbuss is a kind of buss. (But see

§§36.2.3-4.)

§1.2. Program

. This brief polem

ical blast will serve to set the tone for

our subsequent formal analysis of the notion of logical im

plication, vari-ously referred to also as "entailm

ent," or "the converse of deducibility" (M

oore 1920), expressed in such logical locutions as "if ... then -,"

"implies," "entails," etc., and answ

ering to such conclusion-signaling logical phrases as "therefore," "it follow

s that," "hence," "consequently," and the like. (The relations betw

een these locutions, obviously connected with the

notion of "logical consequence," are considered, in some cases obliquely, in

the Gram

matical Propaedeutic, and those w

ho are worried about som

e of the m

ore fashionable views m

ay look there for ours.) W

e proceed to the formal analysis as follow

s: In the next subsection, we use natural deduction (due originally, and in-

dependently, to Gentzen 1934 and Jaskow

ski 1934), in the especially per-spicuous variant of Fitch 1952, in order to m

otivate the choice of formal

rules for "->" (taking the arrow

as the formal analogue of the connective

"that ... entails that _"). The reSU

lting system, equivalent to the pure

implicational part

of Heyting's intuitionistic logic (§IA

), is seen to have som

e of the properties associated with the notion of entailm

ent. In the next tw

o sections we argue that, in spite of this partial agreement,

is deficient in two distinguishable respects. First, it ignores considera-

tions of necessity associated with entailm

ent; in §2, modifications of

are introduced to take necessity into account, and these are show

n to lead to the pure im

plicational fragment of the system

S4 of strict implication (Lew

is and Langford

1932). Second, is equally blind to considerations of

relevance; modifications of

in §3, designed to accomm

odate this im-


Ch. I

§1

portant feature of the intuitive logical "if ... then -," yield a calculus

equivalent to the implicational part of the system

of relevant implication

first considered by Moh 1950 and C

hurch 1951. W

ith §4 we are (for the first tim

e) home: com

bining necessity and rele-vance leads naturally and plausibly to the pure calculus

of entailment.

§5 proves that really does capture the concepts of necessity and relevance

in certain mathem

atically definite senses, and with this w

e complete the

main argum

ent of the chapter. The remaining sections present a num

ber of related results: in §6 we define an even stricter form

of entailment, called

"ticket-entailment," answ

ering to a conception of entailment as an '''in-

ference-ticket"; in §7 we sketch consecution calculuses in the style of G

ent-zen for various system

s; and §8 collects odd bits of information (and a few

questions) about the system

s thus far considered. Let us pause briefly to fix som

e notational matters. W

e remind the

reader that in this chapter we are considering only pure im

plicational system

s, leaving connections between entailm

ent and other logical notions until later. C

onsequently we can describe the languages we are considering

as having the following structure. In the first place, we suppose there is an

infinite list of propositional variables, which w

e never display. But w

e shall often use

p, q, r, S,

etc., as variables ranging over them. Thenform

ulas are defined by specifying that all propositional variables are form

ulas, and that whenever A and B

are, so is (A->B). A

s variables ranging over formulas we em

ploy

A,B

, C,D

,

etc., often with subscripts. W

e warn the reader that for the purpose of the

present discussion we use the arrow ambiguously in order to com

pare various proposed form

alizations of entailment.

As a further notational convention, w

e use dots to replace parentheses in accordance w

ith conventions of Church 1956: outerm

ost parentheses are om

itted; a dot may replace a left-hand parenthesis, the m

ate of which is to

be restored at the end of the parenthetical part in which the dot occurs

(otherwise at the end of the form

ula); otherwise parentheses are to be

restored by association to the left. Example: each of (A->

.B->C

)->.A->

B

->.A

->C

and A->

(B-->C)-->

.A-->B->

.A-->C abbreviates

((A-->(B-->C))--> ((A-->

B)->(A --> C

))).

§1.3. N

atural deduction. The intuitive idea lying behind system

s of natural deduction is that there shonld be, for each logical connective, one rule justifying its introduction into discourse, and one rule for using ("elim

i-

§1.3 N

atural deduction 7

nating") the connective once it has been introduced. For extended discus-sions of this m

otivation see Curry 1963, Popper 1947, or K

neale 1956, and for hazards attendant on careless statem

ents of the leading ideas, see Prior 1960-61 and B

elnap 1962. Since w

e wish to interpret "A

-+B" as "A

entails B," or "B is deducible from

A," we clearly w

ant to be able to assert A-->B whenever there exists a

deduction of B from A, i.e., we will w

ant a rule of Entailm

ent Introducaon, hereafter "-->1," having the property that if

A hypothesis (hereafter "hyp")

B [conclusion]

is a valid deduction of B from A

, then A->

B shall follow

from that deduction.

(This sentence contains a lapse from gram

mar, the first of m

any. If you did not notice it, or if it did not bother you, please go on; only if our solecism

irritates you, consult the G

ramm

atical Propaedeutic for a statement and

defense of our policy of loose gramm

ar.) M

oreover, the fact that such a deduction exists, or correspondingly that an entailm

ent A->

B holds, w

arrants the inference of B from A

. That is, w

e expect also that a rule of m

odus ponens or Entailment Elim

ination, hence-forth

will obtain

in the sense that whenever

is asserted we shall be entitled to infer B from

A. '

So much is sim

ple and obvious, and presumably not open to question.

Problems arise, how

ever, when we ask w

hat constitutes a "valid deduction" of B from

A. How

may we fill in the dots in the proof schem

e above? A

t least one rule Seems as sim

ple and obvious as the foregoing. Certainly

the supposition that A warrants the (trivial) inference that A; and if B has

been deduced from A, we are entitled to infer B On the supposition A. T

hat is, w

e may repeat ourselves:

A hyp

B ?

j B

i repetition (henceforth "rep")

This rule leads imm

ediately to the following theorem

, the law o(identity:

2 3

hyp I rep 1-2 ->1


Ch. I §1

We take the law

of identity to be a truth about entailment; A-->A represents

the archetypal form of inference, the trivial foundation of all reasoning, in

spite of those who w

ould call it "merely a case of stuttering." Straw

son 1952 (p. 15) says that

a man w

ho repeats himself does not reason. But it is inconsistent to

assert and deny the same thing. So a logician w

ill say that a statement

has to itself the relationship [entailment] he is interested in.

Strawson has got the cart before the horse: the reason that A and;;: are

inconsistent is precisely because A follows from

itself, rather than con-versely. (W

e shall in the course of subsequent investigations accumulate a

substantial amount of evidence for this view

, but the most convincing argu-

ments w

ill have to await treatm

ent of truth functions and propositional quantifiers in connection w

ith entailment. For the m

oment we observe that

the difference between Straw

son's view and our ow

n first emerges form

ally in the system

E of Chapter IV

, where we have A

-->A-->A

&A

but not A&A-->.A-->A, just as we have A-->B-->A&B but not A&B-->.A-->B).

But obviously m

ore than the law of identity is required if a calculus of

entailment is to be developed, and we therefore consider initially a device

contained in the variant of natural deduction of Fitch 1952, which allow

s us to construct within proofs of entailm

ent, further proofs of entailment

called "subordinate proofs," or "subproofs." In the course of a deduction, under the supposition that A (say), we m

ay begin a new deduction, w

ith a new

hypothesis:

hyp

hyp

The new subproof is to be conceived of as an "item

" of the proof of which

A is the hypothesis, just like A or any other formula occurring in that proof.

And the subproof of w

hich B is hypothesis might itself have a consequence

(by -->1) occurring in the proof of w

hich A is the hypothesis. W

e next ask whether or not the hypothesis A holds also under the as-

sumption B. In the system

of Fitch 1952, the rules are so arranged that a step follow

ing from A in the outer proof m

ay also be repeated under the assum

ption B, such a repetition being called a "reiteration" to distinguish it from

repetitions within the sam

e proof or subproof: I I

§1.3 N

atural deduction 9

A hyp

C ?

j n hyp

k i reiteration ("reit")

We designate as

the system defined by the five rules, -->1, -->E, hyp, rep,

and reit. A proof is categorical if all hypotheses in the proof have been dis-

charged by use of -->1, otherwise hypothetical; and A is a theorem

if A is the last step of a categorical proof. These rules lead naturally and easily to proofs of intuitively satisfactory theorem

s about entailment, such as the

following law of transitivity.

A-->B hyp

2 B-->C

hyp

3 A-->B

1 reit 4 1:-'

hyp 5

3 reit 6

45 -->E 7

B-->C

2 reit 8

C 67 -->E

9 A-->C

4-8 -->1

10 B-->C

-->.A-->C

2-9 -->1 II

A-->B-->,B-->C-->.A-->C

1-10 -->1

Lewis indeed doubts w

hether this proposition should be regarded as a valid principle of deduction: it w

ould never lead to any inference A-->C w

hich would be questionable

when A----'?B and B---'tC are given prem

isses; but it gives the inference B-->C

-->.A-->C w

henever A-->B is a premiss. Except as an elliptical state-

ment for "(A-->B)&(B-->C

)-->.A-->C and A-->8 is true," this inference is

dubious. (Lewis and Langford 1932, p. 496.)

On the contrary, A

ckermann 1956 is surely right that "unter der V

oraus-setzung A-->B ist der Schluss von B-->C

auf A-->C logisch zw

ingend." The m

athematician is involved in no ellipsis in arguing that "if the lem

ma is

deducible from the axiom

s, then this entails that the deducibility of the theorem

from the axiom

s is entailed by the deducibility of the theorem from

the lem

ma."


Ch.l

§1

The proof method sketched above has the advantage, in com

mon w

ith other system

s of natural deduction, of motivating proofs: in order to prove

A->B (perhaps under som

e hypothesis or hypotheses), we follow

the simple

and obvious strategy of playing both ends against the middle: breaking up

the conclusion to be proved, and setting up subproofs by hyp until we find one w

ith a variable as last step. Only then do w

e begin applying reit, rep, and ->E.

As a short-cut we allow

reiterations directly into subproofs, subsub-proofs, etc., w

ith the understanding that a complete proof requires that

reiterations be performed alw

ays from one proof into another proof im

-m

ediately subordinate to it. As an exam

ple (step 6 below), w

e prove the self-distributive law

(H42, below

):

A -> . B-> C hyp

2 A->

B hyp

3 A

hyp 4

A->B

2 reit 5

B 3 4->

E

6 A->

.B->C

1 reit (tw

ice) 7

B->C

3 6 ->E

8

C 5 7 ->E

9

A--->C 3-8 --->1

10 A -> B->

. A -> C 2-9 ->1

11 (A -> .B->

C)->

. A -> B-> . A -> C 1-10 ->1

§.1.4. Intuitionistic im

plication (H4 ).

Fitch 1952 shows (essentially)

that the set of theorems of FH

4 stemm

ing from these rules is identical w

ith the pure im

plicational fragment H

4 of the intuitionist propositional calculus of H

eyting 1930 (called "absolute implication" by C

urry 1959 and else-w

here), which consists of the follow

ing three axioms, w

ith ->E as the sole

rule: H41

A->.B->

A H

42 (A->

.B->C

)->.A->

B->.A->

C

H43

A->A

(F ormulations like that of H

4 just above, defined by axioms and rules -

often just -

we refer to som

etimes as H

ilbert systems or form

ulations, som

etimes as

or axiomatic form

ulations. Observe that

among H

41-3, H4 3 is redundant.)

In order to introduce terminology and to exem

plify a pattern of argument

which w

e shall have further occasion to use, we shall reproduce Fitch's proof that the tw

o formulations are equivalent.

§1,4 Intuitionistic im

plication 11

To see that the subproof formulation FH

4 contains the Hilbert form

ula-tion H

4 , we deduce the axioms of H

4 in FH4 (H

42 was just proved and

H41 is proved below

) and then observe that the only rule of H4 is also a

rule of FH4 . It follow

s that FH4 contains H

4 . To see that the axiom

atic system H

4 contains the subproof formulation

FH

., we first introduce the notion of a quasi-proof in FH4 ; a quasi-proof

differs from a proof only in that we m

ay introduce axioms of H

4 as steps (and of course use these, and steps derived from

them, as prem

isses). We

note in passing that this does not increase the stock of theorems of FH

4 , since we m

ay think of a step A, inserted under this rule, as corning by reiteration from

a previous proof of A in FH4 (w

hich we know exists since

FH4 contains H

4 ); but we do not use this fact in our proof that IL con-

tains FH4 .

Our object then is to show

how subproofs in a quasi-proof in FH

4 may

be systematically elim

inated in favor of theorems of H

4 and uses of ->E, in such a w

ay that we are ultimately left w

ith a sequence of formulas all of

which are theorem

s of H4. This reduction procedure alw

ays begins with an

innermost subproo!, by w

hich we m

ean a subproof Q w

hich has no proofs subordinate to it. Let Q

be an innermost snbprbof of a quasi-proof P of

FH4 , w

here the steps of Q are A

I, ... , A" let Q

' be the sequence AI->A

j, AI->

A2, ... , Aj->A", and let P' be the result of replacing the subproof Q

of P by the sequence Q

' of formulas. O

ur task is now to show

that P' is convertible into a quasi-proof, by show

ing how to insert theorem

s of H4

among the wffs of Q

', in such a way that each step of Q

' may be justified

by one ofhyp, reit, rep, ->E, or axiomhood in H

4 (the case ->1 will not arise

because Q is innerm

ost). A

n inductive argument then show

s that we may justify steps in Q

' as follow

s: A

I->Al is justified, by H

43. If Ai w

as by rep in Q, then, by the inductive hypothesis, A

l->A

i is by rep in Q

'. If Ai w

as by reit in Q, then in Q

' insert Ai->.Aj->

Ai (H4 1) and use->E

to get A

j->A

i (the minor prem

iss heing an item of the qnasi-proof in P

to which Q

is subordinate, hence also preceding Q' in P').

If Ai was by ->E

in Q, w

ith premisses A

j and Aj->A

i , then in Q' w

e have A

r-.Aj and A

j->.A

j->A

i . Then insert H42 and use ->E

twice to get A

j->A

i as required.

If Ai was an axiom

-recall w

e are dealing with quasi-proofs -

then insert Ai->

.Aj->Ai (H

41) and use -->E to obtain Al--->Ai. So every step in Q

' is justified. Now

notice that we can conclude that every step in all of P' is justified, for P' is exactly the sam

e as P except that Q

(in P) has been replaced by Q'. The only possible trouble m

ight be


Ch. I §I

if some step in P w

ere justified through -.1 by' a reference to the now

absent Q; but such a step can be justified in P' by rep, w

ith a reference to the last line of Q

'. R

epeated application of this reduction then converts any proof in into a sequence of form

ulas all of which are theorem

s of hence the

latter system contains the form

er, and the two are equivalent. N

otice in-cidentally that the choice of axiom

s for m

ay be thought of as motivated

by a wish to prove

and equivalent: they are exactly w

hat is required to carry out the inductive argum

ent above. (W

e retain the concepts of quasi-proof and innermost subproof, w

ith som

e sophistications, for use in later arguments w

hich are closely similar

to the foregoing.) The axiom

s of also enable us to prove a slightly different form

of the result above. W

e consider proofs with no subproofs, but w

ith multiple

hypotheses, and we define a proof of B from

hypotheses AI, ... , A" (in the O

fficial way) as a sequence CI, ... , Cm, B of form

ulas each of which is

either an axiom, or one of the hypotheses A

i ,or a consequence of predeces-sors by -.E

. Then we arrive by very sim

ilar methods at the O

fficial form of

the DE

DU

CT

ION

THEO

REM.

If there exists a proof of B from the hypotheses

AI, ... , A", then there exists a proof of A"-.B

on the hypotheses AI, ... , An_I; and conversely.

We return now

to consideration of w

hich is proved in as

follows:

2 3 4 5 A

-'.B-.A

hyp hyp 1 reit 2-3 -.1 1-4 -.1

Thus far the theorems proved by the subordinate proof m

ethod have all seem

ed natural and obvious truths about our intuitive idea of entailment.

But here w

e come upon a theorem

which shocks our intuitions (at least our

untutored intuitions), for the theorem seem

s to say that anything whatever

has A as a logical consequence, provided only that A is true; if the formal

machinery is offered as an analysis or reconstruction of the notion of en-

tailment, or form

al deducibility, the principle seems outrageous -

such at least is alm

ost certain to be the initial reaction to the theorem, as anyone

who has taught elem

entary logic very well know

s. Formulas like A

-'.B-.A

and A

-'.B-.B

are of course familiar, and m

uch discussed under the heading of "im

plicational paradoxes."

§1,4 Intuitionistic im

plication (H4-) 13

Those whose view

s concerning the philosophy of logic comm

it them to

accept such principles are usually quick to point out that the freshman's

objections are founded on confusion. For example, Q

uine 1950 (p. 37) says that a confusion of use and m

ention is involved, and that (in effect) although

A implies (B im

plies A)

may be objectionable,

if A then if B then A

is not. We have dealt w

ith this sort of gramm

atical point in the Gram

matical

Propaedeutic at the end of this volume. B

ut it is worth rem

arking here that even if Q

uine and his followers are correct about the gram

mar of English

(or any other natural language), it is still true that the naive freshman objects

as much to the second of the tw

o formulations as to the first. So do we.

And C

urry 1959 explains that the arrow of

does not lay any claim to

being a definition of logical consequence. "It does not pretend to be any-thing of the sort" (p. 20). The claim

is supported by an argument to the

effect that "far from

being paradoxical," is, for any proper im-

plication, "a platitude." A "proper" im

plication is defined by Curry as any

implication w

hich has the following properties: there is a proof of B from

the hypotheses AI, ... , A"_I, A" (in the O

fficial sense of "proof from hy-

potheses") if and only if there is a proof of A"-.B

from the hypotheses

AI, ... , A"_I. On these grounds A

-'.B-.A

is indeed a platitude: there is surely a proof of A from

the hypotheses A, B; and hence for any "proper" im

plication, a proof of B-. A from

the hypothesis A; and hence a proof w

ithout hypotheses of A->.B->

A. C

urry calls this a proof of A->

.B-.A

"from nothing." W

e remark that

this expression invites the interpretation "there is nothing from

which A->

.B->A is deducible," in w

hich case we w

ould seem to have done little

toward show

ing that it is true. But of course C

urry is not confused on this point; he m

eans that A->

.B-.A

is deducible "from" the null set of prem

isses -

in the reason-shattering, Official sense of "from

." (These arguments

deserve to be taken more seriously than our tone suggests; we w

ill try to do so w

hen the matter com

es up again in connection with the notion of rele-

vance, in §3.) C

urry goes on to dub the implicational relation of

"absolute implica-

tion" on the grounds that is the m

inimal system

having this property. B

ut we notice at once that is "absolute" only relatively, i.e., relatively

to the Official definition of "proof from

hypotheses." From this point of

view, our rem

arks to follow m

ay be construed as arguing the impropriety of

accepting the Official definition of "proof from

hypotheses," as a basis for defining a "proper im

plication"; as we shall claim

, the Official view

captures


Ch. I §1

neither "proof" (a matter involving logical necessity) nor "from

" (a matter

requiring relevance). But even those w

ith intuitions so sophisticated that A->

.B->A seem

s tolerable might still find som

e interest in an attempt to

analyze our initial feelings of repugnance in its presence. W

hy does A->.B->

A seem so queer? W

e believe that its oddness is due to tw

o isolable features of the principle, which we consider forthw

ith.

§2. N

ecessity: strict implication (S44)'

For more than tw

o millennia

logicians have taught that logic is aformal m

atter, and that the validity of an inference depends not on m

aterial considerations, but on formal considera-

tions alone. We here approach a m

ore accurate statement of this condition

in several steps, first noting that it amounts to saying that the validity of a

valid inference is no accident of nature, but rather a property a valid in-ference has necessarily. Still m

ore accurately: an entailment, if true at all,

is necessarily true. Because true entailm

ents are necessarily so, we ought to grant, as w

e do, that truths entailed by necessary truths are them

selves one and all necessary; and we then see im

mediately that A->

.B->A violates this plausible condition.

For let A be contingently true, and B necessarily true; then given A->.B->

A, A leads to B->

A, and now we have a necessity entailing a contingency, w

hich is nO good. That is to say, for such an instance of A->

.B->A, the antecedent

is true, and the consequent false. Note that this argum

ent is equally an argum

ent against the weaker A:=l.B-'J.A, w

here now the horseshoe is m

aterial "im

plication"; i.e., A is true w

hile B---}A is false. (W

e thank Routley and R

outley 1969 for pointing out a howler [see

§20.2] in the version of this argument in A

nderson and Belnap 1962a; C

olfa straightens us out on the matter in §22.1.2.)

It might be said in defense of A->

.B->A as an entailm

ent that at least it is ""safe," in the sense that if A is true, then it is alw

ays safe to infer A from an

arbitrary B, since we run no risk of uttering a falsehood in doing so; this thought ("Safety First") seem

s to be behind attempts, in a num

ber of elem

entary logic texts, to justify the claim that A->

.B->A has som

ething to do w

ith implication. In reply we of course adm

it that if A is true then it is "safe" to say so (i.e., A->

A). But saying that A is true on the irrelevant

assumption that B, is not to deduce A from

B, nor to establish that B implies

A, in any sensible sense of "implies." O

f course we can say "A

ssume that

snow is puce. Seven is a prim

e number." But if w

e say "Assum

e that snow is

puce. Itfollows that (or consequently, or therefore, or it m

ay validly be inferred that) seven is a prim

e number," then we have sim

ply spoken falsely. A m

an w

ho assumes the continuum

hypothesis, and then remarks that it is a nice

day, is not inferring the latter from the form

er -even if he keeps his sup-

position fixed firmly in m

ind while noting the w

eather. And since a (true)

A does not follow from

an (arbitrary) B, we reject A->.B->

A as expressmg

§2 Necessity (S4

4)

15

a truth of entailment or im

plication, a rejection which is in line w

ith the view

(shared even by some w

ho hold that A->.B->

A expresses a fact about "if ... then _

") that entailments, if true at all are necessarily true.

How

can we modify the form

ulation of H4 in such a w

ay as to guarantee that the im

plications expressible in it shall reflect necessity, rather than contingency? A

s a start, picture an (outermost) subproof as exhibiting a

mathem

atical argument of som

e kind, and reflect that in our usual mathe-

matical or logical proofs, we dem

and that all the conditions required for the conclusion be stated in the hypothesis of a theorem

. After the w

ord "PR

OO

F:" in a mathem

atical treatise, mathem

atical writers seem to feel

that no more hypotheses m

ay be introduced; and it is regarded as a criticism

of a proof if not all the required hypotheses are stated explicitly at the out-set. O

f course additional machinery m

ay be invoked in the proof, but this m

ust be of a logical character, i.e., in addition to the hypotheses, we may

use in the argument only propositions tantam

ount to statements of logical

necessity. These considerations suggest that we should be allowed to im

port into a deduction (i.e., into a subproof by reit) only propositions w

hich, if true at ail, are necessarily true: i.e., w

e should reiterate only entailments. O

f course the illustration directly m

otivates the restriction only for outermost

subproofs, but the same reasoning justifies extending the restriction to all

subproofs: if at any stage of an argument one is attem

pting to establish, under a batch of hypotheses, a statem

ent of a logical character -in our

case, an entailment -

then one should be allowed to bring in from

the outside (by reiteration) only those steps w

hich themselves have the appro-

priate logical character, i.e., entailments. A

nd indeed such a restriction on reiteration w

ould imm

ediately rule out A->.B->

A as a theorem, w

hile countenancing all the other theorem

s we have proved thus far. We call the

system w

ith reiteration allowed only for entailm

ents FS44, and proceed to prove it equivalent to the follow

ing axiomatic form

ulation, which we call

S44' since it is the pure strict "implicational" fragm

ent of Lewis's S4. (See

Hacking 1963).

S44 1

A->A

S44 2

(A->.B->

C)->

.A->B->

.A--+C

S4

4 3 A->

B->.C

->.A->

B

It is a trivial matter to prove the axiom

s of S44 in FS44 , and the only rule of S4 4 (->E

) is also a rule of FS44 ; hence FS44 contains To establish

the converse, we show how

to convert any quasi-proof of a theorem A in

FS44 into a proof of A in S44.

THEOREM.

Let AI, ... , A" be the item

s of an innermost subproof Q

of a quasi-proof P, and let

Q'

be the sequence Ar-..."Al, ... , A1--'!-An , and

16 N

ecessity (S4_) Ch. I

§2

finally let P' be the result of replacing the subproof' Q in P by the sequence

of formulas Q

'. Then P' can be converted into a quasi-proof.

PROOF. First w

e prove that each step of Q' can be justified, by induc-

tion on n. For n = 1 w

e note that A,->

A, is an instance of S4_1. Then,

assuming the theorem

for all i < n, consider A,->

A,.

CASE 1. A

, is by repetition in Q of Ai. Then treat A

,->A

, in Q' as a

repetition of A,->Ai.

CASE 2. A

, is a reiteration in Q of B. Then B

has the form C->D

, by the restriction on reiteration. Insert C

->D

->.A,->

.C->

D in Q

' by S4_3, and treat A,->

.C->

D (i.e., A,->

A,) as a consequence of C->D

(i.e., B), and S4_3 by ->E

.

CASE 3. A

, follows in Q

from Ai and Ai->

A, by ->E. Then by the in-

ductive hypothesis we have A,->Ai and A,->

.Ai->A, in Q

'. Then A,->

A,

is a consequence of the latter and S4_2, with tw

o uses of ->E.

CASE 4. A

, is an axiom. Then A

, has the form B->

C; so it follow

s from

S4_3 that A,->

A, is a theorem

.

Now

we m

ay conclude that P' is a quasi-proof. For a step A,->

A" re-

garded as a consequence of Q in P, m

ay now be regarded as a repetition

of the final step A,->

A, of Q

' in P'. H

ence P' is convertible into a quasi-proof. And repeated application of

this technique to P' eventually leads to a sequence P" of formulas each of

which is a theorem

of S4_. Hence S4_ includes FS4_, and the tw

o are equivalent.

A deduction theorem

of the more usual sort is provable also for S4_:

THEOREM.

If there is a proof of B on hypotheses A" ... , A

, (in the O

fficial sense), where each Ai, 1 ::; i ::; n, has the form

C->D, then there is

a proof of A,->B on hypotheses A"

... , A,_,. (B

arcan Marcus 1946; see

also Kripke 1959a.)

Notice again that as in the case of H

_, the choice of axioms for S4_ m

ay be thought of as m

otivated exactly by the wish to prove an appropriate

deduction theorem.

The restriction on reiteration suffices to rem

ove one objectionable feature of H

_, since it is now no longer possible to establish an entailm

ent B->A

when A is contingent and B is necessary. B

ut of course it is well know

n that the "im

plication" relation of S4 is also paradoxical, since we can easily establish that an arbitrary irrelevant proposition B "im

plies" A, provided

§3 Relevance (R_)

17

A is a necessary truth. A->A is necessarily true, and from

it and S4_3 follow

s B->.A->

A, where B m

ay be totally irrelevant to A->A. O

bserve that B->

.A->A does not violate the intuitive condition laid dow

n at the outset of this section as a basis for dism

issing A--+.B--+A; we cannot by the sam

e de-vice assign values to A and B so that the antecedent of B->

.A->A com

es out true, and the consequent false. The presence of B->

.A->A therefore leads us

to consider an alternative restriction on H ... , designed to exclude such

fallacies of relevance.

§3. Relevance: relevant im

plication (R_). For m

ore than two m

illennia logicians have taught that a necessary condition for the validity of an in-ference from

A to B is that A be relevant to B.V

irtually every logic book up to the present century has a chapter on fallacies of relevance, and m

any contem

porary elementary texts have follow

ed the same plan. N

otice that contem

porary writers, in the later and m

ore formal chapters of their books,

seem explicitly to contradict the earlier chapters, w

hen they try desperately to bam

boozle the students into accepting strict "implication" as a "kind" of

implication relation, in spite of the fact that this relation countenances

fallacies of relevance. But the denial that relevance is essential to a valid

argument, a denial w

hich is implicit in the view

that "formal deducibility,"

in the sense of Montague and H

enkin 1956 and others, is an implication re-

lation, seems to us flatly in error.

Imagine, if you can, a situation as follow

s. A m

athematician w

rites a paper on B

anach spaces, and after proving a couple of theorems he con-

cludes with a conjecture. A

s a footnote to the conjecture, he writes: "In addition to its intrinsic interest, this conjecture has connections w

ith other parts of m

athematics w

hich might not im

mediately occur to the reader. For

example, if the conjecture is true, then the first order functional calculus is

complete; w

hereas if it is false, then it implies that Ferm

at's last conjecture is correct." The editor replies that the paper is obviously acceptable, but he finds the final footnote perplexing; he can see no connection w

hatever be-tw

een the conjecture and the "other parts of mathem

atics," and none is indicated in the footnote. So the m

athematician replies, "W

ell, I was using

'if ... then -' and 'im

plies' in the way that logicians have claim

ed I was:

the first order functional calculus is complete, and necessarily so, so any-

thing implies that fact -

and if the conjecture is false it is presumably im

-possible, and hence im

plies anything. And if you object to this usage, it is

simply because you have not understood the technical sense of 'if ... then

-' w

orked out so nicely for us by logicians." And to this the editor coun-

ters: "I understand the technical bit all right, but it is simply not correct. In

spite of what m

ost logicians say about us, the standards maintained by this

journal require that the antecedent of an 'if ... then -' statem

ent must be

18 Relevance (R_,)

Ch. I

§3

relevant to the conclusion drawn. A

nd you have given no evidence that your conjecture about B

anach spaces is relevant either to the completeness theo-

rem or to Ferm

at's conjecture." N

ow it m

ight be thought that our mathem

atician's footnote should be re-garded as true, "if ... then -

" being taken m

aterially or (more likely)

strictly -but sim

ply uninteresting because of its triviality. But notice that

the editor's reaction was not "'But heavens, that's trivial" (as the contention

that the mathem

atical "if ... then -" is the sam

e as material "im

plication" w

ould require); any such reaction on the part of an editor would properly

be judged insane. His thought w

as rather, "I can't see any reason for think-ing that this is true."

No, the editor's point is that though the technical m

eaning is clear, it is sim

ply not the same as the m

eaning ascribed to "if ... then -" in the pages

of his journal. Furthermore, he has put his finger precisely on the difficulty:

to argue from the necessary truth of A to if B then A is sim

ply to comm

it a fallacy of relevance. The fancy that relevance is irrelevant to validity strikes us as ludicrous, and we therefore m

ake an attempt to explicate the notion of

relevance of A to B. For this we return to the notion of proof from

hypotheses (in standard axiom

-cum---+E form

ulations), the leading idea being that we want to infer

A--+B from "a proof of B from

the hypothesis A." A

s we pointed out before, in the usual axiom

atic formulations of propositional calculuses the m

atter is ll.andled as follow

s. We say that A

I, ... , A. is a proof o

f B from the hy-

pothesis A, if A =

Al, B = An, and each Ai is either an axiom

or else a con-sequence of predecessors am

ong AI, ... , A

. by one of the rules. But in the

presence of a deduction theorem of the form

: from a proof of B on the hy-

pothesis A, to infer A--+B, this definition leads imm

ediately to fallacies of relevance; for if B is a theorem

independently of A, then we have A--+B where

A may be irrelevant to B. For exam

ple, in a system w

ith A--+A as an axiom,

we have

I B

2 A--+A

3 B--+.A--+A

hyp axw

m

1-2, deduction theorem

In this example we indeed proved A--+ A

, but, though our eyes tell us that we proved it under the hypothesis B, it is crashingly ·obvious that we did not prove it from

B: the defect lies in the definition, which fails to take seriously

the word "from

" in "proof from hypotheses." A

nd this fact suggests a solu-tion to the problem

: we should devise a technique for keeping track of the steps used, and then allow

application of the introduction rule only when A

is relevant to B in the sense that A is used in arriving at B.

§3 Relevance

19

As a start in this direction, we suggest affixing a star (say) to the hypothe-

sis of a deduction, and also to the conclusion of an application of --+E just in case at least one prem

iss has a star, steps introduced as axioms being un-

starred. Restriction of

to cases where in accordance with these rules

both A and B are starred would then exclude theorem

s of the form A--+B,

where B is proved independently of A. In other w

ords, what is w

anted is a system, analogous to

and for

which there is provable a deduction theorem

to the effect that there exists a proof of B from

the hypothesis A if and only if A--+B is provable. And we

now consider the question of choosing axiom

s in such a way as to guarantee

this result. In view of the rule -+E

, the implication in one direction is trivial;

we consider the converse. Suppose we have a proof

AJ* hyp

A, ?

An* ?

of A" from the hypothesis A

I, in the above sense, and we wish to convert this

into an axiomatic proof of A 1

-+ An. A

natural and obvious suggestion would

be to consider replacing each starred A, by Al--+A, (since the starred steps are the ones to w

hich Al is relevant), and try to show

that the result is a proof w

ithout hypotheses. What axiom

s would be required to carry the in-

duction through? For the basis case we obviously require as an axiom

A->A

. And in the

inductive step, where w

e consider steps Ai and of the original proof,

four cases may arise.

(I) Neither prem

iss is starred. Then in the axiomatic proof, A"

A,--+Aj, and A

j all remain unaltered, so -+E

may be used as before.

(2) The minor prem

iss is starred, and the major one is not. Then in the

axiomatic proof w

e have A1-+A

j and Aj-+A

j ; so we need to be able to infer

Al--+Aj from these (since the star on A

, guarantees a star on Aj in the

original proof). (3) The m

ajor premiss is starred, and the m

inor one is not. Then in the axiom

atic proof we have At-+.A

i-+Aj and A

i, so we need to be able to infer

Al--+Aj from these.

(4) And finally both m

ay be starred, in which case we have Al--+.A,--+A

j and Al--+A, in the axiom

atic proof, from w

hich again we need to infer A

t-+Aj.

20 C

h. I §3

Summ

arizing: the proof of an appropriate deduction theorem w

here relevance is dem

anded would require the axiom

A.-7 A together with the

validity of the following inferences:

from A.-7B and B.-7C

to infer A.-7C;

from A.-7.B.-7C

and B to infer A.-7C;

from A.-7.B.-7C

and A.-7B to infer A.-7C.

It then seems plausible to consider the follow

ing axiomatic system

as capturing the notion of relevance:

A.-7A A.-7B.-7.B.-7C

--;.A--;C

(A.-7.B--;C)--;.B.-7.A--;C

(A--;.B.-7C

).-7.A.-7B.-7.A--;C

(identity) (transitivity) (perm

utation) (self-distribution)

And w

ithout further proof we state that for this system R

.' (R_ gets de-fined below

) we have the following

THEOREM.

A.-7B is a theorem of R

_' just in case there is a proof of B from

the hypothesis A (in the starred sense).

Equivalent systems have been investigated by M

oh 1950 and Church

1951. (See also Kripke 1959a.) C

hurch calls his system the "w

eak positive im

plicational propositional calculus," and uses the follow

ing axiom

s:

A.-7A A--;B--;.C

.-7A.-7.C--;B

(A.-7.B--;C)--;.B--;.A--;C

(A --;.A --; B)--;.A --; B

(identity) (transitivity) (perm

utation) (contraction)

Following a suggestion w

hich Bacon m

ade to us in 1962, we think of this as a system

of "relevant implication," hence the nam

e "R .... ," since relevance of

antecedent to consequent, in a sense to be explained later, is secured thereby. The sam

e suggestion was also m

ade by Prawitz. first in a m

imeographed

version of Prawitz 1964 distributed to those attending the m

eeting at which

the abstracted paper was read, and then in the m

ore extended discussion in Praw

itz 1965. The proof that R

_' and R_ are equivalent is left to the reader. A

generalization of the deduction theorem above w

as proved by both M

oh 1950 and Church 1951; m

odified to suit present purposes, it may be

stated as follows:

§3 Relevance (R_)

21

THEOREM.

If there exists a proof of B on the hypotheses AI, ... , A" in

which all of A

l, ... , An are used in arriving at B, then there is a proof of A,--;B from

AI, ... , A

,_I satisfying the same condition.

So put, the result acquires a rather peculiar appearance: it seems odd that

we should have to use all the hypotheses. One w

ould have thought that, for a group of hypotheses to be relevant to a conclusion, it w

ould suffice if some

of the hypotheses were used -

at least if we think of the hypotheses as taken conjointly (see the Entailm

ent theorem of §23.6). The peculiarity

arises because of a tendency (thus far not comm

ented on) to confound

with

... -'>.An-)B

We w

ould not expect to require that all the Ai berelevantto B in order for the first form

ula to be true, but we shall give reasons presently, deriving from

another formulation of R_, for thinking it sensible that the truth of the

nested implication requires each of the Ai to be relevant to B; a feature of the

situation which w

ill lead us to make a sharp distinction betw

een the two

formulas (see §22.2.2). It is presum

ably the failure to make this distinction

which leads C

urry 1959 (p. 13) to say of the relation considered in Moh's

and Church's theorem

above that it is one "which is not ordinarily con-

sidered in deductive methodology at all." (H

e's right; it's not. But it ought

to be, for there is where the heart lies.)

We feel that the star form

ulation of the deduction theorem m

akes clearer w

hat is at stake in R_. On the other hand the deduction theorem

of Moh and

Church has the m

erit of allowing for proof of m

ultiply nested entailments in

a more direct w

ay than is available in the star formulation. O

ur next task therefore is to try to com

bine these approaches so as to obtain the ad-vantages of both.

Returning now

to a consideration of subordinate proofs, it seems natural

to try to extend the star treatment, using som

e other symbol for deductions

carried out in a subproof, but retaining the same rules for carrying this sym

-bol along. W

e might consider a proof of contraction in w

hich the inner hy-pothesis is distinguished by a dagger rather than a star:

* t hyp hyp

22 Relevance (R_)

Ch. I §3

the different relevance marks reflecting the initial' assum

ption that the two

formulas, as hypotheses, are irrelevant to each other (or, equivalently, our

initial ignorance as to whether they are irrelevant to each other). Then

generalizing the starring rules, we m

ight require that, in application of --->E, the conclusion B m

ust carryall the relevance marks of both prem

isses A and A--->B, thus:

1 I A--->.A--->B 2 r A 3

A--->.A--->B 4

A--->B 5

B

* hyp

t hyp

* 1 reit

*t 2 3--->E

*t 2 4--->E

To motivate the restriction on --->1, w

e recall that, in proofs involving only stars, it w

as required that both A and B have stars, and that the star was dis-

charged on A--->B in the conclusion of a deduction. This suggests the follow-

ing generalization: in drawing the conclusion A--->B by --->1, w

e require that the relevance sym

bol on A also be present among those of B, and that in the

conclusion A--->B the relevance symbol of A (like the hypothesis A itself) be

discharged. Two applications of this rule then lead from

the proof above to

6 A--->B

7 (A --->.A ---> B)--->.A ---> B

* 2-5 --->1 1-6 --->1

But of course the easiest w

ay of handling the matter is to use classes of

numerals to m

ark the relevance conditions, since then we m

ay have as many

nested subproofs .as we w

ish, each with a distinct num

eral (which w

e shall w

rite in subscripted set-notation) for its hypothesis. More precisely w

e allow

that: (1) one may introduce a new

hypothesis Alkl, where k should be differ-

ent from all subscripts on hypotheses of proofs to w

hich the new proof is

subordinate; (2) from Aa and A--->Bb w

e may infer B

aUb

; (3) from a proof of

Ba from the hypothesis Alkl, w

e may infer A--->B._Ikl, provided k is in a; and

(4) reit and rep retain subscripts (where a, b, c, range over sets of num

erals). As an exam

ple we prove the law o

f assertion:

1 r AIlJ 2 I A--->B121 3

AlII

4 BIl.21

5 A--->B--->BIII

6 A--->.A--->B--->B

hyp hyp 1 reit 2 3--->E 2-4 --->1 1-5 --->1

To see that this generalization of the *t notation, which results in the sys-

tem w

e call FR_, is also equivalent to R_, observe first that the axiom

s of R_ are easily proved in FR

_; hence FR_ contains R_. The proof of the converse

§4.1 N

atural deduction 23

involves little more than repeated application, beginning w

ith an inner-m

ost subproof, of the techniques used in proving the deduction theorem

for R_; it will be left to the reader. (W

e call attention in §4 to some of the

modifications required by the presence of subscripts.) If the subscripting device is taken as an explication of relevance, then it is

seen that Church's R_ does secure relevance since A--->B is provable in R

_ only if A is relevant to B. B

ut if R_ is taken as an explication of entailm

ent, then the reqnirem

ent of necessity for a valid inference is lost. Consider the

following special case of the law

of assertion, just proved:

A---> .A---> A---> A.

This says that if A is true, then it follows from

A--->A. But it seem

s reason-able to suppose that any logical consequence of A--->A should be necessarily true. (N

ote that in the familiar system

s of modal logic, it is intended that

consequences of necessary truths be necessary.) We certainly do in practice

recognize that there are truths which do not follow

from any law

oflo

gic-

but R_ obliterates this distinction. It seems evident, therefore, that a satis-

factory theory of entailment w

ill require both relevance (like R_) and neces-sity (like S4_).

§4. N

ecessity and relevance: entailment (E_).

We therefore consider the

system w

hich arises when w

e recognize that valid inferences require both necessity ,and relevance.

§4.1. The pure calculus of entailm

ent: natural deduction formulation.

Since the restrictions are most transparent as applied to the subproof for-

mat, we begin by considering the system

FE_ which results from

imposing

the restriction on reiteration (of FS4_) together with the subscript require-

ments (of FR

_). We sum

marize the rules of FE_ as follow

s: (1) H

yp. A step m

ay be introduced as the hypothesis of a new subproof,

and each new hypothesis receives a unit class {k} of num

erical subscripts, w

here k is new.

(2) Rep. A, m

ay be repeated, retaining the relevance indices a. (3) R

eit. (A--->B). may be reiterated, retaining a.

(4) --->E. From Aa and (A--->B)b to infer B

aUb •

(5) --->1. From a proof of B

. on hypothesis Alkl to infer (A--->B)._lkJ, pro-vided k is in a.

It develops that an axiomatic counterpart of FE_ has also been considered

in the literature, FE_ in fact being equivalent to a pure implicational cal-

culus derived from A

ckermann 1956. In §8.3.3 w

e consider various formula-

tions of this system, and in C

hapter VIII discuss various aspects of A

cker-m

ann's extraordinarily original and seminal paper, w

hich served as the point

alan ross anderson and nuel d.belnap, jr. entailment the ...entailment the logic of relevance and...

Documents