Relevant Logic - University of St Andrews

200 Pages·2012·1.07 MB·English

Relevant Logic
A Philosophical Examination of Inference
Stephen Read
February 21, 2012
i
(cid:13)c Stephen Read
First published by Basil Blackwell 1988
Corrected edition (incorporating all errata noted since 1988) (cid:13)c Stephen
Read 2010
ii
To my wife, Gill,
without whose encouragement and support
it could not have been completed
Contents
Acknowledgements vi
Introduction 1
1 Scepticism and Logic 6
1.1 Scepticism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Fallacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Descartes’ Test . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Defeasibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.5 Demonstrativism . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.6 Fallibilism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 Classical Logic 19
2.1 The Classical Account of Validity . . . . . . . . . . . . . . . . 19
2.2 The Deduction Equivalence . . . . . . . . . . . . . . . . . . . 20
2.3 Material Implication . . . . . . . . . . . . . . . . . . . . . . . 22
2.4 Truth and Assertibility . . . . . . . . . . . . . . . . . . . . . . 24
2.5 Three Morals . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.6 The Lewis Argument . . . . . . . . . . . . . . . . . . . . . . . 31
2.7 Three Senses of Disjunction . . . . . . . . . . . . . . . . . . . 32
2.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3 An Intensional Conjunction 36
3.1 Conjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2 The Deduction Equivalence . . . . . . . . . . . . . . . . . . . 38
3.3 Bunches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.4 Consequence . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.5 Relevant Logic . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.6 Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.7 Antilogism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
iii
CONTENTS iv
3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4 Proof-Theory for Relevant Logic 51
4.1 The Bunch Formulation . . . . . . . . . . . . . . . . . . . . . 52
4.2 A Deductive System for DW . . . . . . . . . . . . . . . . . . 55
4.3 Extensions of DW . . . . . . . . . . . . . . . . . . . . . . . . 58
4.4 Derivations in DW . . . . . . . . . . . . . . . . . . . . . . . . 61
(cid:3)
4.5 A Deductive System for R and R . . . . . . . . . . . . . . . 70
(cid:3)
4.6 Derivations in R . . . . . . . . . . . . . . . . . . . . . . . . 73
5 Worlds Semantics for Relevant Logics 78
5.1 Worlds Semantics . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.2 Relevant Semantics . . . . . . . . . . . . . . . . . . . . . . . . 80
5.3 Formal Semantics for DW . . . . . . . . . . . . . . . . . . . . 82
5.4 Formal Results for DW-Semantics . . . . . . . . . . . . . . . 84
5.5 Extending the Semantics . . . . . . . . . . . . . . . . . . . . . 90
(cid:3)
5.6 Semantics for R and R . . . . . . . . . . . . . . . . . . . . . 94
Appendix 100
5A.1 Completeness for DW . . . . . . . . . . . . . . . . . . . . . . 101
5A.2 Extending DW . . . . . . . . . . . . . . . . . . . . . . . . . . 107
(cid:3)
5A.3 Completeness for R . . . . . . . . . . . . . . . . . . . . . . . 109
6 Relevance 114
6.1 The Pre-History of Relevance . . . . . . . . . . . . . . . . . . 114
6.2 The Later History of Relevance . . . . . . . . . . . . . . . . . 118
6.3 Variable-Sharing . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.4 Derivational Utility . . . . . . . . . . . . . . . . . . . . . . . . 124
6.5 Logical Relevance . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.6 Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
7 Logic on the Scottish Plan 131
7.1 Geach’s Challenge . . . . . . . . . . . . . . . . . . . . . . . . 131
7.2 The Dialetheic Response . . . . . . . . . . . . . . . . . . . . . 132
7.3 The Australian Plan . . . . . . . . . . . . . . . . . . . . . . . 134
7.4 Truth-functionality . . . . . . . . . . . . . . . . . . . . . . . . 135
7.5 The American Objection . . . . . . . . . . . . . . . . . . . . . 137
7.6 Classical Relevant Logic . . . . . . . . . . . . . . . . . . . . . 139
7.7 Local Consistency . . . . . . . . . . . . . . . . . . . . . . . . 140
CONTENTS v
7.8 The Scottish Plan . . . . . . . . . . . . . . . . . . . . . . . . 141
8 Deviant Logic and Sense of Connectives 144
8.1 The Deviant’s Predicament . . . . . . . . . . . . . . . . . . . 144
8.2 Rigid Designation . . . . . . . . . . . . . . . . . . . . . . . . . 146
8.3 Rigid Connectives . . . . . . . . . . . . . . . . . . . . . . . . 148
8.4 The Homophonic Principle . . . . . . . . . . . . . . . . . . . 150
8.5 A Metalanguage for R . . . . . . . . . . . . . . . . . . . . . . 152
8.6 Logical Consequence . . . . . . . . . . . . . . . . . . . . . . . 154
8.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
9 Semantics and the Theory of Meaning 159
9.1 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
9.2 Tonk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
9.3 Conservative Extension . . . . . . . . . . . . . . . . . . . . . 163
9.4 Assertion-conditions . . . . . . . . . . . . . . . . . . . . . . . 165
9.5 Proof-conditions . . . . . . . . . . . . . . . . . . . . . . . . . 166
9.6 Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
9.7 Negation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
9.8 Normalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
9.9 Tonk Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . 178
10 Conclusion 180
Index 184
Acknowledgements
In working on the matters discussed in this book, I have received help,
advice, criticism and encouragement from many students, friends and col-
leagues. I would particularly like to thank my colleagues in the Department
of Logic and Metaphysics at St Andrews for allowing me to take study leave
in 1987, during which time this book was completed. I was also fortunate to
be invited to the Australian National University to a three-month Visiting
Fellowship during this time. I am especially grateful to Michael MacRobbie
forarrangingfortheinvitation,andforprovidingmewithsuchastimulating
research environment. I am indebted to the Australian National University,
the British Academy, the British Council and the University of St Andrews
for financial assistance which made the visit possible.
For their helpful and searching comments on earlier drafts of the present
material I would like to thank by name: Ross Brady, Jack Copeland, Max
andMaryCresswell,RoyDyckhoff,JohnFox,AndrFuhrmann,AllenHazen,
Reinhard Hlsen, Frank Jackson, Peter Lavers, Errol Martin, Bob Meyer,
Joan Northrop, Charles Pigden, Graham Priest, Peter Schroeder-Heister,
John Slaney, Richard Sylvan and Neil Tennant. I am especially indebted to
John Slaney, who first introduced me to the idea of the bunch formulation
of a natural deduction calculus. John and I ran joint seminars in Edinburgh
and St Andrews from 1984 to 1987, from which I learned a great deal. I am
also grateful to Stephan Chambers of Basil Blackwell Ltd. for his help and
encouragement in arranging publication.
Stephen Read
St Andrews
December 1987
vi
Introduction
The logician’s concern is with validity, with the relation of consequence be-
tween premises and conclusion. In order to justify an assertion, we may
adduce other statements, from which we claim the assertion follows. But
what is the criterion by which to decide if the conclusion really does fol-
low? The question has two aspects: concretely, to decide in particular cases
whether the conclusion follows from the premises—in technical language,
whether a consequence relation holds; and abstractly, to understand in gen-
eral what the relation between premises and conclusion in a valid argument
is.
The purpose of this book is to explore and defend a particular answer
to this abstract question, an answer which will be characterised in chapter
6 as the Relevant Account of Validity. Relevant logic evolved as a general
framework of logical investigation in the early 1960s, out of work by Alan
Anderson and Nuel Belnap, themselves extending ideas of Wilhelm Acker-
mann,AlonzoChurchandothers(seechapter6below). Inavalidargument,
the premises must be relevant to the conclusion. My aim in this book is to
show why this must be so, and what the content of this claim is.
Interestinrelevantlogichasgrownenormouslyinthelastdecade,follow-
ing publication of the first volume of Anderson and Belnap’s encyclopedic
work Entailment: the logic of relevance and necessity.1 Nonetheless, in-
terested readers will experience some difficulty in finding out what relevant
logic involves. Entailment itself is a large work containing a wealth of detail.
Yet, for example, it includes nothing on the possible worlds semantics (see
chapter 6 below) for relevant logic developed in the early 1970s, that topic
being reserved for the second volume, whose publication has been badly,
and sadly, delayed. Details of the semantics can certainly be unearthed
from Relevant Logics and their rivals by Richard Routley and others,2 but
1A.Anderson and N.Belnap, Entailment, vol.I (Princeton, 1975).
2F.R. Routley et al., Relevant Logics and their rivals (Atascadero, Calif., 1982).
1
INTRODUCTION 2
many readers, particularly those not already well-versed in the subject, will
find that work an uncompromising and difficult one. By far the best in-
troduction so far to the subject is Michael Dunn’s survey article, ‘Relevant
Logic and Entailment’.3
The present book does not attempt to compete with Dunn’s survey. I
have set myself two aims here. The first is to lay forth what I believe are the
correct philosophical reasons for rejecting classical logic and adopting a rel-
evant logic as a correct description of the basis of inference. These are not,
in general, the reasons which led historically to the development of the sub-
ject, and are not those emphasised in the writings of Anderson, Belnap and
Dunn. In many ways relevance as a core notion is displaced in the present
framework by direct consideration of inference itself. Secondly, I have tried
to keep the work intelligible to anyone who has some training in logic and
in philosophical analysis, but no more than the average second or third year
undergraduatewithanintroductorycourseonformalandphilosophicallogic
behind them. Anyone who has read and understood Lemmon’s Beginning
Logic or Guttenplan’s The Languages of Logic should have no difficulty with
the technical material in chapters 4 and 5. The most difficult material, on
the completeness proof, I have relegated to an appendix to chapter 5. Later
chapters do not presuppose that readers have mastered the material in this
appendix. I hope, however, that chapters 4 and 5 themselves do not con-
tain any gratuitous technicality. One cannot reason clearly and correctly
about logic without proper acquaintance with the technical development of
the subject. Equally, neither can one develop the formal material usefully
without embedding it in the appropriate philosophical analysis.
I start in chapter 1 by looking at the general nature of theories, and I
approachthistopicbyconsideringthenotionofknowledge. Doesknowledge
requireaninfallibleanddemonstrablycertainproof, asisfoundonlyinlogic
and mathematics? I show that this conception of knowledge is mistaken. It
followsthatnotonlyisscepticismrefuted,butthetruenatureofalltheories,
in science, mathematics and logic, is one of essentially fallible explanatory
models and generalisations. These can both be justifiably asserted on the
basis of appropriate grounds, and equally rejected when countervailing evi-
dence is discovered.
What then is the correct model and theory of validity? It is a plausible
thought that an argument is valid if and only if it is impossible for the
3J.M.Dunn,‘RelevantLogicandEntailment’,inHandbook of Philosophical Logic,vol.
III:Alternatives to Classical Logic,edd.D.GuenthnerandD.Gabbay(Dordrecht,1986),
pp. 117-224.
INTRODUCTION 3
premisestobetrueandtheconclusionfalse. However,thisClassicalAccount
of Validity entails that Ex Falso Quodlibet, that is, the inference from P
and ‘∼P’ to Q, is valid. Also plausible is the Deduction Equivalence, that
is, that a conditional follows from some other premises if and only if its
consequent follows from them in conjunction with its antecedent. Adding
this to the Account of Validity, it follows that the conditional is material,
or truth-functional. Since such a view validates invalid arguments, we are
forced to reject or revise the Classical Account, and in passing to conclude
that disjunction is ambiguous.
Further reflection on conjunction shows that it exhibits an ambiguity
matching that discovered for disjunction. The intensional sense of conjunc-
tion so discerned (we call it, ‘fusion’) allows a suitable revision to the Classi-
cal Account of Validity and also to the Deduction Equivalence, avoiding the
unacceptable consequences of chapter 2. These principles—both plausible,
and of great explanatory power—are, therefore, not simply rejected, but re-
vised following proof-analysis. An argument is valid if and only if the fusion
of its premises and the opposite of its conclusion cannot be true; and a con-
ditional follows from some other propositions if and only if its consequent
follows from the fusion of its antecedent and those other propositions.
The distinctions and logical notions informally mapped out in chapters
2 and 3 can now be elucidated in a formal treatment. First, the inferential
structure of a non-truth-functional conditional together with negation and
the various conjunctions and disjunctions is given. The basic inferential
notion of a derivable sequent captures the idea of a conclusion following
from a combination of premises, a combination making use of both types
of conjunction as ways of putting premises together. Secondly, the logic is
extended to embrace a ’necessity’-operator, and so is capable of expressing
entailment itself.
The resulting formal system contains several non-truthfunctional (i.e.
intensional) operators. When formal systems with such operators (modal or
relevant) were first proposed, they were challenged to produce a semantics
which would show them to be coherent and viable logics. I show how pos-
sible worlds, or indexed, semantics answered this challenge, and how this
semantics can be adapted to provide a semantics for the systems of chapter
4. These systems are proved sound and, in an Appendix, complete with
respect to their semantics.
What lies behind the challenge to the Classical Account of Validity
in chapter 2 is the objection that that account validates inferences whose
premises are in some way irrelevant to their conclusions. Many attempts
have been made to explicate the notion of relevance needed here, in partic-