Table Of ContentPrelims-N51623.fm Page i Wednesday, May 23, 2007 4:29 PM
Handbook of the History of Logic
Volume 8
The Many Valued and
Nonmonotonic Turn in Logic
Prelims-N51623.fm Page ii Wednesday, May 23, 2007 4:29 PM
Handbook of the History of Logic
Volume 8
The Many Valued and
Nonmonotonic Turn in Logic
AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK • OXFORD
PARIS • SAN DIEGO • SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO
North-Holland is an imprint of Elsevier
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Handbook of the History of Logic
Volume 8
The Many Valued and
Nonmonotonic Turn in Logic
Edited by
Dov M. Gabbay
Department of Computer Science
King’s College London
Strand, London, WC2R 2LS, UK
and
John Woods
Philosophy Department
University of British Columbia
Vancouver, BC Canada, V6T 1Z1
and
Department of Computer Science
King’s College London
Strand, London, WC2R 2LS, UK
and
Department of Philosophy
University of Lethbridge
Lethbridge, Alberta, Canada, T1K 3M4
AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK • OXFORD
PARIS • SAN DIEGO • SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO
North-Holland is an imprint of Elsevier
Prelims-N51623.fm Page iv Wednesday, May 23, 2007 4:29 PM
North-Holland is an imprint of Elsevier
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CONTENTS
Preface vii
List of Authors xi
Many-valued Logic and its Philosophy 13
Grzegorz Malinowski
Preservationism: A Short History 95
Bryson Brown
Paraconsistency and Dialetheism 129
Graham Priest
The History of Quantum Logic 205
Maria Luisa Dalla Chiara, Roberto Giuntini
and Miklos R´edei
Logics of Vagueness 285
Dominic Hyde
Fuzzy-set Based Logics — An History-oriented Presentation of 325
their Main Developments
Didier Dubois, Francesc Esteva, Llu´ıs Godo and Henri
Prade
Nonmonotonic Logics:APreferential Approach 451
Karl Schlechta
Default Logic 517
Grigoris Antoniou and Kewen Wang
Nonmonotonic Reasoning 557
Alexander Bochman
Free Logics 633
Carl J. Posy
Index 681
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PREFACE
From its inception nearly 2500 years ago, logic has taken a dominant interest in
the notion of logical consequence. It is widely agreed that there exists a basic con
ception of deductive consequence, which Aristotle called “necessitation”, in which
sentences or sets of sentences serve as inputs, generating output-sentences in a
truth-preserving way. What is not widely agreed is that truth-preserving neces
sitation is the sole relation of consequence, or the only form of it that merits the
serious interest of logicians. Aristotle himself introduces a further — and, for his
interests, the more important — idea of syllogistic consequence. Syllogistic conse
quence is a restriction of necessitation, got by imposing further conditions which,
among other things, require that syllogisms not have redundant premisses. This is
clearly a constraint on inputs, and it is easy to see that it has the effect of making
syllogistic consequence strongly non-monotonic. For if an argument is a syllogism,
then the result of adding new sentences to its premisses cannot itself be a syllo
gism. Aristotle also imposed constraints on the outputs of syllogistic consequence.
Syllogisms are required to have only single propositions as conclusions.
The question of the constraints it may be desirable or necessary to impose on
a stripped down, merely truth-preserving deduction relation brings into conver
gence the two main themes of the present volume of the Handbook of the History
of Logic. Twentieth and twenty-first century logic is well-stocked with affirmative
answers to this question. Many valued logicians gravitate to the idea that a re
alistic characterization of consequence requires the admissibility of inputs which,
owing to vagueness, temporal or quantum indeterminacy, or reference-failure, can
not be considered classically bivalent. Consider the sentence “The present king
of France”. On the many valued approach, the non-existence of the present king
of France denies the sentence a referent, which in turn denies it a classical truth
value. In classical logic, such sentences may not be admitted to the consequence
relation. In many valued systems, the admissibility of such sentences is secured
by the postulation of one of more additional truth values. Free logics offer an
interesting alternative to many valued treatments of reference-failure. Instead of
allowing many-valued sentences as inputs, free logicians retain the classical truth
values while restricting output. In particular, the quantifier introduction rule
Fa (cid:1) ∃x(Fx) fails when Fa is guilty of reference failure. In a variation of free
logic, the necessity to multiply truth values is also averted, provided we are pre
pared to add to the logic’s standard domain of discourse a non-standard domain,
in which singular terms such as “the present king of France” pick up a referent,
notwithstanding that they lack a referent in the standard domain. Either way,
viii Preface
whether the many valued way or the free logic way, something has to be added if
sentences such as these are to be admitted to the consequence relation. Either an
additional truth value must be introduced or an additional domain of discourse.
Many valued logics are non-bivalent. This generates a problem for classical va
lidity. An argument is valid classically if and only if there is no valuation making
its premisses true and its conclusion false. So any argument containing at least
one non-bivalent sentence — whether in basic systems such as K3 or in quantum
logics or vagueness logics — is trivially valid classically. Accordingly, many valued
consequence has to be contrived so as to avert these promiscuous validities. Many
valued logics also impose constraints on outputs. In classical logic, the Law of Ex
cluded Middle is a logical truth, hence is a consequence of any sentence whatever.
But since in virtually all systems of many valued logic Excluded Middle is not a
logical truth, it is not output for every input.
Non-monotonic logicians, like Aristotle before them, are constrainers of inputs;
and often they lay restrictions on outputs as well. The leading intuition of non-
monotonic logics is that there exist consequence relations in which reasonably
drawn conclusions from a given set of premisses can be upset by the addition of
new information, and that such consequence relations should be taken seriously
by logicians. Default logics are a standard affirmative response to this intuition.
When the conclusion drawn from some given premisses is a “default”, then it
obtains provisionally and only in the absence of information to the contrary. Since
indications to the contrary might subsequently present themselves in the form of
new information, such consequences are non-monotonic.
There are also many valued approaches in which it seems appropriate or nec
essary directly to constrain the consequence relation itself. Dialetheic logics are a
case in point. Dialetheic logics are systems in which selected sentences are allowed
to be both true and false at once. Such sentences, while true, aren’t true only,
since they are also false, hence not false only. Accordingly being both true and
false is a third truth value, which makes dialetheic systems many valued. If conse
quence were allowed to operate classically in these logics, then any input carrying
the third truth value would have every sentence whatever as output. To avert this
explosion, consequence has to be a paraconsistent relation, that is, one that does
not generate this unfettered output. Accordingly, dialetheic logic is also a para-
consistent logic. The converse does not hold, however. There are paraconsistent
logics that muffle the explosiveness of classical consequence without the necessity
to posit true contradictions. A dominant move in paraconsistent circles is to con
strain explosiveness by restricting the application of the output rule, Disjunctive
Syllogism, when inputs contain inconsistencies, whether deemed true or not.
A further development — also a many valued one — are fuzzy logics, which are
purpose-built to accommodate vague sentences both as inputs to and outputs of
the consequence relation. The founding insight of these logics is not that vague
sentences require additional truth values, but that the classical truth and falsity
will do provided that we allow the values of sentences to be degrees (or slices) of
them. So seen, “Harry is bald” might be either true or false, or neither; and if
Preface ix
neither, it might be somewhat true, or true to degree n, where n is fairly high; or
more false than true, or false to degree m, where m is higher than any degree to
which the sentence is true.
The ten chapters of The Many Valued and Non-Monotonic Turn in Logic are
designed to give readers a detailed, expert and up-to-date appreciation of the
character and importance of the main expression of the volume’s twin themes.
Once again the Editors are deeply and most gratefully in the debt of the vol
ume’s very able authors. The Editors also warmly thank the following persons:
Professor John Beatty, Acting Head of the Philosophy Department, and Pro
fessor Nancy Gallini, Dean of the Faculty of Arts, at the University of British
Columbia; Professor Michael Stingl, Chair of the Philosophy Department, and
Professor Christopher Nicol, Dean of the Faculty of Arts and Science, at the Uni
versity of Lethbridge; Professor Andrew Jones, Head of the Computer Science
Department at King’s College London; Jane Spurr, Publications Administrator
in London; Carol Woods, Production Associate in Vancouver and our valued col
leagues at Elsevier, Senior Publisher, Arjen Sevenster, and his successor Donna
Weerd-Wilson, and Production Associate, Andy Deelen.
Dov M. Gabbay
King’s College London
John Woods
University of British Columbia
and
King’s College London
and
University of Lethbridge
