Table Of ContentCOLLECTED PAPERS OF STIG KANGER
WITH ESSAYS ON HIS LlFEAND WORK
Vol. I
SYNTHESE LIBRARY
STUDIES IN EPISTEMOLOGY,
LOGIC, METHODOLOGY, AND PHILOSOPHY OF SCIENCE
Managing Editor:
JAAKKO HINTIKKA,Boston University, U.S.A.
Editors:
DIRK VANDALEN, University ofUtrecht, TheNetherlands
DONALD DAVIDSON, University ofCalifornia, Berkeley, U.S.A.
THEO A.F.KUIPERS, University ofGroningen, TheNetherlands
PATRICKSUPPES, Stanford University, California, U.S.A.
JAN WOLENSKI, Jagiellonian University,Krakow. Poland
VOLUME 303
COLLECTED PAPERS OF
STIG KANGER WITH
ESSAYS ON HIS LIFE
ANDWORK
VoI. 1
Edited by
GHITA HOLMSTROM-HINTIKKA
Boston University, Boston, USA.
STEN LINDSTROM
Umea University, Umea, Sweden
and
RYSIEK SLIWINSKI
Uppsala University, Uppsala, Sweden
SPRINGER SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-1-4020-0022-5 ISBN 978-94-010-0500-5 (eBook)
DOI 10.1007/978-94-010-0500-5
Printed on acid-free paper
Ali Rights Reserved
© 2001 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 2001
Softcover reprint of the hardcover 1s t edition 200 1
No part of the material protected by this copyright notice may be reproduced or
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TABLE OF CONTENTS
Introduction vii
PURE LOGIC
A Note on Partial Postulate Sets for Propositional Logic 3
Provability in Logic 8
The Morning Star Paradox 42
A Note on Quantification and Modalities 52
On the Characterization of Modalities 54
A Simplified ProofMethod for Elementary Logic 58
EquivalentTheories 65
An Algebraic Logic Calculus 70
Equational Calculi and Automatic Demonstration 76
Entailment 82
The Paradox ofthe Unexpected Hanging, Regained Again 94
APPLIED LOGIC: OBLIGATIONS, RIGHTS AND ACTION
New Foundations for Ethical Theory 99
Rights and Parliamentarism 120
Law and Logic 146
Some Aspects on the Concept of Influence 170
On Realization ofHuman Rights 179
Unavoidability 186
Unavoidability. Appendix 192
APPLIED LOGIC: PREFERENCE AND CHOICE
Preference Logic 199
A Note on Preference-Logic 209
Choice and Modality 211
v
vi
Choice Based on Preference 214
Decision by Democratic Procedure 231
PHILOSOPHY OF SCIENCE
Measurement: An Essay in Philosophy of Science 239
The Notion of a Phoneme 274
Published Writings of Stig Kanger 279
Index of Names 285
Subject Index 289
Photo:Rune Kanger
Stig Kanger
INTRODUCTION
Stig Kanger (1924-1988) made important contributions to logic and formal
philosophy. Characteristic ofKanger as a philosopher was his firm convic
tion that philosophical problems can be clarified - and sometimes even
solved - by means of exact logical and mathematical methods. His most
substantial, and groundbreaking, contributions were in the areas ofgeneral
proof theory, the semantics of modal and deontic logic, and the logical
analysis of the concept of rights. But he contributed significantly to action
theory, preference logic and the theory ofmeasurement as well.1
Kanger was Professor of Theoretical Philosophy at Uppsala University
from 1968until his death in 1988. He was borninChina where helived with
his parents, the Swedish missionaries Gustav and Sally Kanger, until he was
thirteen. He received his higher education in Stockholm and obtained his
Ph. D. from Stockholm University in 1957 under the supervision ofAnders
Wedberg. Before being appointed to the Chair ofTheoretical Philosophy in
Uppsala, Kanger was Docent at Stockholm University and Professor of
Philosophy at Abo Academy inFinland.
FordetaileddiscussionsandanalysesofvariousaspectsofKanger's work seethecritical
essaysinVolumeIIofthepresentcollection.FortreatmentsofKanger'sworkingeneralproof
theoryseeG.Sundholm, "TheProofTheoryofStigKanger:APersonalRecollection"andK.
B.Hansen,"Kanger'sIdeasonNon-well-foundedSets:SomeRemarks".Hiscontributionsto
theareaofeffectiveproofproceduresandautomatedreasoningarediscussedinD.Prawitz, "A
Note on Kanger's Work on Efficient Proof Procedures" and in A. Degtyarev and A.
Voronkov,"Kanger'sChoicesinAutomatedReasoning".JaakkoHintikka'spaper"TheProper
TreatmentofQuantifiers inOrdinaryLogic" concerns Kanger's formalization of first-order
logic as analgebraic logic calculus. Kanger's semantics for modal logic is discussed in S.
Lindstrom, "AnExpositionandDevelopmentofKanger's EarlySemanticsfor Modal Logic"
andhisapproach todeonticlogicinR.Hilpinen's "StigKangeronDeontic Logic".Kanger's
theory of rights is dealt with in L. Lindahl, "Stig Kanger's Theory of Rights" and in L.
Aqvist, "StigKanger'sTheoryofRights:BearersandCounterparties,Sources-of-law,andthe
HanssonPetaluma Example".Kanger's contributions tothetheoryofactionaredescribed in
G. Holmstrom-Hintikka, "Stig Kanger's Actions and Influence". Finally, Kanger's con
tributions to the logic of preference are taken up by S. O. Hansson, "Kanger's Theory of
Preference and Choice" andW. Rabinowicz, "Preference Logicand Radical Interpretation:
Kanger MeetsDavidson".
ix
G.Holmstrom-Hintikka,S.Lindstromand R.Sliwinski Ieds.},Collected PapersofStig KangerwithEssays
onhisLifeandWork,Vol.I. ix-xiv,
© 2(0) AllRightsReserved.PrintedliyKluwerAcademicPublishers,theNetherlands.
x INTRODUCTION
Kanger's dissertation, Provability inLogic, 1957, was remarkably short,
only 47 pages, but very rich in new ideas and results. By combining
a
Gentzen-style techniques from prooftheory witha model theory laTarski,
Kanger gave a novel and elegant proofof Godel's completeness theorem for
classical first-orderpredicate logic(withoutidentity).Fromthecompleteness
proofhe could extract a simple semantical proofofGentzen's Hauptsatz as
wellasaneffective proofprocedurefor predicate logic.The dissertation also
contained thefirstfullydevelopedmodel-theoreticsemantics formodallogic.
The basic idea of Kanger's completeness proof - an idea that was also
developed around the same time in various versions by Beth, Hintikka and
Schutte - is to view a proofofa logically valid formula as an unsuccessful
attempt to find a counter-model to it. Kanger applied this idea directly to
Gentzen's cut-free sequent calculus for classical predicate logic: given a
sequent T - 11, the rules of the sequent calculus are appliedbackwards ina
systematic search for a counter-model, i.e., a model in which all the for
r
mulas of are true and all the formulas of 11 are false. The backwards
application of the rules yields a possibly infinite tree: all the formulas
occurring in the sequents of the tree are subformulas of formulas in the
r -
original sequent. Ifthe sequent 11 is notprovable, the resulting tree has
a maximal branch inwhich eachsequent ~I - ~2 satisfies thecondition ~I n
~2 = 0. Acounter-modeltothesequentI'- il canthen beconstructed from
such a branch, essentially by assigning the value trueto an atomic sentence
if itoccurs intheantecedent of a sequent inthe branch and the valuefalse if
itoccurs inthe succedent. If, on the other hand, the sequent isprovable, the
systematic search for a counter-model will be frustrated: each branch will
terminate inan axiom of the form rl, A, r2 - ill' A, il2, and the resulting
tree will be a proofin the Gentzen calculus of the initial sequent r il.
=>
As an immediate corollary ofthe completeness theorem for the cut-free
sequent calculus, Kanger obtained a simple model-theoretic (and non
constructive) proof of Gentzen's Hauptsatz; i.e., the statement that any
sequent that is provable inthe sequent calculus with the cut rule:
r - r - r
From il, A and A, il, infer -il
is also provable in the calculus without the use of this rule. It is easily
verified that thecut rule isasemantically valid rule. Hence, the system with
thecut rule issemantically sound. Suppose nowthatr -il isprovable inthe
system with the cut rule. Then the sequent is valid by the soundness ofthat
r -
system. It then follows by Kanger's completeness theorem that il is
provable without the cut rule.
INTRODUCTION xi
As another byproduct of the completeness theorem, Kanger obtained a
proofprocedure for classical logic that iseffective inthe sense ofproviding
an algorithm for finding a proof of any given logically valid sequent (or
formula): To construct a proof of a valid sequent r = fl, we start from
below with the given sequent and construct a tree of sequents above it by
meansofrepeated backwards applicationsoftherules ofthecut-free sequent
calculus. We continue until theprocess terminates and we have reached an
axiom atthe topofeach branch inthetree. The resulting tree isthena proof
of the valid sequent that we started with. Kanger's completeness proof
guarantees that theprocess terminates after finitelymany applications ofthe
rules, provided, ofcourse,thatthesequent westarted withwasindeedvalid.
Ifnot, the search for a proofmay go on indefinitely. Of course, Kanger's
proofprocedure does not provide us with an effective method for deciding
the validity ofany given formula offirst-order predicate logic. By the well
known theorem due to Alonzo Church, we know that no such decision
method exists. In the paper "A Simple Proof Procedure for Elementary
Logic" Kanger describes how the proof procedure can be extended to
predicate logic withidentityandhowitcan bemade more efficientfor actual
implementation on a computer.
The part ofKanger's dissertation that had the greatest impact, however,
was the fifteen pages devoted to modal logic. There he gave the first
development ofaviablemodel-theoreticsemantics formodallogic.Kanger's
semantics has close affinities to the various versions of so-called possible
worlds semantics developed by Jaakko Hintikka, Saul Kripke and Richard
Montague. But there are also important differences between the various
approaches. For one thing, Kanger's semantic ideas are closer in spirit to
Tarski'sthantothemetaphysicallymoreloaded interpretations ofKripke and
David Lewis. The idea of "metaphysically possible" possible worlds was
certainly foreign to Kanger.
Kanger's ambition wastoprovide alanguage L of quantified modal logic
a
withamodel-theoretic semantics laTarski. For thispurpose he introduced
thenotionofasystem. Asystemisanorderedpair S = <Do,v>,where Do
isa designated (non-empty) domainandv isa function which for every non
empty domain D assigns an appropriate extension inD to every non-logical
constant ofL
InKanger's dissertation appears, forthe firsttimeinprint, asemanticsfor
modal operators in terms ofwhat we nowadays call accessibility relations.
Each modal operator 0 is associated with an accessibility relation Ro
between systems in terms ofwhich the semantic evaluation clause for 0 is
spelled out:
