Don Pigozzi on Abstract Algebraic Logic, Universal Algebra, and Computer Science

476 Pages·2018·3.32 MB·English

by  Janusz Czelakowski

Outstanding Contributions to Logic 16
Editor
Janusz Czelakowski
Don Pigozzi on
Abstract Algebraic
Logic, Universal
Algebra, and
Computer Science
Outstanding Contributions to Logic
Volume 16
Editor-in-chief
Sven Ove Hansson, Royal Institute of Technology, Stockholm, Sweden
Editorial Board
Marcus Kracht, Universität Bielefeld
Lawrence Moss, Indiana University
Sonja Smets, Universiteit van Amsterdam
Heinrich Wansing, Ruhr-Universität Bochum
More information about this series at http://www.springer.com/series/10033
Janusz Czelakowski
Editor
Don Pigozzi on Abstract
Algebraic Logic,
Universal Algebra,
and Computer Science
123
Editor
JanuszCzelakowski
Institute of Mathematics andInformatics
University of Opole
Opole
Poland
ISSN 2211-2758 ISSN 2211-2766 (electronic)
Outstanding Contributions to Logic
ISBN978-3-319-74771-2 ISBN978-3-319-74772-9 (eBook)
https://doi.org/10.1007/978-3-319-74772-9
LibraryofCongressControlNumber:2017964448
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Ad multos annos, Don!
Preface
1. In a most surprising way logic is connected with algebra. Each language,
whether natural or artificial, possesses a complex algebraic structure. In the
simplest case this structure is revealed in the languages of propositional log-
ics as certain absolutely free algebras. The above linguistic perspective es-
tablishes the first bridge between logic and algebra. There are more links
between these two domains and they exist on a deeper level. George Boole
(1854) proved that the “laws of thought” can be framed algebraically as
identities of an algebra. Thus, thinking is also “algebraizable”. Boole’s work
was extended in various directions by a number of researchers, beginning
with William Stanley Jevons. Charles Sanders Peirce integrated his work
with Boole’s during the 1870s. Other significant figures were Augustus de
Morgan, Platon Sergeevich Poretskii, William Ernest Johnson, and Ernst
Schr¨oder.TheconceptionofaBooleanalgebrastructureonequivalentstate-
ments of a propositional calculus is credited to Hugh MacColl (a four part
article from 1877–1879, see (MacColl, 1906), preceding Gottlob Frege’s Be-
griffschrifft). The Fregean principle of compositionality, that the meaning of
any complex expression is a function of the meanings of its constituents, to-
getherwithotherprinciplesiscentralinformalsemantics.Roughlyspeaking,
these principles establish a homomorphism between the algebraic structure
of each language and the algebraic structure constituted by meanings of the
expressions of this language.
The above discoveries gave rise to algebraic logic. As a result the link be-
tween logic and algebra is cemented and becomes inseparable. Studying the
dependenceexistingbetweenthemisstilloneofthevitalareasofscientificac-
tivity.Itisalsointhisverycontextthatthedifferentiationoftheprofessional
belonging of the individual researcher—a logician or an algebraist—becomes
blurred.
2. Professor Don Pigozzi, together with his colleagues who cooperated with
himattheturnofthe1980sandthe1990s,managedtoeffectachangeofthe
paradigm of algebraic logic. This new situation can best be illustrated with
vii
viii Preface
thisanalogy:atthebeginningofthe20thcentury,mathematicianscarriedout
extensivestudiesintheareaoffunctionspaces,e.g.,thespacesofcontinuous
functions on compact topological spaces, the spaces of absolutely integrable
functions on measure spaces, etc. A series of deep and detailed results was
obtained regarding this sphere. Still, there was a lack of the right key to the
general theory which would allow ordering the research field and deriving
well-knownparticularcasesfromafewnotionsandtheorems.Obviously,the
notion of a Hilbert space did order a certain section of the field. However,
what was obtained then was rather a set of facts loosely connected with one
another, though each separately was an original and deep mathematical the-
orem. It was not until the theory of Banach spaces was put forward, along
with its apparatus of notions and key theorems, that the gathered research
material could be ordered and adequate mathematical arsenal was provided
tomakeitpossibletolayfoundationsoffunctionalanalysis.Wecametodeal
with a similar situation in algebraic logic in the second half of the last cen-
tury.Thecontinuumoflogicalsystems,groupedintosomecategories(modal
systems,temporalsystems,systemofdynamiclogic,relevantlogics,etc.),en-
teredthestageforgood.Thethenliteratureofthesubjectaboundedindiffi-
cultandsophisticatedmetalogicalresultswhichcharacterizedvariousaspects
of individual systems. In the study of the systems, new semantic tools, like
relational semantics, neighborhood semantics, and the like, are made use of.
Still,sometypesofsemanticshavelimitations,sincenoteverysystemcanbe
adequately semantically characterized in terms of an appropriate complete-
nesstheorem;thereappearsthephenomenonofsemanticalincompleteness—
wearefamiliarwith,e.g.,normalmodalsystems,notpossessinganadequate
Kripke-style semantics. It was known that the path from a logical system
to algebraic semantics leads through Lindenbaum-Tarski algebras (Linden-
baum(1929),Tarski(1930),L ukasiewiczandTarski(1930)).Thatwasawell-
markedandreliablerouteforclassicallogic,intuitionisticlogic,andaseriesof
other logics. Helena Rasiowa, in her pioneering monograph (Rasiowa, 1974),
introducedimplicativesystems,thoughttobeabroadclassoflogicalsystems
forwhichthegeneralizationoftheLindenbaum-Tarskimethod,whichshein-
vestigated,did“work”.Here,thekeywasthenotionofanimplication,viewed
asabinaryconnectivesatisfyingnaturalassumptions,analogoustotheprop-
erties of the implication of classical or intuitionistic logics. Nevertheless, the
class of S-logics investigated by Rasiowa does not encompass many elemen-
tary intensional systems, like the main modal logics. If one goes through the
relevant literature of the 1970s, the scenery was saturated with millions of
logical systems, each of which being somehow important, individually exam-
inedanddescribed.However,therewerenomethodologicaltoolsavailableof
sufficient generality which would allow framing theses systems from a uni-
formresearchperspective.Again,thekeywhichallowedintroducinganorder
were the notions of the Leibniz operator and of a protoalgebraic logics, the
latter employing the above-mentioned operator. Both notions were explicitly
defined by Wim Blok and Don Pigozzi (1986), (1989). Although the both
Preface ix
notions had been known in the literature under other names (e.g., W´ojcicki
(1988)inhisearlierworkswroteaboutthelargestmatrixcongruenceswhich
are the same objects as Leibniz congruences, and Czelakowski (1985, 1986)
isolatedtheclassofnon-pathologicallogics,thestudieslaunchedbyWimBlok
andDonPigozziintroducedauniversalnotionalnetwork,aswellasgaverise
to systematic investigations conducted in the new language of abstract al-
gebraic logic (AAL, in short). It is considered that the most important and
indisputable achievements of them are the introduction of the notion of an
algebraizablelogicinthepioneeringmonographentitledAlgebraizable Logics
andalsogivingthekeypropertiescharacterizingthisclass.Algebraizabilityis
arigorousmathematicalnotionsystematicallyinvestigatedbymanylogicians
since then. (We omit its definition here.)
3. It can be argued that studies in the field of logic, which were carried out
in the 1960s and the 1970s, whose nucleus was the notion of a consequence
operation,didnotmeetwithabroaderinterestatthattime.Theconsequence
theory, whose fundamentals had already been laid by Alfred Tarski after
the World War I (see Tarski (1956) remained rather underestimated. It was
mainly Tarski’s concept of truth in formalized languages which attracted
interest as his first-rate achievement, especially in the context of later model
theoretic and algebraic applications. Apparently it was so. A few causes of
suchastateofthingscanbeindicated.Firstly,thethentrendsintheworld’s
logic were different and the foundations of mathematics laid, in particular,
byKurtG¨odelandhisdiscoveriesdevelopedrecursiontheoryandsettheory,
especially upon introducing the notion of forcing by Paul Cohen. In many
universities the methods of relational semantics for intensional logics were
developed.Asitwasmentioned,intheWarsawschoolofAndrzejMostowski,
Helena Rasiowa and Roman Sikorski the algebraic foundations were created
for a broad class of non-classical logics, viz. the extended implicative calculi.
Roman Suszko, Ryszard W´ojcicki and their disciples were also active at that
time.
What is a logical system then? In the literature, we will find several def-
initions. Here, we will limit ourselves to two of them only. Logic (on the
propositional level) is most often understood as a set of formulas closed with
respect to substitutions and certain rules of inference. For example, an array
of modal logics is defined in this way. In another approach, the basic no-
tional frame is composed by structural and finitary consequence operations
defined on pertinent languages. This is a more general approach than the
former one. The following problem is connected with it: if one accepts the
notion of logic, viewed as a certain invariant set of formulas, to be the ba-
sis, what consequence operation should be attached to this set? On the level
of normal modal systems, as e.g. S4, one can associate, in a natural way,
two different consequence operations with each such a system: the first—the
so-calledweak consequence,determinedbythegivensystem,andthedetach-
ment rule, as a primitive rule of inference, as well as the so-called strong
consequence, formed out of the weak one by adjoining the G¨odel rule as a
x Preface
primitiverule(andtherebyapplicableinallderivations!).Whichconsequence
shouldbechosen?Thisdependsontheresearchpreferencesofthegivenlogi-
cian.Forinstance,withintheframeworkoftherapidlydevelopingunification
theory for propositional and equational logics, one investigates structurally
completeoralmoststructurallycompletesystems.Naturally,therearesolely
strong modal consequences in sight there.
Comprehensive studies of logical systems, chiefly intensional ones, such as
modal or temporal logics, etc., which have been conducted in recent years,
have caused the consequence theory to be placed in the focal point. The cre-
ation of the theory of entailment relation (this term can be translated as
a multi-conclusion consequence relation), hybrid logics, substructural logics
and the growing interest in non-monotonic reasonings has exercised a strong
impact on consequence theory. The existence of a continuum of logical sys-
tems, of various references and applications, be it in informatics, philosophy,
or theory of language or others, resulted in the need of ordering the research
field,findingafewcommonprinciples,accordingtowhichonecannotionally
embrace the complex matter of contemporary logic.
TheintroductionoftheLeibnizandSuszkooperatorsmarkedanewstage
for metalogic. It was discovered that a uniform classification scheme encom-
passingalllogicalsystemscanbebasedonsomeplausiblepropertiesofthese
operators. As a result, abstract algebraic logic (AAL) has emerged. AAL
offers a transparent and natural hierarchy of logical systems (and not only
propositional ones); each level of the hierarchy is determined by a simple
and concrete property of the above-listed operators, such as monotonicity or
order-continuity.Letusunderlinethatthenotionofaconsequenceoperation
is of key importance in the AAL.
4.ThecentralissuethatunderlayAALwastogetthegistofthedependence
between logic and algebra from the mathematical viewpoint; the point is
aboutthequestionwhichforover150yearshasbeenpermeatingthehistoryof
the both disciplines.
AALtakesamoreabstractandgeneralapproachthanthetraditionalalge-
braic logic. In contrast to algebraic logic, where the focus is on the algebraic
forms of specific deductive systems, AAL is concerned with the process of al-
gebraization itself. AAL investigates degrees of algebraizability of deductive
systems,makinguseofrigorousmathematicaltools.Thedegreeofalgebraiz-
ability of a system is determined by the place of the system in the hierachy
of logics based on the Leibniz operator.
The problem area of AAL and the issue of algebraizability of logical sys-
tems in particular, have set new tasks to algebra and logic, viz. to describe
the algebraic semantics for deductive systems. The relation between algebra
and logic is the strongest in the case of algebraizable systems. Speaking in
the most general way, algebraizability of a system L consists in determining
the conditions which allow replacing the process of deduction of logical for-
mulas by the process of congruence generation on appropriate algebras in an
equivalent manner on the ground of this equational system.