Tarski, Filozofia, Filozofia - Artykuły
[ Pobierz całość w formacie PDF ]Tarski, Truth and Model Theory
Author(s): Peter Milne
Source: Proceedings of the Aristotelian Society, New Series, Vol. 99 (1999), pp. 141-167
Published by: Blackwell Publishing on behalf of The Aristotelian Society
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VII*TARSKI, TRUTHANDMODELTHEORY
by PeterMilne
ABSTRACT
As Wilfrid Hodges has observed, thereis no mentionof the notion
truth-in-a-model in Tarski's article 'The Concept of Truth in Formalized
Languages'; nor does truth
make many appearancesin his papers on model
theoryfrom the early 1950s. In laterpapersfromthe same decade, however,
this
reticence is cast aside. Why should Tarski, who defined truthfor formalized
languagesandprettymuchfoundedmodeltheory,havebeen so reluctantto speak
of truthin a model?Whatmightexplainthe changein his
practice?
The
answers,
I believe, lie in Tarski's views on truth simpliciter.
In
asurveyof modeltheorybefore1945RobertVaught
explained
the mathematicalmotivationbehindTarski'sdefinitionof
truth
in 'The
Concept
of Truthin Formalized
Languages' (henceforth
CTFL).It was, he said, Tarski'sdissatisfactionwith the notion of
truthas it was then
being
used in what we would now call model
theory.As Vaughtrelates:
Since the notion 'ayis truein 3' is highly intuitive(and
perfectly
clearfor anydefinitea), it hadbeenpossibleto go even as faras the
completeness theorem [for first-order logic] by treating truth
(consciously or unconsciously)essentially as an undefinednotion
-one with many obvious properties.[...] [In CTFL]Tarskimade
the needed analysis of truth.For one thing,he discussedjust what
axioms are neededif truthis taken(as above)to be undefined.But
his majorcontributionwas to show thatthe notion 'a is truein 91'
can simply be
defined
insideordinarymathematics,forexample,in
ZF. [...] Tarskiinferredthat variousother semanticalnotions, for
example,
the notion of
logical consequence(in, say, eithera first-
orderor a second-orderlanguage)can also be defined in
ordinary
mathematics.
This is the mathematicalmotivation.Vaughtadmitsthattherewas
more:
The 'motivation' sketched above for Tarski'swork on truthwas
given entirely from the point of view of mathematics and
mathematicallogic. Inhistoricalfact,thiswas only a partof Tarski's
motivation,forhe wasalso verymuchconcernedwith the
positions
*Meeting of the Aristotelian Society, held in Senate House, University of London, on
Monday, 25th January,1999 at 8.15 p.m.
142
PETERMILNE
andattitudeson thenotion
of truthtaken
by
various
philosophers
(suchasWittgenstein,
Given on the one hand Tarski's known association with the
philosophers of Warsawand the Vienna Circle and on the other
Tarski'stremendousrole in the developmentof model theory,this
is all very plausible. The fly in the ointment is an observation
WilfridHodges madein a paperreadto this society
in
1986: there
is no mentionof the notiontruthin a modelin CTFL.Hodges goes
on to point out that the notion only begins to appearin Tarski's
writings on model theoryin the 1950's.2Even then truthis rarely
mentioned in the earlier writings of this period. For example,
Tarskicould give an informalgloss on the subjectmatterof model
theory withoutmentioningtruth:
I
shouldlike to pointout
a new-directionof meta-mathematical
research-the studyof
the relationsbetweenmodelsof formal
systemsandthe syntacticalpropertiesof thesesystems(in other
wordsthesemanticsof formalsystems).Theproblemsstudiedin
this domainareof the followingcharacter:Knowingthe formal
structureof an axiom system, what can we say about the
mathematical
propertiesof themodelsof thissystem;conversely,
givena class of modelshavingcertainmathematical
properties,
1. RobertVaught, 'Model Theorybefore 1945', in Leon Henkinet al. (eds.), Proceedings
of the Tarski Symposium,Proceedings of
Symposia in Pure Mathematics, Vol. XXV
(Providence RI, AmericanMathematicalSociety, 1974) 151-172, pp. 160-1.
Cf.
similar
claims in Vaught, 'AlfredTarski'sWorkin Model
Theory',Journalof SymbolicLogic,
51
(1986), 869-882, p. 871. Wilfrid Hodges, 'Truth in a Structure',Proceedings of the
Aristotelian Society, LXXXVI (1986), 135-51, p. 136, notes that the earliest use he has
come across of anything like 'true in 21'is in Thoralf
Skolem, 'Uber die Unmoglichkeit
einer vollstandigen Charakterieserungder Zahlenreihe
mittels eines endlichen Axiomen-
systems', Norsk MatematiskForenings Skrifter,ser. 2 no. 10 (1933), 73-82. My own
unsystematic researches have thrown up an
earlier occurrence, namely John von
Neumann's 'Eine Axiomatisierung der Mengenlehre' Journal fur
die reine und
angewandte Mathematik, 154 (1925), 219-240, Correction, ibid., 155 (1926), 128,
translatedas 'An Axiomatizationof Set Theory' in Jeanvan Heijenoort(ed.), From Frege
to Godel:A Source Book in MathematicalLogic, 1879-1931 (fourthprinting,Cambridge,
Harvard University Press, 1981), pp. 393-413, where, on p. 411 in a discussion of
categoricity, we find 'a geometric propositionthat is true in
one system satisfying the
axioms of [properlyaxiomatized]geometryis truealso in anyothersuchsystem' and '...the
continuumproblemmight (if categoricityis lacking) be truein one system satisfying the
axioms [of set theory] and false in another'. Hodges notes occurrences of 'verifie'
propositions containing undefined symbols being verified in interpretations-in
Alessandro Padoa's 'Essai d'une theorie algebriquedes nombresentiers, precede d'une
introductionlogique a une theorie deductive quelconque', in Bibliotheque du Congres
internationalde philosophie, Paris, 1900, Vol. III (Paris,ArmandColin, 1901), pp. 309-
324, partially translated as 'Logical Introductionto Any Deductive Theory', in
van
Heijenoort(ed.),
op. cit., pp. 118-23.
2. Hodges, op. cit., p. 137.
to nameonlyone).1
TARSKI,TRUTHANDMODELTHEORY
143
of postulatesystemsby
meansof whichwe candefinethisclassof models.3
Similarly, in the first chapter, written by Tarski (Andrzej
Mostowski and Raphael Robinson collaborated on the second
chapter),of the 1953 study UndecidableTheories,
a chapterthat
introducesmodel-theoreticconcepts,thereis only one unqualified
use of the word 'true'.Significantly,as we shall see, this occursin
the context of remarkson first-orderarithmetic.However, this
reticence in employing the notion truthdisappearsby the time of
the 1957 paper 'ArithmeticalExtensions of Relational Systems'
writtenjointly
with Vaught.Here we find truth spoken of quite
explicitly, withoutqualificationor reservation:
..it is mucheasierto definethetruthof sentencesin termsof the
satisfactionof formulas-insteadof definingthe notionof truth
directly. [...] The notions of satisfaction and truth will play an
essentialpartin ourdiscussionand
thereforewill be definedin a
formal
way. [...]
DEFINITION
1.3
A
sentence a
is
said
to be
true
in the
relation
system
91
=
<A, R>
if everysequencex
E
A(01)
satisfies
a
in 91.
(Under
the
same
conditionswe
say
that91is a model
of a.)4
Here, as readersof CTFL would expect, truthis defined in terms
of satisfaction. But what is to explain the absence of truth-in-a-
model in CTFLitself and the paucity of referencesto truthin the
earlier
papers
on model theory?Answeringthese questions sheds
light on Tarski'sconception of truthand, in particular,what he
achieved in CTFL with his definitions of truth for formalized
languages. This is a topic on which there has been much debate
andnot a little misunderstanding.
Incidentally,a reticence in using terms like 'true in a model'
was, it seems, common in model theoretic investigations.
Throughoutthe 1950's and even in a survey
article from 1960
AbrahamRobinsonscrupulouslyspeaksof sentencesholding in a
model, never of theirbeing true in a model.5 A fairly systematic
3. Tarski[1954],
pp.
19-20. Referenceto Tarski's publicationsis by date;for details see
the bibliographyat the end of this article.
4. Tarskiand Vaught [1957], p. 81.
5. AbrahamRobinson, 'RecentDevelopmentsin Model Theory', in
ErnestNagel, Patrick
Suppes and
Alfred
Tarski
(eds.), Logic, Methodology and Philosophy of Science:
Proceedings of the 1960 InternationalCongress(StanfordCA, StanfordUniversityPress,
1962), 60-79, passim.
whatcanwe sayabouttheformalstructure
144
PETERMILNE
trawl through the pages of the Journal of Symbolic Logic,
Fundamenta Mathematicae and
Transactionsof the American
Mathematical Society yields few unhedged uses of 'true in a
model' and not many heuristic or informal uses either, most
authorsspeakingof sentencesbeing satisfiedorholdingin models.
Quine is an exception (not the only one): in 1954 we find him
statingthe Lowenheim-Skolem Theoremthus:
The celebrated
theoremof L6wenheimandSkolemtells us that
everyconsistent
set
of
sentencesSof
quantificational
schemata
(i.e.,
formulasof thelower
predicate
calculus
admittingof a true
interpretation)
admits a true numerical
letterssuchthat
all schemataS come
outtruewhenthevariablesof quantification
areconstruedasrangingoverpositive
integers.)6
(I'll come back to Quine.)
One might well say, 'What'sin the use of a word?', but I think
it is worth askingjust why Tarskiin particularshould have quite
so scrupulouslyavoided use of the phrase 'true in a model' and
like terms,especially when, as Vaughtinformsus, it had
been used
informallypriorto Tarski'sdefinitionof truthand is the obvious
expression to use.
(i.e., aninterpretation
of predicate
Take a simple language
S
with, say, two
sentences, X1 and X2,
meaning, respectively, 'grass is green' and 'snow is white'.
Tarski'smaterialadequacy
constraintsays that any definition of
6. WillardVan Ornan Quine,
'Interpretationsof Sets of Conditions',Journalof Symbolic
Logic, 19 (1954), 97-102, p. 97. (This paper
is reprintedin Quine,Selected Logic Papers
(enlargededition, CambridgeMA, HarvardUniversityPress, 1995), pp. 205-211.) Other
exceptions occur in Czes}aw Ryll-Nardzewski, 'The Role of
the Axiom of Inductionin
Elementary Arithmetic', Fundamenta
Mathematicae, 39 (1952), 239-263, and Leon
Henkin,
'Some Interconnectionsbetween Modem Algebra and MathematicalLogic',
Transactions of the American Mathematical Society, 74 (1953),
410-427. Compare
Quine's statementof the
Lowenheim-Skolem Theoremwith thatin Helena Rasiowa and
Roman
Sikorski, 'A Proof of the Skolem-Lowenheim Theorem', Fundamenta
Mathematicae,38 (1951), 230-232, p. 230:
Every
consistent set A of formulaeof the classical (first order)functionalcalculus is
simultaneouslysatisfiable
in the domain I of positive integers.
It's surprisingto find thateven some modern
textbooksin logic still abjureis truein favour
of is satisfied or
holds; see, e.g., Heinz-Dieter Ebbinghaus,Jorg Flum and Wolfgang
Thomas,
MathematicalLogic (New York,Springer,1984). J.L. Bell andMoshe Machover,
A Course in
Mathematical Logic (Amsterdam,North-Holland, 1977), betrays a certain
schizophrenia, truth happily putting in an appearancein the Tarskian 'Basic
Semantic
Definition' but not in the chapterdevoted to model theory.
everyset of well-formed
interpretation
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