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From Intentionality to Formal Semantics (From Twardowski to
Tarski)Author(s): Jan WoleskiReviewed work(s):Source: Erkenntnis
(1975-), Vol. 56, No. 1, The Legacy of the Lvov-Warsaw School
(2002), pp.9-27Published by: SpringerStable URL:
http://www.jstor.org/stable/20013104 .Accessed: 04/02/2013
21:19
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JANWOLE?SKI
FROM INTENTIONALITY TO FORMAL SEMANTICS (FROM TWARDOWSKI TO
TARSKI)1
Formal semantics deals with rigorous investigations of semantic
properties of linguistic expressions via logical and mathematical
methods. Interests
concerning language and its features were always very vivid in
philosophy since its inception, but became a characteristic mark of
philosophy in the 20th century, particularly within the analytic
camp. This can be regarded as an answer to a prophetic remark of
Bertrand Russell (Russell 1903, p. 42): The study of grammar, in my
opinion, is capable of throwing far more light on philosoph? ical
questions than is commonly supposed by philosophers.
This claim, linked with Russell's general attitude to logic as
the heart and
very centre of philosophy, constituted a part of the general
background in which semantics developed as a branch of logic and
philosophy of lan?
guage. The second general factor contributing to this process
consisted in
working out suitable formal methods coming from mathematical
logic and some other fields of mathematics, like set theory and
algebra. Thus, formal semantics was a child of philosophy and
mathematics. Alfred Tarski was the person who integrated both
parents and created the first mature se?
mantic theory. This paper tries to explain why this happened in
Poland, although systematic formal semantics could arise in many
other places.
In fact, formal semantic concepts, like reference, model,
satisfaction, completeness, truth, validity, etc. can be found
before Tarski, particu? larly in writings (I mention here only some
classics of logic and analytic
philosophy in 20th century; formal semantic concepts were
earlier used
by Bolzano) of Gottlob Frege, Russell, Ludwig Wittgenstein,
Leopold L?wenheim, David Hubert or Kurt G?del. Frege's distinction
(see Frege 1891) of Sinn and Bedeutung was semantic in its
character; the same con? cerns Russell's theory of descriptions
(see Russell 1903) and his theory of logical types (see Whitehead
and Russell 1910) having a clear se?
mantic dimension. Although Wittgenstein was hostile to semantics
(see below), his ideas of tautology as a sentence true in all
circumstances or representation of facts by propositions (see
Wittgenstein 1922) are fairly semantic. L?wenheim 1915 and Skolem
1920 contain proofs of a celeb
?* Erkenntnis 56: 9-27, 2002. jr% ? 2002 Kluwer Academic
Publishers. Printed in the Netherlands.
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10 JAN WOLENSKI
rated semantic result known as the L?wenheim-Skolem theorem.
Hilbert's
sharp separation of mathematics and metamathematics in the 20's
and, in
consequence, of language and metalanguage, was of the utmost
import? ance for all future research in the foundations of
mathematics, including formal semantics. The Hilbertians also used
the concept of the domain of individuals and the concept of
satisfaction, both fundamental for formal semantics. The history of
the completeness problem, a typical logical ques? tion involving a
semantic idea is very instructive. This issue was clearly observed
in Principia Mathematica in the following condition imposed on
every correct logical system (p. 12):
[... ] the system must embrace among its deductions all those
propositions which we believe to be true and capable of deduction
from logical premises alone.
However, Russell did not consider the completeness problem as
something to be proved.2 It was stated as an open question for
first-order logic in Hu?
bert and Ackermann 1928. It was important for the realization of
Hilbert's
program and solved in G?del 1930, where the completeness theorem
for first-order logic was proved in the form: Every valid formula
of first-order
logic is provable; the semantic factor is represented here by
the concept of validity. G?del used informal semantic arguments in
the intuitive ex?
planation of his famous incompleteness theorem (G?del 1931, p.
151; page-reference to reprint; my italics): "The method of proof
just explained can clearly be applied to any formal system that,
first,
when interpreted as representing a system of notions and
propositions, has at its disposal sufficient means of expression to
define the notions occuring in the argument above (in particular,
the notion "provable formula") and in which, second, every provable
formula is true in the interpretation considered. The purpose of
carrying out the above proof with full
precision in what follows is, among other things, to replace the
second of the assumptions
just mentioned by a purely formal and much weaker one.
The replacement mentioned in this quotation consisted in
substituting syntactic concepts of consistency and ^-consistency.
Particularly interest?
ing in this respect is the recently published early book by
Rudolf Carnap, Untersuchungen zur allgemeinen Axiomatik (see Carnap
2000), which contains a lot of formal semantic ideas, including an
anticipation of the
G?del-Malcev theorem (it is another form of the completeness
theorem), that every consistent set of sentences has a model.
However, Carnap aban? doned his further study in the theory of
models due to the incompleteness results that demolished the idea
advanced in Untersuchungen zur allge?
meinen Axiomatik that the categoricity of models is a sufficient
device for
proving consistency of mathematics. Now I am going to present a
brief account of the history of ter?
minological matters concerning the word 'semantics', because it
is also
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FROM INTENTIONALITY TO FORMAL SEMANTICS 11
illuminating for our question (see papers on Semantik and
Semiotik in Ritter und Gr?nder 1995 for further information).
According to a com? mon opinion, the word 'semantics' (precisely:
its French counterpart
's?mantique'), derived from the Greek word semantikos (= having
mean? ing, denoting), appeared for the first time (at least in
modern times) in the book Essai de s?mantique, science de
significations by M. J. A. Br?al (1897). However, Quine says in his
lectures on Carnap delivered in 1934 (see Quine 1990, p. 168):
As used by C. S. Peirce "semantic" is the study of the modes of
denotation of signs: whether a sign denotes its object through
causal or symptomatic connection, or through imagery, or through
arbitrary convention and so on. This sense of semantic, namely
a
theory of meaning, is used also in empirical philology:
empirical semantic is the study of historical changes of meanings
of words.3
For Br?al, semantics was a branch of general linguistics. In
particular, semantics was occupied with the so-called lexical
meaning and its changes through time. Thus, semantics in this sense
belonged to what was called "the diachronic treatment of language".
This tradition is still fairly alive in
contemporary linguistic theory. Quine's description of the word
'semantic' in Peirce corresponds, as Quine explicitly states, to
its use in philology.
However, some linguists ascribe a more theoretical role to
linguistic se? mantics. Karl B?hler 1934 is an example. He says (p.
33; page-reference
to Eng. tr.) that a theory of semantic functions of language is
a part of theory of language. This account is to be found also
among philosoph? ers. It is also rather obvious that Peirce did not
limit his semantic only to empirical studies. Linguists (and
sometimes philosophers) also use the
word 'semasiology' instead of 'semantics'; B?hler 1934, p. 34
proposed
the term 'sematology' for a general theory of symbols.
The word 'semantics' became popular in philosophy in 1930's
thirties. Earlier, it was used only occasionally, for example Ogden
and Richards
(1923) mentioned the science of Semantics as dealing with the
relation between words and facts.4 Incidentally, the fact that
Quine used 'semantic' as a noun, and not as an adjective, gives
evidence that there was no estab? lished jargon at the time.
Another interesting point is that Rudolf Eisler's
W?rterbuch der Philosophische Begriffe has no entry on
semantics, even in its 4th edition (Eisler 1930). This dictionary
was certainly an expression of fairly common philosophical
experience. The lack of the word 'semantics' indicates that this
term was hardly used by philosophers.
Poland was an exception in this respect. In the twenties, Polish
philo? sophers began to use the word 'semantyka' (the Polish
counterpart of 'semantics') for considerations on the
meaning-aspect of language. In
particular, a very influential book by Tadeusz Kotarbi?ski,
Elements of
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12 JAN WOLENSKI
Theory of Knowledge, Logic and Methodology of Sciences
(Kotarbi?ski 1929) begins with the chapter "Such Semantic Relations
as Expression,
Denotation, etc.". A general characterization of semantics in
Kotarbi?ski
(p. 15; page-reference to Eng. tr.) takes "semantic ideas" as
"referring to this aspect of the language, which is concerned with
meanings". At the same time, Stanislaw Lesniewski introduced the
term 'semantic cat?
egories' for what Edmund Husserl understood by
Bedeutungkategorien. Kazimierz Ajdukiewicz employed the term
'semantics' in his review of the above mentioned book by
Kotarbi?ski.5 The content of the relevant section shows that
Ajdukiewicz considered semantics to be occupied with various
functions of language (meaning, denotation, etc.). In another paper
(Ajdukiewicz 1931), Ajdukiewicz discusses semantic functions of
which
meaning is an example. The same author delivered a course in
logical semantics in Lvov in the academic year 1930/31. It seems
that it was the first occurrence of the name
'logical semantics'. Semantic categories (in Lesniewski's sense)
and logical antinomies were the main subject of this course. In
fact, Ajdukiewicz considered syntactic problems (supplemented by
some remarks on the use of expressions) rather than semantic (at
least in the later sense of "semantics") ones.6
How was it in Tarski's writings? In Tarski 1930-1931, which is
the first note on his definition of truth, we find only the
adjective 'heterose
mantic' (Tarski probably took this word from Lesniewski - the
adjective
'heterological', derived from German 'heterologisch' introduced
by Kurt
Grelling and Leonard Nelson in 1908, is much more popular;
Tarski used it in his later papers in the context of the Grelling
antinomy). Next, Tarski 1932 employs the term 'Semasiologie' and
says that the concept of truth is of the semasiological character.
It was probably the first time when the
concept of truth was characterized as semantic in the present
sense. As far as I know, there is no explicit statement in Polish
literature before Tarski that the concept of truth belongs to
semantics although almost every Pol? ish philosopher accepted the
classical (Aristotelian) truth-definition, which
was later commonly interpreted, at least in Poland, as an
anticipation of the semantic approach to the concept of truth.
Further, Tarski 1932 announces that further results concerning the
concept of truth (in particular, the con? struction of a correct
definition of truth by formulating it in metalanguage) can be
extended to other semasiological notions. According to Tarski, this
fact opens a way to building the semasiology of any language except
the natural one. Tarski mentions satisfaction as another important
semasiolo?
gical concept, and adds that this concept can help us in working
out a correct treatment of further notions. On the base of his
later writings, we know that he had the concept of denotation
(reference) in his mind. We
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FROM INTENTIONALITY TO FORMAL SEMANTICS 13
can conclude, although indirectly, that semasiology in Tarski's
sense deals with the relation between language and what language
refers to.
Tarski 1933, that is, his seminal paper on truth contains the
official
explanation of the meaning of 'semantics'(p. 252; page-reference
to Eng. tr.):
[... ] we attempted to go further and to construct [... ]
definitions of concepts belonging to semantics of a language
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i.e., such concepts as satisfaction, denoting, truth,
definability, and so on. A characteristic feature of the semantical
concepts is that they give expression to certain relations between
the expressions of language and the objects about which these
expressions speak, or that by means of such relations they
characterize certain classes of
expressions or other objects. We could also say (making use of
the suppositio materialis) that these concepts serve to set up the
correlation between the names of expressions and the expressions
themselves."
Moreover, Tarski contrasts semantics of language with its
morphology for which the concept of consequence is the most
important; of course,
morphology is what is now considered as syntax. The above
explanations are repeated in Tarski's programmatic paper on
the foundations of semantics (Tarski 1936, p. 401;
page-reference to Eng. tr.):
The word 'semantics' is used here in a narrower sense than
usual. We shall understand by semantics the totality of
considerations concerning those concepts, which roughly speak? ing,
express certain connexions between the expressions of a language
and the objects and states of affairs referred to by these
expressions. As typical examples of semantic concepts
we may mention the concepts of denotation, satisfaction, and
definition [... ] The concept of truth - and this is not commonly
recognized
- is to be included here, at least in this classical
interpretation, according to which 'true' signifies the same as
'corresponding with
reality'.
Tarski's ideas of semantics in a narrower sense, as contrasted
with se? mantics as considerations of various functions of language
as well as
syntax (morphology) and of truth as a semantic concept were
fairly novel. They decisively went beyond all earlier, including
Polish, characterizations of semantics. The early Carnap, as I
already mentioned, understood se?
mantics as metalogic or syntax. The situation does not change in
Carnap 1934 (see p. 9; page reference to Eng. tr.), where he speaks
on se?
mantics only in connexion with the views of Leon Chwistek and
says that Chwistek's semantics has the same aim as syntax. Carnap
was obviously aware of the linguistic meaning of the word
'semantics' and other above
mentioned proposals like 'semasiology' or 'sematology'. He also
used a
hybrid word 'quasi-syntactic' for concepts expressing relations
of words to objects, but having complete syntactic
translations.
Tarski's ideas were not only novel, but also revolutionary for
many philosophers and they became the turning point in the
philosophical career
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14 JAN WOLE?SKI
of semantics. Tarski 1936 was based on his talk before the
international
philosophical congress in Paris (1935). Due to this fact, a
fairly large group of philosophers was informed about the meaning
of semantic proposed by Tarski. Sociologically speaking, Tarski's
paper impressed many philosoph? ers, although others (notably, Otto
Neurath) were sceptical about semantics and its philosophical
importance. The positive attitude is documented by
Ayer's recollections (Ayer 1977, p. 116):
Philosophically, the highlight of the Congress was the
presentation by Tarski of a paper which summarized his theory of
truth".
Since 1936 the word 'semantics', as used in logic and philosophy
of lan?
guage, denotes considerations about relations holding between
expressions and their objectual references. Although the word
'semantics' does not oc? cur in Carnap 1936, he entirely accepted
the spirit of Tarski's explanations. In 1938, Charles Morris
revived the word 'semiotic'. Morris thought about semiotic as a
general theory of signs and divided it into pragmatics, se?
mantics and syntax; semantics was understood as in Tarski. This
tripartite division was adopted in Carnap 1939. The canonical
description is perhaps best formulated in Carnap 1942 (p. 9), the
first philosophical monograph in which the term 'semantics' appears
as a term-word:
If in an investigation explicit reference is made to the
speaker, or, to put it in more general terms, to the user of a
language, then we assign it to the field of pragmatics [...] If we
abstract from the user of the language and analyze only the
expressions and their designata,
we are in the field of semantics. And if, finally, we abstract
from the designata also and
analyze only the relations between the expressions, we are in
(logical) syntax. The whole science of language, consisting of the
three parts mentioned, is called semiotic.
In order to complete the discussion of development of the
concept of semantics and the related terminology, let me finally
mention a very influential description given by Quine 1953 (p.
130):
When the cleavage between meaning and reference is properly
heeded [...], the problems of what is loosely called semantics
become separated into two provinces so fundamentally distinct as
not to deserve a joint appellation at all. They may be called the
theory of
meaning and the theory of reference. 'Semantics' would be a good
name for the theory of meaning, were it not for the fact that some
of the best works in so-called semantics,
notably Tarski's, belong to the theory of reference. The main
concepts in the theory of
meaning, apart from meaning itself, are synonymy (or sameness of
meaning), significance (or possession of meaning), and analyticity
(or truth in virtue of meaning). Another is
entailment, or analyticity of the conditional. The main concepts
in the theory of reference
are naming, truth, denotation (or truth-of), and extension.
Another is the notion of values of variables.
Introducing semantics to logic and philosophy was and has been
re?
ceived as an essential change. The theory of models (formal
semantics of
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FROM INTENTIONALITY TO FORMAL SEMANTICS 15
formal languages) changed logic considerably. Almost all
philosophy of language and much philosophy of science is today
strongly influenced by semantics. Semantical methods are also
employed in epistemology, onto?
logy, ethics, and aesthetics, for example in discussions on
realism, possible worlds, normative reasoning or literary fictions.
Important philosophers, like Ajdukiewicz, Carnap and Popper
radically changed their essential views under the influence of
semantics. On the other hand, this revolution took rather a long
time. Still in Church 1956 (p. 67; my italics) we can read:
In concluding this Introduction let us observe that much of what
we have been saying has been concerned with the relation between
linguistic expressions and their meaning, and therefore belongs to
semantics [... ] From time to time in the following chapters we
shall interrupt the rigorous treatment of a logistic system in
order to make an informal semantical aside.
Thus, even in the middle fifties, leading logicians were not
quite convinced that semantics could be a primary concern in logic.
Today, fundamental role of semantics in logic is fairly
unquestionable.
The above considerations show that Polish mathematicians and
philo? sophers were pioneering on the conceptual and terminological
level.
However, it does not explain by itself why it was so. It is even
obvious that achievements of Polish logicians and philosophers in
particular semantic
(or semiotic) problems were, before Tarski, rather modest in
comparison with earlier writings, particularly those of Frege and
Russell. Frege's dis? tinction of Sinn and Bedeutung, or Russell's
theory of descriptions could be starting points of advanced
semantic theories at the time of their for? mulation, but it did
not happen. Similarly, one could expect that early works in logical
model theory (L?wenheim, Skolem, etc.) would be gener? alized to
general semantic theory, but it did not appear before Tarski
either.
Something what did not influence Polish philosophy had to block
the de?
velopment of formal semantics before the thirties. And something
had to stimulate Polish philosophers toward semantics. The
anti-semantic style of thinking is very well explained by a
distinction, introduced by Jaakko Hintikka (see Hintikka 1988) and
elaborated by Martin Kusch (see Kusch
1989), between the conception that considers language as a
universal me? dium (LUM, for brevity) and the conception that
regards it as calculus (LAC, for brevity).7 Here is a concise
comparison of both conceptions (the second one is also called the
model-theoretic conception of language (after
Kusch 1989, pp. 6-7):
(LUM1) Semantics is inaccessible; (LAC1) Semantics is
accessible;
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16 JAN WOLE?SKI
(LUM2) Different systems of semantic relations are
inconceivable; (LAC2) Different systems of semantic relations are
conceivable; (LUM3) Model theory is rejected; (LAC3) Model theory
is accepted; (LUM4) Semantic Kantianism (the view that the
linguistic resources are
prior to any experience) is adopted; (LAC4) Semantic Kantianism
is rejected; (LUM5) Metalanguage is illegitimate; (LAC5)
Metalanguage is legitimate; (LUM6) Truth as correspondence is not
intelligible; (LAC6) Truth as correspondence is intelligible;
(LUM7) Formalism is linked with the thesis that semantics is
not
cessible; (LAC7) Formalism is linked with the thesis that
semantics is accessible.
Thus, LUM, accepted by Frege, Russell and Wittgenstein, places
the users of language, so to speak, inside the linguistic system.
This internal relation to the language is responsible for the fact
that we cannot make statements about language and its relation to
the world which are crucial for doing any semantics, including
formal one. Wittgenstein formulated this view in a particularly
radical way (see Wittgenstein 1922, 4.121, 5.6):
[... ] That which mirrors itself in language, language cannot
represent [... ] That which expresses itself in language, we cannot
represent by language. [...]. The limits of my language mean the
limits of my world. (5.6)
Although Frege and Russell admitted a more external position to
language and its relation to the world, their semantic comments to
logical theories
were actually "an informal aside" only. On the other hand, the
LAC con?
ception or the model-theoretic tradition, represented by Edmund
Husserl, L?wenheim and then developed by Tarski, considered
language as a re
intepretable calculus which was used for description of various
formal and informal structures.
What blocked the rise of formal semantics?8 LUM certainly did.
It
probably prevented Frege and Russell from noticing the
importance of sys? tematic metalogical studies which immediately
lead to semantics; it is why
Russell saw the completeness problem, but not as something to be
formally proved. This position was strengthened by Wittgenstein who
influenced the Vienna Circle. However, the Viennese philosophers
did not want to consider philosophical theses as nonsense found a
way out in the syntactic (formal) way of speaking. This explains
why Carnap intended to reduce semantics to syntax and considered
semantic concepts, for example truth,
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FROM INTENTIONALITY TO FORMAL SEMANTICS 17
as syntactic or, at best, quasi-syntactic (see Carnap 1934). The
method of arithmetization introduced by G?del seemed to justify
this approach because it showed how to interpret metalogical
properties of a language in its syntax.9 Thus, logical empiricists
had reason for their hope that a mood of speech is possible which
(a) was about the language (contra Wittgen? stein and LUM), and (b)
did not concern the world (pace Wittgenstein and
LUM).10 However, the influence of LUM does not explain why
semantic theory did not arise with works of L?wenheim, Skolem or
Hubert. There is a simple answer that things always require time
and semantics could not
appear at once. This answer is proper perhaps for the case of
L?wenheim and Skolem (additionally, both worked on a particular
problem without being interested in generalizing their result to
general formal semantics). Yet this explanation does not fit Hubert
and his school. The Hilbertians had several mathematical devices to
do semantics, used the concept of model
informally, sharply saw problems with semantic flavour (the
completeness theorem) which should be rigorously proved and
accepted rather LAC than
LUM. A plausible answer for the case of Hilbert was given by
G?del in two
following passages (the first is quoted in Feferman 1988, p.
107, the second in Hao Wang 1974, p. 9):
However in consequence of philosophical prejudices of our times
[... ] the concept of objective mathematical truth as opposed to
demonstrability was viewed with greatest suspicion and widely
rejected as meaningless.
Non-finitary reasoning in mathematics was widely considered to
be meaningful only to the extent to which it can be
'interpreted' or 'justified' in terms of a finitary
metamathematics [...]. This view almost unavoidably, leads to an
exclusion of non-finitary reasoning from
metamathematics. [... ] my objectivistic conception of
mathematics and metamathematics in general, and of transfinite
reasoning in particular, was fundamental also in my other
work in logic.
In the first fragment, G?del certainly alludes to the Vienna
Circle. The second passage points out finitism and constructivism
as sources of anti semantic thinking. It explains why semantics was
and could be only something auxiliary in logical research of the
Hilbertians. However, we can also speculate a little about a
possibly wider significance of G?del's
diagnosis. Frege, Russell, Wittgenstein, L?wenheim, Skolem and
Carnap were constructivists and this can be taken as an additional
explanation that
they were hostile or at least insensitive to semantics.11 The
same applies to the intuitionistic school in the foundations of
mathematics, although it
was not interested in semantics by explicit philosophical reason
which led the intuitionists to constructivism.
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18 JAN WOLE?SKI
What about G?del himself? It is really a very interesting case.
His opin? ions quoted above can be supplemented by further ones
(the first is quoted in Hao Wang 1974, p. 9, the second comes from
G?del 1931, p. 181): [... ] it should be noted that the heuristic
principle of my construction of undecidable num?
ber theoretical propositions in the formal system of mathematics
is the highly transfinite
concept of 'objective mathematical truth' as opposed to
demonstrability [... ] with which it was generally confused before
my own work and Tarski's work.
As it will be shown in Part II of this paper, the true reason
for the incompleteness inherent
in all formal systems of mathematics is that the formation of
ever higher types can be
continued into the transfinite [...], while in any formal system
at most denumerably many of them are available. For it can be shown
that the undecidable propositions constructed
here become decidable whenever appropriate higher types are
added [... ] An analogous situation prevails for the axiom system
of set theory.
All above quoted passages from G?del clearly show that he saw
truth and
non-finitary reasonings as important conceptual and inferential
devices.
On the other hand, G?del stopped at their heuristic and informal
role, but he did not develop a semantic theory. Why? It seems that
G?del himself was partly limited by the philosophical background of
the Hilbert program and perhaps even of the Vienna Circle. Since I
discussed the matter else?
where (see Wole?ski 1991, Wole?ski 1998; see also Murawski
1998), I
only repeat my conclusion about the main difference between
G?del and
Tarski as far as the matter concerns the (un)definability of
truth. G?del, due to his background, regarded truth as not
subjected tout court to math? ematical treatment and thereby
undefinable, but Tarski succeeded in its correct mathematical
definition and proved it indefinability under certain
conditions, that is, for languages of infinite order or such
that arithmetic is expressible in them (this second version was
achieved under G?del's influence).
I will argue that five factors, three general and two
particular, decided that formal semantics in the form of the
mathematical treatment of the
truth-predicate arose in Poland. The general factors include:
intentionality, LAC, and a free admission of non-constructive
methods. Extensionality and the claim for exact definitions are
more particular factors. Let me
begin with the latter. The principle of extensionality was a
dogma among Polish logicians. It is perhaps best seen in Jan
Lukasiewicz's construction of many-valued and modal logic where all
logical values behave exten
sionally and modalities are truth functions. In general,
non-extensional
(intensional) contexts were regarded as defective from the
logical point view. This view was particularly strongly stressed by
Lesniewski who de?
manded that all intensional contexts should be eliminated from a
correct
language.12 Certainly, the principle of extensionality resulted
in a limita?
tion of the scope of logical investigations in Poland, because
it banished the
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FROM INTENTIONALITY TO FORMAL SEMANTICS 19
systems, nowadays called "intensional logics", from the scope of
formal research. On the other hand, this attitude considerably
facilitated the se?
mantic theory, because it could be based on compositionality of
semantic values. By contrast, Frege and Russell were puzzled by
oddities of inten? sional contexts, particularly by the behaviour
of identity in them. Thus, the
way to a unified semantic theory was obscured in the case of
Frege and Russell from the beginning, although they fully
appreciated extensional?
ity on the level of syntax. This story shows that sometimes a
minor and accidental (or even dogmatic) view can have important
consequences.
In Kreisel 1987 (p. 122), we find the following story: An
historical titbit: an objection not foreseen in the dissertation
[G?del 1930
- J. W.]. Ac?
cording to Mostowski, in a conversation in Tarski's presence,
the latter and his students had no confidence in G?del paper when
they saw the relevant issue of the Mhfte [Monatshefte
f?r Mathematik und Physik - J. W] in Warsaw. Why? G?del had not
formally defined
validity! Anybody who is surprised by this knows ispso facto
that he simply has no feeling for the subject.
Kreisel's comment sounds ironic. It is possible that G?del "has
no feeling for the subject". If so, it additionally explains why he
used the concept of validity in an informal manner. On the other
hand, G?del precisely defined other important concepts occuring in
his writings, for example, that of recursive function. If so, it
means that he had no feeling for the subject of defining semantic
concepts. Why? Certainly, it was exaggeration, if the
story was reported correctly, to have doubts concerning G?del's
paper only because he did not define validity formally. But perhaps
the issue was this: it would be better if validity had been
formally defined by G?del.
According to the standards of conceptual clarity and linguistic
sharpness shared in Polish analytic philosophy, definitions play a
very important role in any science. And if someone uses or
introduces a new concept, he or she should do it by a definition.
Perhaps it is not very important when a
particular separate problem is investigated, but the issue
becomes espe? cially significant for the development of new fields.
To repeat: semantic
concepts were informally used before Tarski and he had a rich
empirical evidence when he decided to construct formal semantic
theory with truth as its central concept. How could it be done
without definitions? In this case, a general methodological
attitude of Polish philosophers who insisted that definitions
should be worked to be productive.
Now I am passing to general points. The idea of intentionality
was in? troduced to philosophy by Franz Brentano. Let me recall the
most famous
passage from Brentano 1874 (p. 88; page-reference to Eng.
tr.):
Every mental phenomenon is characterized by what the Scholastics
of the Middle Ages called the intentional (mental) inexistence of
an object, and what we might call, though
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20 JAN WOLE?SKI
not wholly unambiguously, reference to a content, direction
toward an object (which is not to be understood here as meaning a
thing), or immanent objectivity. Every mental phenomenon includes
something as object within itself, although they do not all do so
in the same way. In presentation, something is presented, in
judgement something is affirmed or denied, in love loved, in hate
hated, in desire desired and so on.
Almost every word from this passage was extensively discussed. I
will
point out only one question, namely the status of intentional
objects. Brentano seems to say that they are parts (more exactly:
metaphysical parts) of mental phenomena. Not everybody in the
Brentanist camp (in? cluding later Brentano) agreed with this view.
In particular, Kazimierz Twardowski defended the view that although
all presentations have objects to which they direct, intentional
objects are real in most cases.13 It was associated with
Twardowski's famous distinction between the content and
object of presentation. Twardowski applied his account of
intentionality to linguistic expressions. In particular, names are
linguistic counterparts of presentations, and the former, like the
latter, have content (meaning) and object (reference). Thus,
linguistic expressions inherit semantic (Twar? dowski did not use
this word) properties from the intentional character of
mental acts. Here we have what Roderick Chisholm (see Chisholm
1986, p. 13) called "the primacy of the intentional".
Twardowski was the real father of Polish analytic philosophy,
including also Polish logical tradition. Let me quote Tarski 1992
(p. 20; this letter
was written in 1930): Almost all researchers who pursue the
philosophy of exact sciences in Poland, are in?
directly or directly the disciples of Twardowski, although his
own work could hardly be
counted within this domain.
Twardowski's original position was not fully acceptable for
Polish lo?
gicians because it could lead to psychologism, radically
rejected in the Polish logical school. However, it was easily
transformed to a conception about language and its properties.
Antipsychologism forced that the inher? itance of semantic
properties from mental acts had to be rejected. Thus, the original
view about the primacy of the intentional over the causal relations
was replaced by the primacy of the intentional by the primacy of
the semantic.14 This step was made by Polish philosophers, particu?
larly Kotarbi?ski and Lesniewski, and then adopted by Tarski as a
general
philosophical background of considerations about language. If we
say that semantics is primary, the immediate question is: to what
is it primary? The answer is clear: semantics is prior to syntax.
Now LAC can be taken as a
simple consequence of Polish semanticism, that is, the view that
semantics is primary, as opposed to Carnap's syntacticism.15 It is
interesting to point out that LAC was popular between students of
Brentano. In particular, as I
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FROM INTENTIONALITY TO FORMAL SEMANTICS 21
already noted, Husserl accepted it, at least in his early phase.
This example also shows that LAC alone is not sufficient for formal
semantics. It must be supplemented by suitable formal methods. I do
not suggest that Husserl
made an error when he did not enrich his philosophy of language,
I only point out that formal semantics is a complex enterprise.
Another outcome of the primacy of the semantic consists in
considering languages as principially equipped with meaning. It is
not so that inter?
preted languages appear when we add semantic valuations to
purely formal constructions, but the typical situation is that we
have the interpretation in advance. Of course, it does not prevent
formalization of large portions of our linguistic resources or
constructing purely formal schemes for these or other tasks. Also,
we can always change the assumed meanings, but mean?
ings are always prior with respect to syntactic properties of
language. This
point was advanced by Lesniewski in one of the most striking
expressions of semanticism (Lesniewski 1929, p. 487-488;
page-reference to Eng. tr.): Having no predilection for various
'mathematical games' that consist in writing out accord?
ing to one or another conventional rule various more or less
picturesque formulae which need not be meaningful, or even
-
as some of 'mathematical gamers' might prefer -
which should necessarily be meaningless, I would not have taken
the trouble to systematize and to often check quite scrupulously
the directives of my system, had I no imputed on its theses a
certain specific and completely determined sense, in virtue of
which its axioms, definitions
and, final directives [... ] have for me an irresistible
intuitive validity. I see no contradiction, therefore, in saying
that I advocate a rather radical 'formalism' in the construction of
my system even though I am an obdurate 'intuitionist'. [... ] I
know no method more effective for acquainting the reader with my
logical intuitions than the method of formalizing any deductive
theory to be set forth. By no means do theories under the influence
of such formalization cease to consist of genuinely meaningful
propositions which are for me
intuitively valid. But I always view the method of carrying out
mathematical deductions on an 'intuitionistic' basis of various
logical secrets as a considerably less expedient method.
As it is widely known, Lesniewski did very much for the
development of semantics. He introduced, at least in Poland, the
language/metalanguage
distinction, formulated the diagnosis of the Liar paradox and
outlined the method of its solution by excluding self-referential
sentences. Yet
Lesniewski did not construct a general semantic theory. He was
even not interested in that, because he did not accept mathematical
methods needed for executing this task.16
Tarski adopted intuitive formalism (this label is better than
"intuition? istic formalism" which can lead to a confusion with
intuitionism as a view in the foundations of mathematics) and
semanticism, and supplemented these general philosophical views by
proper mathematical tools. He was the first who perfectly
understood that formal semantics requires non
finitary rules, not only as heuristic devices but as its
mathematical methods. After years, he remarked (Tarski 1954, p.
713; page-reference to reprint):
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22 JAN WOLE?SKI
As an essential contribution of the Polish school to the
development of metamathematics one can regard the fact that from
the very beginning it admitted into metamathematics all
fruitful methods, whether Unitary or not."17
It is interesting that Tarski first defined the concept of
satisfaction and truth in a paper (see Tarski 1931) about definable
sets of real numbers. The main issue was the concept of
definability, not truth itself. He observed (p. 119; page-reference
to Eng. tr.) that:
Each particular set of numbers with which we are concerned in
mathematics is a definable
set, inasmuch as we have no other means of introducing any set
individually into mathem?
atics than by constructing the sentential function which
determines it, and this construction
is itself the proof of the definability of the set. On the other
hand, it is easily seen that the
family of all definable sets (just of the functions which
determine them) is denumerable, while the family of all sets is
not. More than that, it is known that, with any denumerable
family x set of numbers, a uniquely determined set x* can be
correlated that does not
belong to x; taking for x the family of all definable sets, we
get x*, and an example of a
set of numbers defined in terms of the metasystem, but not
definable in the system itself.
Thus, the diagonal reasoning which is non-finitary is deeply
connected with definability defined via the concept of
satisfaction. And a more philo? sophical account is expressed in
the following way (Tarski 1933, p. 253; page reference to Eng. tr.;
it is a comment on the undefinability of truth for
languages of the infinite order): In the course of our
investigation we have repeatedly encountered similar phenomena:
the
impossibility of grasping the simultaneous dependence between
objects which belong to infinitely many semantic categories; lack
of terms of 'infinite order'; the impossibility of
including in one process of definition, infinitely many
concepts, and so on. [... ] I do not believe that these phenomena
can be viewed as a symptom of the formal incompleteness of
the actually existing languages - their cause is to be sought
rather in the nature of language
itself; language, which is a product of human activity,
necessarily possesses a 'finitistic'
character, and cannot serve as an adequate tool for the
investigations of facts, or for the
constructions of concepts of an eminently 'infinitistic'
character.
The simplest reading of these deeply philosophical remarks is
this: due to
the finitistic nature of any language considered as a syntactic
object, we should apply non-finitary semantic tools in order to
catch infinite concepts like truth. Today we have an impressive
wording of this fact: the concept of truth exceeds the arithmetical
hierarchy of objects definable by arithmet? ical predicates. Since
syntax can be arithmetized, as G?del showed, and
provided that finitistic properties are recursively definable
and equating 'syntactically definable' with 'recursively definable'
we obtain the precise
formulation of the main moral coming from formal semantics: for
any lan?
guage sufficient for arithmetic, its semantics transcends its
syntax. At this
point, semanticism (the primacy of the semantic) meets the
non-finitary nature of semantic concepts. The explanation why
formal semantics arose
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FROM INTENTIONALITY TO FORMAL SEMANTICS 23
in Poland is now simple: semanticism and non-finitary
mathematical meth? ods were consciously combined by Tarski.
However, it was the path from Twardowski to Tarski that let to this
end.18
NOTES
1 I use in this paper some material from Wole?ski 1998. 2 Emil
Post proved the completeness theorem for propositional calculus in
the form:
Every formula of the system is either provable in it or leads to
inconsistency. It is a
purely syntactic formulation that was suggested by Post himself:
"We have consistently
regarded the system of "Principia" and generalizations thereof
as purely formal develop? ments [...]." (Post 1921, p. 164-165;
page-reference to reprint). Hence, I do not include the Post result
into my sketch of the history of formal semantics. Later it was
proven that only
propositional calculus admits the purely syntactic version of
the completeness theorem. 3
Unfortunately, I was not able to identify a place in which
"semantic" occurs in Peirce.
Hanna Buczy?ska-Garewicz, an expert in Peirce, informed me that,
according to her
knowledge, this word was never used by him. 4
Ogden and Richards mention a work by Dr. Postgate (1896), but I
was not able to check whether he used the word 'semantics'. 5 This
review is included into Eng. tr. of Kotarbi?ski 1929, pp. 515-536
(the section on semantics is on pp. 522-529); the Polish original
was published in 1930. 6 Also Carnap used at that time (1930-1932,
in his unpublished manuscripts) the word 'Semantik' as a synonym
for 'Syntax' or 'Metalogik'. In Poland, this use was proposed
by
Leon Chwistek. 7 The idea that language is a reinterpretable
calculus does not occur in descriptions of
LAC offered by Hintikka and Kusch. I added this point because it
seems to me that it is not
enough to say that this approach consists in regarding language
as an abstract formalism with an added interpretation. Tarski
always insisted that formal semantics is intelligible only if
formalized languages are interpreted. We can, of course, change the
assumed in?
terpretation but doing semantics for uninterpreted languages is
impossible. Hintikka, in the mentioned paper, argues that Tarski
accepted LAC for formal languages, but LUM for colloquial speech.
It seems not quite accurate. Tarski excluded natural language as
a
subject of semantics for its antinomial character. In other
words, there is no external point of view from which all possible
languages can be consistently investigated. It means that LAC for
all languages taken together is impossible, but not that LUM is
accepted. 8 I omit here the fear of antinomies, the circumstance
very strongly stressed by Tarski (see
Tarski 1936, p. 20). It seems that the role of this factor was
always exaggerated, similarly as the significance of
set-theoretical antinomies for the development of the foundations
of mathematics. Let me add, however, that our perspective is
different from that held by Tarski. Thus, what I can say in this
note is that the matters look differently from our point of view
than from the perspective of the early 30s.
Carnap's syntacticism (the word introduced by Thomas Oberdan)
and his way to se? mantics are extensively studied in Coffa 1991,
Oberdan 1993 and Cirea 1994. I sum up this discussion in Wole?ski
1998. 10 This remark must be properly understood. It does not imply
that language does not con?
cern the world. Frege, Russell and Wittgenstein were very far
from that. LUM in hands of
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24 JAN WOLE?SKI
Wittgenstein and the Vienna Circle claims that a correct,
meaningful metalanguage speak?
ing about relations between the language and the world is
impossible. For Wittgenstein, we
have no chance to improve this situation, but for the Vienna
Circle, the syntactic mode of
speech is a solution. Later, Neurath and some other logical
empiricists accused semantics
of introducing metaphysics into philosophy. I omit this problem
which concerns more the
question of how semantics was received than how it arose. 11
According to G?del (see Wang 1987, p. 182), Skolem did not prove
the completeness theorem for first-order logic, because he did not
accept non-finitary methods of inference.
Thus, the speculation about the negative influence of
constructivism on the development of semantics is not groundless.
12 Lesniewski was Tarski's teacher. In early Tarski's writings we
find many traces of
Lesniewski's influence, also in the question of extensionality.
13 The so called objectless presentations have special objects; in
fact, Twardowski rejec? ted objectless presentations, but this way
of speaking is convenient, although it may be
misleading. My account of Twardowski's view is simplified and
based on Twardowski
1894. 14 It does not mean that the primacy of the intentional
must be rejected at all. It only means that it is not relevant for
logic and semantics, but only for pragmatics. 15 I do not claim
that all Polish philosophers, even students of Twardowski, shared
LAC.
The matter is controversial concerning Ajdukiewicz's early views
associated with his radical conventionalism. 16
Henryk Hiz informed me that Lesniewski explicitly departed
himself from the semantic definition of truth given by Tarski. He
(Lesniewski) used to say: "Please, do not join me with this
construction."
17 Tarski was inclined to empiricism and nominalism,
philosophical views not quite co?
herent with non-finitary reasonings. However, he considered his
philosophical opinions as
somehow "private". In particular, they could nor decide about
the admissibility of math?
ematical methods "fruitful" for metamathematics. The attitude of
Hilbert or Carnap and
many other logicians was exactly contrary. Let me add that
semanticism and admission of
non-finitary methods resulted with realism in semantics. It is
contrasted with antirealistic
semantics usually based on constructive logic. 18 I am indebted
to an anonymous referee for many fruitful comments about the
earlier
draft of this paper.
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FROM INTENTIONALITY TO FORMAL SEMANTICS 27
Institute of Philosophy Jagiellonian University Grodzka 52
31-044 Krakow
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E-mail: [email protected]
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Issue Table of ContentsErkenntnis (1975-), Vol. 56, No. 1, The
Legacy of the Lvov-Warsaw School (2002), pp. 1-122Volume
InformationFront MatterEditorial [pp. 1-6]Preface [pp. 7-8]From
Intentionality to Formal Semantics (From Twardowski to Tarski) [pp.
9-27]Philosophical Background and Philosophical Content of the
Semantic Definition of Truth [pp. 29-62]Kotarbiski as a Scientific
Realist [pp. 63-82]What Difference Does It Make: Three Truth-Values
or Two Plus Gaps? [pp. 83-98]Reasoning on a Tight Budget:
Lesniewski's Nominalistic Metalogic [pp. 99-122]Back Matter