From Intentionality to Formal Semantics (From Twardowski to Tarski)

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.


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


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


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



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

-

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


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



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;


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,



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.


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



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


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



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):


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



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


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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Institute of Philosophy Jagiellonian University Grodzka 52 31-044 Krakow

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Article Contentsp. [9]p. 10p. 11p. 12p. 13p. 14p. 15p. 16p. 17p. 18p. 19p. 20p. 21p. 22p. 23p. 24p. 25p. 26p. 27

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

From Intentionality to Formal Semantics (From Twardowski to Tarski)

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