Abstract:
In this lecture I will comment about Tarski's work, mainly his work in syntax and semantics, both from the point of view of linguistics and a little from a philosophical perspective. It is astonishing how many of Tarski's writings are relevant to linguistics and how many more to philosophy. Beside Warsaw he had close links with Wienerkreiss and Popper
Tarski contributed significantly to the development of linguistics. He was the first to provide what is called today a generative grammar of a language, that means a recursive definition of a sentence in structural descriptive terms. And he showed how to fashion semantics in a rigorous and profound way. In these two efforts he strictly kept apart syntax and semantics. Of course, syntax must precedes semantics, but only semantics deals with the relation between language and reality. In semantics Tarski's main achievements are the definition of truth and the definition of consequence.
Tarski's definition of a sentence in a language, in his first case, the language of the elementary theory of classes, is completely independent of semantics. It starts with elementary sentences of the kind xÕy and proceeds to all permissible combinations with logical connectives and quantifiers. It is a perfect case of generative grammar
Tarski's success is largely due to the fact that he is dealing only with mini-languages. Therefore his technical problems are also mini in comparison with those in colloquial speech. I do not mean to imply that all languages with which Tarski deals in his work are simple. Some of them: present considerable complications, for instance languages of infinite order or those languages in which the substitution range of a variable is not identical with a semantic category. Against any natural language Tarski advances a vigorous battle. .
Semantics is introduced with the relation of satisfaction of a sentential function by a sequence of objects; in the case of the theory of classes, Tarski uses a sequence of classes. This invites problems of methodology and of philosophy. To base semantics on both denotation and truth may seem overdone. One of them should determine the other. Truth often implies denotation. If I know that Dublin is in Ireland, then I know what Dublin is and what is Ireland. To specify individuals, Tarski has the difficulty which often faces a set theoretician when asked to state what are the members of his most elementary sets.Tarski always felt most comfortable working in the framework of set theory. However, he also recognized individuals. He wrote: "In the intuitive interpretation of the language which I always have in mind here, the variables represent names of classes of individuals" (C161)1. In another place, he writes: "Individuals are objects which are not sets". This calls for some comments. As Mostowski wrote (in The Encyclopedia of Philosophy), Tarski was a nominalist, i.e., he considered that general concepts are names, just names. Individuals are singular. In a true sentence of the form 'a is b' LeÊniewski's systems allowed for the subject 'a' names of singular objects only. So that it must be provable that if x is a and y is a, then x=y. Predicates may have singular or plural denotations. Of course, the class of all b may name a singular object on a higher level. But what are the singular objects which are not classes?
Let me venture a proposal in what seems the spirit of the school. Individual persons are individuals. Here, it matters that a part of a man is not a man. Neither is a man a part of another individual. Perhaps we agree to include among individuals our dog. It has a personality. Perhaps other organisms: birds, fish, maybe trees. But to call a desk an individual is an abuse. It is anthropomorphic to call desks, shoes, stones, planets individuals. However, in the practice of speech we make exceptions. A carpenter works with nails. A nail is a nail, but when it breaks, no half of a nail is a nail. The carpenter also considers his finished desk an object and he can count desks he made. Similarly, a grammarian speaks about an occurrence of a word as an individual. But LeÊniewski warned not to confuse an occurrence of a word with the concept of a word. Instead we can say that the first word in your copy of the program for our conference conforms with the first word in my copy. And there is no such object as the program of our conference, only individual copies which conform to one another. Tarski inherited from LeÊniewski this approach to words and to other language fragments. Language fragments are individual material objects and concatenation is putting one such object after another, making a new object. Tarski's axiomatization of concatenation is a glorious summary of these views. These ideas in lingustics created the distinction between a sequence of speech fragments and a string of the same fragments. A string divided in any way will be the same string. But not so for sequences. Incidentally, let me mention here two unusual views about objects. KotarbiŸski claimed that the boundary between organic and inorganic matter is not sharp, opening the door to panvitalism. LeÊniewski considered the mereological sum of his left hand and the Mont Blanc mountain to be an object. A fragment of the picture in his dream was also an object for LeÊniewski, who under the influence of Hussrl and Ingarden was inclined to some phenomenology.
Two writen strings conform to each other if one of them can be taken as the result of someone's copying of the other. That was the explication of conformity by LeÊniewski. Speech fragments are called utterances. Utterances, like words or sentences, are not exactly individuals in Tarski's sense. They are events, but a linguist treats them as a carpenter does his nails. Just as to copy a word you do not have to make your inscription the same size as the original, or use the same color of ink, so the vocal repetition does not have to preserve the pitch or the speed of the original performance. In our culture there is a good sense of freedom in copying inscriptions and in repeating utterances. A repetition has to convey only the same information as the original does.
Just as we copy or reprint inscriptions, we all repeat what others say. This banal fact is fundamental for linguistics. We study particular cases of the schema: a says that p . In it, 'a' is a name of any utterance, any kind of name, a structural name, or concatenation of fragments of a, a phonemic description of a, or an occasional phrase, as Edward just used x. . In 'p' we put an utterance which is a repetition of a.
Notice, forming a name of an utterrance by quotation marks is not a very happy choice. For, besides all the complications pointed out by logicians from Tarski on, as a graphic device a quotation mark applies to a writen language, but not equally well, if at all, to a spoken language. Here is someone's typical report of such a fact: Edward said that some women do not like Wagner. {Note that here Edward himelf is a.} Someone else reports: Edward said that not every woman likes Wagner. A third person reports: Edward said that Wagner is not liked by some women. And the fourth: Edward said that it is not the case that Wagner is liked by every woman. The first two are recorded by logicians as equivalent, but not the third nor the fourth. At least not directly. We may suppose that any of the four sentences are correct repetitions of Edward's utterance. There were some other people present and none objected. Note that we usually repeat not verbatim From the repetitions we can not infer what Edward actually uttered. But it does not matter. We know what his utterance said and that is what we are after. In repetition, the unrecoverability of the original is significant. Linguistics is the science of the hearer, not of the speaker. We do not care what the speaker had in mind, only how he is heard. And this will be relevant to my version of Tarski's Convention T.
Linguistically, it is important to know parphrases of a sentence2. It is also important that some good sentences which have the same words are not paraphrases of the four metioned. We mark them by 'o'. E.g. o Women do not like every Wagner.
Some words are found in texts on many, if not all, topics and in different styles. We take them to be grammatical constants, similar to the logical constants, which are used throughout the language. The choice of logical constants is partly arbitrary, and the choice affects the study of grammar. Grammatical constants occurring In our examples are some, every, do, not, -ed, by, and -s . In between the constants other words can occur. In distinction from constants we call them content words. With the same arrangement of constants we can find other content words, e.g.
`Not every dog bites the neighbors
Some dogs do not bite the neighbors
The neighbors are bitten by some dogs
It is not the case that the neighbors are bitten by every dog
oNeighbors bite every dog _
To generalize, let 'N' stand for words like dog or neighbor and 'O' for bite, like, and similar. Now the schema will be
` Not every N1 O N2
Some N1do not O N2
N2 are O by some N1
It is not the case that N2 are O by every N1
oN2O every N1__
Such schemata are called batteries of paraphrastic transformations. Between the elements of a battery the relation of a transformation holds. A transformation allows to add or delete a grammatical constant, to change the order of words, to repeat a word, or to delete any repetition, provided that the result is a sentence. Besides paraphrastic transformations, there are also non-paraphrastic transformations, e.g. , negation, consequence, generalization and changing tense or mode 3.
Linguists seldom, if ever, study which sentences are true. Their focus is on information. In repeating, we don't take responsibility for the truth of the speaker's utterance. But we ask whether the repetition preserves informantion without taking away or adding anything to it. It is important that Tarski's definition of a sentence uses no semantic concept like truth or satisfaction. Each of the steps leading to definition of a sentence (C178 def.12) is a purely mechanical, syntactic, or, to use Tarski's phraseology, structural description. And so is the description of a transformation. What is the import of a given transformation is another matter. This requires a semantic judgement, e.g. whether the result of the transformation is a negation, consequence, generalization or, perhaps, nothing clear and thus should not be used.
Batteries of transformations and their combinations constitute the grammatical scaffolding of the language. Part of it is the logical scaffolding, part specific to a given language.
Language exhibits a double structure: transformations and the cooccurrence of words. Words are coming and going. Many English words are forgotten, change their usage, no longer cooccur with the same words. New words enter every language all the time and cooccur with other words in unpredictable ways. The behavior of words depends partly on our changing knowledge. Words are the most unstable lot in language. But transformations and the structures of paraphrastic sets stay unchanged for a long time . New words easily find their proper places in transformations. And, except for banalities, grammar will not decide which sentences are true. At best, it can help show which sequences of sentences are well formed strings. And, with logic within its machinery, grammar can contribute to the analysis and critique of reasonings in a natural language. It is unquestionable that in reasonings within a natural language, people use some of the rules of inference stated by logicians, e.g., modus ponens, syllogism, operating with quantifiers and negation as did the audience of Edward. But not all of it is due to the wisdom of logicians. I think that several schemata of reasoning were practiced with good results long before Hindu grammarians or Greek philosphers registered them. Modus ponens had substanial empirical grounding. Similarly, a linguist finds transformations when they are practiced by the speakers. He does not invent them, he just registers them.
Let us reflect for a moment upon the relation says that and its arguments. In a categorial notation: S (N _ S), in Herris notation Ons. Not every nominal phrase can stand as the subject of say . For instance rye , a rib, coal-tar, seven can not be so used (except if there is a special symbolism, as when a rib is used as a coat of arms of a family). Somebody may observe, and with good reason, that say that requires three arguments as in Edward said to me that Š (though more fluent may be Edward told me that Š ). For, we normally, speak to somebody. In our example Edward said it to those who later reported what he said. But there is a difficulty here. Edward told me Š may imply that he manifestly addressed me. But for a general semantic consideration the fact of address may be disregarded. Also, it is of marginal interest that after says there may be a noun: He said grace, mass, oath or similar ritualistic recitations. Of course, when the subject of say that is a personal name, or plural, say accepts all forms of a finite verb. But in an abstract discourse it remains a constant says.
So far we don't know whether Edward spoke in English. Perhaps he used a Polish sentence Nie wszystkie kobiety lubià Wagnera and by that he said that not every woman likes Wagner. We state that the Polish utterance says that not every woman likes Wagmer. Abstracting from the speakers, we ask What did the Polish utterance say? Possible correct answers were given before. The language in which the answers are given and in which says that is coined, is the now used language, or U-language, as it was called by Haskell Curry and I will preserve that term, in his memory. Saying that can be modified in a variety of ways. E.g, Edward said tentatively thatŠ, says with laughter that Š, says with commitment that Š, said whispering that Š , or simply whispered that Š, The court ruled that Š or other illocutionary variations of verba dicendi. Some require gramatical change in the subordinate phrase; The coach odered us to run, but excused Mary from running 4.
We mark sentences that depart from ordinary usage but are still grammatically correct by a dagger: Ý17 is blue. I call them rough utterances. It may happen that for a rough sentence you can find a context in which it sounds natural, smooth: I attend County School 17. Our basketball team wears blue shirts. 16 is red, 17 is blue. For the famous ÝColorless green ideas sleep furiously saving context may be Students are drab and naive. But those colorless, green ideas sleep furiously during my lecture. Such contexts are called saving legends. Rough sentences form a partial order; some are rougher than others. Thus, 17 is prime and blue is rougher than Ý17 is blue. To give a rough arithmetic sentence,17 divided by 0 = 1. Rough setences should be distinguished from ungrammaical strings, such *Wear blue team basketball shirts our or *A girl walk to school. In the schema a says that p the fragment that p is what some philosophers called "a proposition which expresses a sentence". While I will not traffic further in propositions, in this talk, the semantics I am proposing does make allowance for them.
Addressing the liar paradox requires some preliminary consideration of the words true and false. Logicians seldom distnguish between two uses of each of these words. One use is predicative about an utterance; the sentence Kiev is in Ukraine is a true sentence and Kiev is in Mongolia is a false one. That predication is in U, though the sentences about which we predicate may be in another language. The second use is with the word that just after true or after false : It is true that Kiev is in Ukraine, and It is false that Kiev is in Mongolia. Here, we do not speak about linguistic utterances, but about geopolitical facts. The entire sentence is in U and the words true and false occur inside the utterances. Accordingly, I will call the second usage internal. And the first external . To recapitulate, true that and false that are in the same U language as the phrase just after that. True and false without that are prediates about the preceding sentences in whatever language they happen to be. Sometimes, for short, I will speak about internal and external truth and falsehood. The external true is adjectival, whereas the internal may be taken to be nominal. In some other languages the difference is more pronounced. In Polish the external are adjectives prawdziwe and fa½szywe, but the internal are clearly nouns, prawda and fa½sz 5. Reading logical formulae in English we are forced to use internal truth and falsehood. The formula CpCNqNCpq is often read as If p then if not q then not if p then q . But this is a rough utterance. Note that in any typical English sentence the negation will occur not at the beginning but in the middle of it. The difficulty is more striking with NC; not if p occurs easily as a corrective after a sentence, e,g, You will be late - Not if my watch is right. NCpq is read more fluently as it is false that if p then q or as it is not true that if p then q. Here and in many internal falsehoods, it is not the case that or similar locutions are also used. Tarski and, following him, most of the philosophical literature, spoke only about external truth, calling it just truth . They insist that true is a metalinguistic predication about a sentence of the object language sentence, where object means 'object of study'. As is usual, in Tarski's work the internal truth is tacit. It may be tempting to identify the external truth with Tarski's truth. But a natural language contains its metalanguage6. Instead of the Tarskian construction of levels of languages, I propose a syntactic distinction of it is true that and the predicative true. To construct formalized languages and analyze their properties is a different task than to study languages spoken in existing communities.
The links between internal and external truth and falsehood are outlined by the following principles:
Principle of two truths
(P2T) If a says that p, then a is true, if and only if it is true that p, where it is true that can be deleted7.
Principle of two falsehoods
(P2F). If a says that p, then a is false, if and only if it is false that p. In plain English (and Polish) one puts negation inside the sentence: If Edward pojecha½ do Kijowa says that Edward went to Kiev then the Polish sentence is false, if and only if Edward did not go to Kiev. Principle of consistency
(PC). If a is true then it is false that a is false. By transposition and cancelling of double negation, if a is false, then it is false that a is true. Generally, normal logic applies to the internal affairs. But we do not accept here that if a is not true, then it is false nor that if a is not false, then it is true. We must leave room for external value gap. Note also that there is no established internal predicate for roughness. (A proposal for such a predicate: it is not clear whetherŠ or not)
Liar paradox8: For some a
1. a says that a is false.
Substituting in P2F a is false for p and detaching 1,
2. If a is false then it is false that a is false .
Applying the tautology CCpNpNp to 2,
3. It is false that a is false.
With the same substitution in P2T and detaching 1,
4. If a is true, then a is false.
Using transposition to 4 and detaching 3,
5. It is not true that ais true.
Being neither true nor false, a is like rough sentences. But here we cannot imagine a legend which could save a or 1. I call 1 a twister .
The twister does not lead to an antinomy. By an antinomy I mean, following LeÊniewski, a situation which a person faces when from the premisses which she accepts as true and by reasonings which she considers valid she concludes two sentences of the forms f and not f .The only solution to an antinomy is to find that a premiss or a reasoning was not correct. No such situation occurs with the twister. That is why I used above the term paroadox, somethig shocking.
Returning to the liar, its strangeness does not consist in self-referentiality. I assert that I am saying the truth, all the truth, and nothing but the truth. is not strange, though it is as self-referential as the liar. It is exactly why it is asserted in important moments. To assert means to assert truth.. Therefore the liar says I assert the truth that I am lying. which is not only logically inconsistent but also psychologically untolerable. Tarski's conclusion differs from what I have just said. He wrote "These antinomies [liar and heterological] seem to provide a proof that every language which is universal and for which the normal laws of logic hold, must be inconsistent". (C 164-5}. This may be an exaggeration. When I reach a contradiction, I know that a premiss was false. Do not blame the language for our mistakes. Anyway, I do not understand an accusation that a language is inconsistent. A language is not a set of statements. It is only a mechanism for making statements. But what statements we choose to hold depends on us.
If I can reconstruct Tarski's thought, a language is inconsistent when it allows to build sentences wich lead to inconsistency. English allows us to build Today is Tuesday and tomorrow will be Friday. With a small part of calendar you obtain a contradiction. But English, while it allowed you to build that sentence, does not force you to admit its truth. I put such talk among rough sentences. Other authors (Van Frassen, Herzberger) put it also in the value-free class Some linguists take the radical step and dismiss such sayings from sentencehood. It is not the self-referentiality that causes the dismissal. The main villain . is the falsehood self-applied
Tarski's animosity toward natural language makes him write surprising statements. To give an example, he writes about a natural language (C164): "It would not be in harmony with the spirit of this language if in some other language a word occurred which could not be translated into it". Well, words usually cannot be translated. sentences can. The Aleutic word igloo does not have an English equivalent. The Oxford dictionary gives a description only: "Eskimo dome-shaped hut, esp. one built of snow". Polish cwaniak is often translated into English sly. But one dictionary gives the translation of You are a cwaniak as So that's your little game. That Tarski could think of translating verbatim reflects badly on Polish linguists of his time. And indeed, they mostly wrote dictionaries.
There are many proposals for a grammar of English as well as for a number of other languages. Each directly or implicitly uses the categorial assignement to words, as did Tarski. Fundamentally, it consists in dividing words of a simple sentence into operators and their arguments. Some words cannot be operators: proper names, sheep, water, the boy, the lake , etc. Usually this class is named 'N'. Operators may be defined as sentence-forming functor. This class is 'O'. Operators are subdivided according to the arguments they require. On are those that take as argument an N, as does freeze in the lake froze. Next is Onn, as read in The boy reads a book. Ono as say in Edward says that it snows, (where p happens to be an O without an argument requirement). Ooo is a conjunction, Oo may be for example a negation, or an adverb with the argument sentence. Teaches in Felice teaches Mark the violin is an Ono, where the second argument (a zeroed play ) is an Onn. etc. But the categorial assignements are soon blurred by transformations. Any obvious or unimportant words or phrases can be dropped: The boy reads, Felice teaches. We also have: Felice is a teacher of violin, Mark is her student, or Mark studies violin with Felice, Mark studies with Felice, She is a violin teacher, Mark studies with Felice, who is a violin teacher. All of those and many other variations must be permitted in your grammar as paraphrases of one another. A working hypothesis claims that the information contained in any sentence x will be conveyed by a sequence of short sentences, each with clear category assignements. The string is then informationally equivalent to the sentence x.
To give a general characterization of Tarski's semantics, its main theme is to connect objects with truth. One who has doubts about objects or about their connection with truth may have difficulty with this work, which is closer to the Aristotelian tradition than to the line of Stoics, Wittgenstein and Wolniewicz. For a semantic theory in this style I don't have enough time
1.The references are to the second edition, edited by John Cocoran of Tarski Logic, Semantics, Meta-Mathematics, translated by J.H.Woodger 1983, Hockett.
2. Linguistic studies of paraphrases are to be found in Richard M. Smaby, Paraphrase Grammars, Dordrecht, Reidel, 1971 and Catherine Fuchs, La Paraphrase, Paris, Presses Universitaires de France, 1982.
3. Maurice Gross showed that every French verb enters a different set of transformations, i.e. for any two French verbs there is a transformation in which one of the verbs is the main O and to the other the transformation does not apply.
4. This was noticed by Maurice Gross and, in a different way, by J. L. Austin.
5. A distinction similar to the internal and external truth appears in a paper by John L. Pollock . The Truth about Truth in The paradox of the liar ed. by Robert L Martin, Yale University Press, 1970. However, Pollock discusses possibilities of giving a definition of the external truth starting from the internal truth as more basic. No such definition was attempted in my talk. The principle of two truths contains the clause If a says that p, which changes the topic.
6. That a natural language contains its own metalanguage in which it describes the language was forcefully stated by Zellig Harris; see especially his Mathematical Structures of Language, New York, Interscience Publishers, 1968. Chapters 5 and 6 discuss the connection between different languages, including scientific discourses, sublanguages of English vis a vis their metalanguages. Also his A Theory of Language and Information. A Mathematical Approach, Oxford, 1991.
7. There is a close affinity between P2T and Convention T of Tarski, ( C pp. 187-9). Although Tarski does not speak explicitly about internal truth he uses it implicitly when he writes 'x ‘ Tr if and only if p'. Here after the last 'if' one can read 'it is true that p'. Tr is the external truth, true that is the internal. Both P2T and Convention T speak about the relaton between the two truths. Neither of the formulae is a definition of truth. P2T can be used as a rule of inference, as I used it in analysis of the twister. In Convention T Tarski requires that for 'x' one substitute a structural - descriptive name of any sentence of the language in question and for the symbol 'p' the expression which forms the translation of this sentence into the metalanguage. For the reason explained above, I prefer to say that that 'p' is a translation, a repetition, a paraphrase of x into the U language or that x says that p. These are minor variations. More substantial differences arose because Tarski studied some formalized languages and I natural languages. They are given and must be studied empirically. And the most important property of a linguistic phrase is its repeatability. Whatever is not repeatable is not in language. Some phrases are saying something. A noun boy can be repeated, but it does not say anything, whereas A boy runs says something, namely that a boy is running. As Tarski said we need a structural-descriptive name of any sentence. He obtained it by the recursive method, today often called generative grammar. At the time of his writing there was no generative grammar of any natural language. Following his metod, today several attempts at such a grammar of English are known. If we limit English to some fragment suitable to a particular discourse, they can be adequate, if not pefect. 8. The formulation presented here differs from many by being independent ofany context or situation . By contrast. e.g. S. Kripke (Outline of a theory of truth, The Journal of Philosophy 72 (1975) shows how much liar may depend on empirical premisses.

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