ERROR LIST BELOW 020214 Please send errors you have spotted using the format below. Some of these corrections and comments could lead to papers, or at least abstracts.
John Corcoran. Errors in Tarski's 1983 truth-definition paper. Bulletin of Symbolic Logic. 19 (2013) 514. ► JOHN CORCORAN, Errors in Tarski's 1983 truth-definition paper. Philosophy, University at Buffalo, Buffalo, NY 14260-4150, USA E-mail:
[email protected] In 1935, at 34 years of age, Alfred Tarski (1901–1983) published a German translation by Leopold Blaustein of the world-shaking truth-definition paper [1, pp. 152–278] begun over six years earlier [1, p. 277]. The main results were achieved when he was 28 years old (ibid.). The German translation was not flawless. Further errors were introduced in the 1956 English translation by J. H. Woodger: Tarski had been unable to approve final page proofs [1, p. x]. In the 1983 second edition Tarski made scores of corrections—some originally suggested by him, some by me, and some by others. But he noted that more extensive rewriting was desirable [1, p. xiv]. The posthumous second, third, and fourth printings each saw further corrections including restoration of a missing 14-word passage on p. 172 noted by Matthias Schirn in 1997 [1, p. viii]. This essay lists and discusses errors that have come to light since 1983. Most are indisputable, but several will be disputed. An unproblematic example: page 156 calls the expression ‘it is snowing’ a name of a sentence: add another pair of quotes. Disputable examples include segmentation errors throughout his paper: for example, he takes commonnoun phrases such as ‘true sentence’ to be proper names—denoting classes. In the same contexts, he takes the incoherent fragment ‘is a’ to be a relational verb coextensive with ‘belongs to’. In short, e.g., he segments is a true sentence as ‘is a’ plus ‘true sentence’, instead of as ‘is’ plus ‘a true sentence’. [1] ALFRED TARSKI, The concept of truth in formalized languages, Logic, Semantics, Metamathematics, revised edition, edited with new introduction and analytical index by J. Corcoran, Hackett, 1983, translator J. H. Woodger, fourth printing 2007, originally Oxford UP, 1956. END OF ABSTRACT Acknowledgements: William Abler, Otávio Bueno, Gabriela Fulugonio, David Hitchcock, Joaquin Miller, Frango Nabrasa, Jose Miguel Sagüillo, Matthias Schirn, Martin Walter, Leonardo Weber, and others. ERRORS, CORRECTIONS, AND COMMENTS NOTATION 156.6 is 6 lines down from the top of page 156. 156.-6 is 6 lines up from the bottom of page 156.
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152.6 FOR For although PUT For, although 152 T speaks of defining the term ‘true sentence’, of defining the concept of truth, and of defining the meaning of the term ‘true sentence’. What is his real goal? See 155 below. 155.2 Now T speaks of a definition of truth for sentences of colloquial language. See 152 above. 155.6 FOR (1) a true PUT (1) A true? COMMENT: Tarski might be giving the clausal form of the sentence—in which case the uppercase is not needed and the period goes with the embedding sentence beginning ‘By this I mean. But, even though he often employs clausal forms, he never says a word about what they are and why they are needed. Neither the Polish nor the German version has a capital letter after ‘(1)’.
155.16 FOR starting-point PUT starting point DELETE HYPHEN. 155.17 Now T speaks of partial definitions of the truth of a sentence. What is this? See 152 above. Comment: Polish has no articles, but the German has the definite article before the German word for truth (‘Wahrheit’).
156.2 FOR individual name PUT proper name OR JUST name. 156.3 FOR individual name PUT proper name OR JUST name.
COMMENT: This is probably not a mistranslation. See the last full sentence before the footnote on 159.Both the Polish and the German use an adjective synonymous with the English adjective ‘individual’. But in English, ‘individual name’ is very peculiar. 156.12 FOR the name ‘it is snowing’ PUT the name ‘ ‘it is snowing’ ’. COMMENT: See 157.14, where T refers to the name “ ‘snow’”, a 6-character expression which denotes a certain 4-chacter expression, viz. ‘snow’. Both the Polish and the German have the double quotation marks. Woodger was sloppy here.
OMISSION: T never explains that names of quote names are made by adding another set of quotes.
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COMMENT: “ ‘snow’” denotes ‘snow’. ‘snow’ denotes snow. Snow does not denote. One could say because it is cold. Nothing cold denotes. Being cold excludes having a denotation. COMMENT: “ ‘snow’” (is *denotes) (‘snow’* snow). ‘snow’ (is *denotes) (‘snow’* snow). Snow does not denote. “ ‘snow’” (is *denotes) a ( 6-* 4-) character word. ‘snow’ (is *denotes) a ( 6-* 4-) character word. ‘snow’ (is *denotes) a (word * form of water). 156.-23 FOR names of PUT general names whose extensions consist of. 156.-11 MISTAKE: There is no way to take shape to play this role: the left paren has the same shape as the right paren, etc. See Corcoran-Frank-Maloney 1974, and Corcoran 2006. Carnap makes a related point in 157.8 FOR For example we could use ‘A’, ‘E’, ‘Ef’, ‘Jay’, ‘Pe’ as names of the letters ‘a’, ‘e’, ‘f’, ‘j’, ‘p’. PUT For example, we could use ‘A’, ‘E’, ‘Ef’, ‘Jay’, and ‘Pe’ as names of the letters ‘a’, ‘e’, ‘f’, ‘j’, and ‘p’, respectively. NAMING TYPES: COMMENT ‘A’ is a name of the letter ‘a’. Thus, A is the first letter of ‘and’. ‘Es’ is a name of ‘s’. Thus, Es is the first letter of ‘snow’. ITERATED QUOTES CONVENTION: The only places I know of where T iterates quotes making a quotes name of a quotes name is 157.14 and 159.-4, where the outer quotes are the double [ “ ]. Normally, I use singles [‘] for all quotes naming. As mentioned above, T iterates quotes at 156.12 in both the Polish and the German versions. At 157.14 the Polish version has a single pair of quotation marks but the German version a double pair—evidently a correction on Tarski’s part. At 159.-4 both the Polish and the German have iterated quotation marks. In both the Polish and the German both pairs of quotation marks are double quotation marks. They are distinguished from one another by subscripting the first left quotation mark and superscripting the second left quotation mark.
COMMENT: In German names begin with uppercase letters. But is there any reason to carry this over in the English translation? We do not capitalize the first letter of proper names of numbers or numerals. In the Polish version the names es, en and so on are not capitalized. The capitalization appears to reflect the German convention of capitalizing nouns. (Of course, the letters being named and
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so the names of them are different in the three languages, the Polish word for snow being ‘Ñnieg’ and the German word ‘Schnee’.)
157.19 FOR (4) an PUT (4) The COMMENT: Both the German and the Polish have a lower-case initial letter of the first word after ‘(4)’.
157.22 FOR seven letters Es, En, O, Double-U, I, En, and Ge (in that order) PUT six letters Es, En, O, Double-U, I, and Ge (in the order Es, En, O, Double-U, I, En, Ge) OR seven-member letter series Es, En, O, Double-U, I, En, Ge COMMENT: We have non-quote conventional names for the Greek letters: alpha, beta, gamma, etc. There is no corresponding convention for English (Latin) letters or for HinduArabic numerals (figures, digits). What is the situation in German? Did Tarski coin these names or was there already in place a generally accepted letter naming convention? See my BSL paper on schemata: Corcoran 2006. David Hitchcock wrote: “The non-quote conventional names for the Greek letters can almost all be found in Plato’s Cratylus. Plato is unlikely to have invented those names, since his characters use them un-self-consciously, without any explanation of the choice of name. Plato may have innovated in putting them in writing—the occurrences in the Cratylus are practically the only occurrences of those names in the surviving corpus of ancient Greek writings. Tarski’s German names for the letters look invented to me, but it needs a native speaker of German to tell us for sure.”
158.20 FOR Nevertheless no rational ground can be given why such substitutions should be forbidden. PUT Nevertheless no rational ground I know of can be given why such substitutions should be forbidden. COMMENT: No such qualification occurs in the Polish or German. 158.-3 FOR the expression ‘it is snowing’ which occurs there twice PUT the expression ‘it is snowing’, which occurs there twice COMMENT: We have an attributive or parenthetical relative claus, which requires commas in English. The assertion beginning 158.-5 contradicts the last paragraph of 159.
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The Polish has no commas on either side of the relative clause, which looks as if it might be the equivalent of ‘occurring there twice’. The German has the equivalent adjective phrase without commas around it: ‘the twice there occurring expression “it is snowing”’.
160.11 FOR not (5) but PUT not (3) but Both the Polish and the German have ‘(3)’. 161.-12 FOR ‘father’ PUT ‘the father of’ The Polish has no equivalents of the English words ‘the’ and ‘of’, just: ‘ojciec’ in the expression ‘ojciec x’a’. The German does have equivalents, and takes ‘Vater’ to be the functor in the expression ‘der Vater des x’. This looks like a slip on Tarski’s part. 164.1ff COMMENT: To his credit and to our benefit, here and throughout, AT avoids the absurd convention of capturing commas with quote marks. For example, where some would write “ ‘not,’ ”, AT writes “ ‘not’, ” to denote the three character string ‘not’ ending with a tee, as opposed to the four character string ‘not,’ ending with a comma.
164.3 FOR sentential calculus (or theory of deduction) PUT sentential calculus (or the socalled theory of deduction) OR sentential calculus [DELETING (or theory of deduction)] COMMENT: Sentential calculus is no theory of deduction. 164.6f FOR definition of a true sentence PUT definition of ‘true sentence’.
164.-13f FOR if we can speak meaningfully about anything at all, we can also speak about it in colloquial language PUT if we can speak about a thing, we can also speak about it in colloquial language THAT IS: FOR anything at all PUT a thing COMMENT anything at all CANNOT BE THE ANTECEDENT OF A PRONOUN. THIS WAS JUST BAD GRAMMAR. 164.-1f DELETE BOTH COMMAS. 165.4F FOR If we analyse this antinomy in the above formulation we reach the conviction that no consistent language can exist for which the usual laws of logic[…] PUT
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If we analyse this antinomy in the above formulation, we reach the conviction that no consistent language can exist for which the usual laws of logic[…] ADD COMMA. 165.-3 FOR definition of a true sentence PUT definition of ‘true sentence’. 165.-2f FOR definition of truth PUT definition of ‘true sentence’. 172.21 FOR variables v1, v2, v3, …) PUT variables ‘v1’, ‘v2’, ‘v3’, …)
173.18 AXIOM 1. ng, sm, un, and in are expressions, no two of which are identical. COMMENT: The clause ‘no two of which are identical’ is completely useless: no two things are identical. What he wants is ‘where ng, sm, un, and in are pairwise distinct, i. e. ng is distinct from sm, un, and in; sm, is distinct from un and in; and un is distinct from in’. FOR no two of which are identical PUT ‘where ng, sm, un, and in are pairwise distinct, i. e., ng is distinct from sm, un, and in; sm, is distinct from un and in; and un is distinct from in’.
174.-8 COMMENT: There are several misleading aspects to this interesting footnote.
177.9 FOR On the other hand the PUT On the other hand, the ADD COMMA Both German and Polish use a single word that is not set off by a comma; in the Polish version the word is embedded in the sentence, which is in fact a clause preceded by a semi-colon.
178 COMMENT: The first occurrence of x in the string ‘(x = 1 & for every x, x = x)’ is free but the second, third, and fourth occurrences are bound. Using Tarski’s terms, is x a free variable of that string? Using Tarski’s terms, is x a free variable of that string?
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186.16 FOR: The extension of the two concepts is thus not identical. PUT: The extensions of the two concepts are thus not identical. COMMENTS: Some philosophers of language ridicule people who write ‘[…] is identical’ or ‘[…] is not identical’, using ‘is identical’ or ‘is not identical’ as a one-place predicate. These two expressions are parts of two-place predicates: ‘is identical to’ and ‘is not identical to’.
194.3 FOR The concept of truth is reached in the following way. PUT The definiens of our definition of the concept of truth is reached in the following way. COMMENT: We already have reached the concept; what we are reaching for is a definition—actually a definiens, or right side. 194.7 FOR free variables PUT free variable occurrences 194.9 FOR free variable PUT free variable occurrence 194.15 FOR The sentences of the first kind […] are the true sentences. PUT The sentences of the first kind […] are the true sentences.[DEITALICIZE] COMMENT This is a conclusion about true sentences, not a definition of true sentences. 194.16 FOR those of the second kind […] are the false sentences. PUT those of the second kind […] are the false sentences.[DEITALICIZE] COMMENT This is a conclusion about false sentences, not a definition of false sentences.
195.-22 FOR we must now relinquish this postulate and regard any many-one relation [...] PUT we must now relinquish this postulate and regard any one-many relation [...] COMMENT Leonardo Weber pointed out that Woodger is translating the words “eindeutige Relation” , which he translates “one-many” everywhere else, e.g. page 171.
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207.21 [T]hose assumptions of the metatheory which determine the cardinal number of the class of all individuals [...] exert an essential influence on the extension of the term 'true sentence'. The extension of this term is different according to whether that class is finite or infinite. In the first case the extension even depends on how big the cardinal number of this class is. FOR [T]hose assumptions of the metatheory which determine the cardinal number of the class of all individuals [...] exert an essential influence on the extension of the term 'true sentence'. PUT [T]he cardinal number of the class of all individuals [...] exerts an essential influence on the extension of the term 'true sentence'.
COMMENT: The first sentence reveals a serious error whose ramifications appear elsewhere in CTFL. This erroneous statement conflicts with other statements Tarski makes. The error can be fixed by deleting the first few words of the sentence, viz. the following words “[T]hose assumptions of the metatheory which determine” and making the verb agree. Otherwise the quote is flawless in intent. The expression ‘in the first case’ means “in case the class is finite”. The assumptions of the metatheory are chosen by a human being [Tarski] who is powerless to determine the cardinal number of the class of all individuals. No assumptions of the metatheory could determine the cardinal number of the class of all individuals. The class of all individuals is whatever it is regardless of what assumptions anyone makes about it. Moreover, Tarski repeatedly says that the axioms should be accepted without proof and that what is taken as an axiom depends on the state of knowledge in the field at the time. This [LSM 207] confuses (the extension of ‘provable’ * provability) with (the extension of ‘true sentence’ * truth). 224.5 FOR free variables occurring PUT variables occurring free OR variables having free occurrences 224.-15 FOR free variables of PUT variables occurring free in OR variables having free occurrences in 224.-14 FOR free variable PUT variable occurring free in OR variable having a free occurrence 249.3.1 FOR quantifiation PUT quantification
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ADD see 249.3.2 FOR function y PUT function ‘y’ ADD quotes 249.5 The variable ‘n’ thus represents names of classes the elements of which are classes of individuals. 258.-7– -2. For this purpose we must nevertheless to some extent sharpen the premises of the theorem by assuming that the class of all provable sentences of the metatheory is not only consistent, but also ω–consistent in the sense of Gödel, K. (22), p. 187, or in other words, that this class remains consistent after a single application of the rule of infinite induction, which will be discussed below. COMMENT: Gödel’s system’s non-logical constants are 0 and f [for successor]; the numerals are strings of effs followed by a zero. He says a set of formulas is ω–inconsistent it it contains the negation of a universal whose numeral instances it contains. For example if it contains ‘not every x has P’ and it contains ‘0 has P’, ‘f0 has P’, ‘ff0 has P’, and so on. 259.1-5 The fact that we cannot infer from the correctness of all substitutions of such a sentential function as [sc. ‘x is not a true sentence or the negation of x is not a true sentence’] the correctness of the sentence which is the generalization of this function, can be regarded as a symptom of a certain imperfection in the rules of inference hitherto used in the deductive sciences. 259. 4 CORRECTION: DELETE THE COMMA. 259.1-5. CRITICISM 1: Here AT is talking about inferring ‘for every x, Fx’ from the set of its substitutions. Without alerting the reader he changes the subject in the next sentence 259. 6-17. There he discusses inferring ‘for every x, if x is an expression, Fx’ from the set of its substitutions provided every expression is named in at least one substitution. 259.1-5. CRITICISM 2: The “inference” mentioned is unsound. For example, take the first-order language L0SN, where ‘0’ denotes zero, ‘S’ is the successor symbol, ‘N’ expresses “is an integer”, and the variables range over the set of real numbers. Every substitution of ‘Nx’ is true, but ‘for every x, x is an integer is false’.
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259.1-5. CRITICISM 3: AT implies that the absence of a rule countenancing inferring a conclusion from infinitely many premises is a defect [258.6]—when he admits [260.5-10] that such a rule could never be applied. He never says why this absence is a defect. Is it a defect in our counting methods that we can never finish counting off the integers? Is it a defect in rulercompass constructions that they do not trisect arbitrary angles? What does he mean by ‘defect’? Not every limitation is an imperfection. Not every limitation is a defect. 259.1-5. CRITICISM 4: It is misleading, confusing, anachronistic, and a disservice to many readers to speak of inferring ‘from the correctness’ of certain sentences ‘the correctness of’ another sentence. What is the correctness of a sentence? AT should keep to his established terminology of inferring sentences from sets of sentences.
QUESTION FOR JC: Does it matter whether instead of the above we take the following/ [sc. ‘it is not that x is a true sentence or it is not that the negation of x is a true sentence’]
COMMENTS ON 259.1-5 1. The sentence 259. 1-5 is longer than necessary. It can be shortened: SHORTENED 259.1-5 The fact that we cannot infer from all substitutions of a one-free-variable sentential function the function’s generalization is an imperfection in the previously used rules of inference.
2. Below it is clear that the variable ranges over a superset of the set of expressions of an object language among whose signs are ‘N’, ‘A’, ‘Π’, and ‘I’ [see 168]. 259.1-5 implies that all results of substituting expression names for the variable are “correct”. 259.6-8. In order to make good this defect we could adopt a new rule, the so-called rule of infinite induction, which in its application to the metatheory may be formulated somewhat as follows:[…]
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