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1 Philippe De Brabanter 1. I Logique & Analyse 248 (2019), 351-378 WHAT DO WE MEAN WHEN WE TALK OF OBJECT-LANGUAGE AND METALANGUAGE IN THE THEORY OF NATURAL LANGUAGES?1 Philippe De Brabanter Abstract In this paper I raise the issue whether it is legitimate to talk about metalanguage and object-language in the context of natural language, as linguists often do. I show that whereas the study of formalised languages allows a clear distinction between metalanguage and object-language, that distinction is muddied by a number of factors when it comes to natural language use. I also show that in the latter context, talk of metalanguage and object-language requires making a further distinction, between object-language and natural language, and that none of the features displayed by metalanguage and object-language in logical theory has a perfect counterpart in the realm of natural languages. In particular a natural ‘object-language’ proves not to be a language after all, while a natural ‘metalanguage’ should arguably be regarded as a use of a language instead of as a language in its own right. Keywords: metalanguage, object-language, formalised language, colloquial language, natural language, quotation, meta-word, metalexicon 1. Introduction Linguists make frequent use of the term metalinguistic, and more occa- sional use of metalanguage, and object-language in discussions of natural- language phenomena.2 In what senses exactly? Do those terms borrowed 1 I would like to thank the audience at the conference Language and metalanguage, logic and meta-logic. Revisiting Tarski’s hierarchy, UCLouvain, May 19, 2016, for useful com- ments and challenges. I am indebted to an anonymous reviewer and to Raf Salkie for point- ing out flaws in an earlier version of this paper, and to Mikhail Kissine for helpful sugges- tions. Finally, I gratefully acknowledge the financial support of the F.R.S.-FNRS research project T.0184.16, 2016-2020. 2 A note on the quoting conventions used in the body of this paper: – single quotes for the introduction of terminology, and for scare quoting – italics for metalinguistic citations, and for emphasis – double quotes for citations of other authors, and for direct discourse more generally – French angle quotation marks for meanings/contents – boldface for parts of examples that deserve special attention (mostly, quotations) These conventions are not necessarily followed in the examples when these are borrowed from external sources (various writers stick to various rules). doi: 10.2143/LEA.248.0.3287321 © 2019 by Peeters Publishers. All rights reserved. 352 PHILIPPE DE BRABANTER from the theory of formalised languages make sense when applied to the theory of natural languages? These are the questions I seek to answer in the present paper. My starting-point is a very conservative picture of the relations between object-language and metalanguage as defined for formalised languages. In particular, I am not concerned with the following issues: – the reasons logicians/philosophers had for making the distinction between object-language and metalanguage; – logicians’/philosophers’ views on the defective design features of natural languages; – the different kinds of metalanguage that can be distinguished in the theory of formalised languages (e.g. syntactic vs. semantic); – more recent proposals in logic for doing away with the (need for a) dis- tinction between object-language and metalanguage. The paper is organised as follows: after this short introduction, I devote Section 2 to a brief outline of the core design features of the metalanguages used in the theory of formalised languages, and how they relate to their object-languages. In 3, I examine at length what happens to these features and relations in the natural-language context. In Section 4, I offer diagrams of the relations between metalanguage, object-language, and natural language. This is followed by a brief conclusion in Section 5. 2. Object-language and metalanguage in the theory of formalised lan- guages: a brief sketch In the 1930s, Alfred Tarski put forward the idea that the truth-predicate of a language L — the semantic predicate par excellence — could not be defined in that very language, on pain of generating semantic paradoxes of self-reference such as the ‘Liar paradox’. His solution was to develop a view according to which there exists a hierarchy of languages, and truth in a language of level n (an object-language, LO) is to be defined in a richer, higher-order language of level n+1 (its metalanguage, LM) (e.g. Tarski 1944: 350). In Tarski’s framework, an LM is always the metalanguage of a particular LO. Such an LM must contain at least the following building blocks: – names of, and variables for, the well-formed formulas of the LO whose semantics is at stake (Tarski 1983:3 172-173; also Carnap 1947: 4; Church 1956: 60). In the cases that will be of greatest interest to us, those 3 This is the translation of a paper originally published in Polish in 1933. OBJECT-LANGUAGE AND METALANGUAGE 353 names are quotations, typically formed by enclosing LO formulas in quo- tation marks. – semantic predicates, such as true (sentence-in-L), satisfy, denote, designate, etc. (Tarski 1983 passim; 1944: 345; also Carnap 1947: 4; Gupta 1998: 266). – all the LO symbols and formulas, or translations of these (Tarski 1944: 350; also Reichenbach 1947: 10-11; Church 1956: 63; Prior 1967: 230; Simmons 2009: 545). Those various building blocks are illustrated below in two instances of Tar- ski’s T-sentences. The first one concerns a formalised language; the second one concerns what Tarski called a ‘colloquial language’, and is only provided because it permits me to exemplify a quotation-as-name. Note, however, that Tarski was highly skeptical of the possibility of giving a definition of truth for colloquial languages, owing to what he called their ‘universality’, i.e. the notion that “if we can speak meaningfully about anything at all, we can also speak about it in colloquial language” (Tarski 1983: 164). (1) 1 2(ı1,2 + ı2,1) is a true sentence (in the language of the calculus of classes, or LCC) if and only if for any classes a and b we have a b or⋂ ⋂b a. (2) ‘Der Schnee ist weiss’ is a true-sentence-in-German if and only if snow⊆ is white.⊆ In (1), true-sentence-in-LCC is a semantic predicate. The expression 4 “ 1 2(ı1,2+ı2,1)” is the name, in the metalanguage, of a sentence of LCC. Here, the metalanguage is a suitably enriched variant of English (cf. also Carnap⋂ ⋂ 1947: 4; Church 1956: 47). Following if and only if, one finds a translation into the metalanguage of the LCC sentence (cf. Tarski 1983: 187). Note that the LCC sentence named and translated in (1) could also have been quoted. This would have been another way of naming it. To see how a sentence can be named by means of a quotation of it, see example (2), where the subject is a quotation-name of a German sentence, true-sentence- in-German is a semantic predicate, and snow is white is an extensionally 5 equivalent translation in EnglishM of a GermanO sentence. Note that if the 4 Exceptionally, for clarity, I use double quotation marks for metalinguistic citation, instead of italics. 5 Where suitable, I use subscripts to specify whether some linguistic object is to be regarded as a metalanguage or an object-language. A caveat is in order, though: in the con- text of natural languages, as we will see, the subscripts are not equivalent in meaning: GermanO means “the German language, considered as an object of description”; EnglishM means “the set of sentences in English that are metalinguistic”, not “the English language, considered as a metalanguage”. 354 PHILIPPE DE BRABANTER metalanguage had been enriched German, an LO sentence itself rather than its translation would have been used in the metalanguage. LO’s and LM’s are constructed by logicians. The features that these lan- guages should display are therefore stipulated. LO’s are formalised lan- guages. LM’s are assembled in order to construct these formalised languages 6 and/or describe their semantics. Each particular LM is so designed as to be expressively richer than its LO. Since it is prohibited to name an LO element within LO itself, any sentence that contains a quotation-name of an LO expression is unambiguously a sentence of an LM of that LO. Likewise with sentences including a semantic predicate. This means that sentences of LO and LM can easily be told apart. Some writers think that it is important that LM include translations of all LO expressions instead of these expressions themselves (Carnap 1947: 4; Church 1956: 63; Simmons 2009: 545. For an extended discussion of what Tarski meant by translation, see Mou 2001; Raatikainen 2003). Others do not seem to care much whether LM includes LO as such or trans- lations (Tarski 1944: 350; Prior 1967: 230). If LM includes translations (and not expressions) of LO, there is no inclusion relation between LM and LO. If (or when) LM includes LO expressions themselves (and not transla- tions), then LM LO (e.g. Tarksi 1944: 350). In the latter case, some sen- tences are “ambiguous as to level of language”, as Reichenbach (1947: 10) 7 put it. What all ⊃logicians agree on, however, is that LM is of a higher order than LO. Below is a recap of the main properties of the LO-LM relation in the theory of formal languages: (i) LM is set up to describe/construct a particular LO; (ii) Both LM and LO are languages in their own right; (iii) LM has its own lexicon (set of building blocks).
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