The Theory of Grammars

Hans-Heinrich Lieb

Integrational Linguistics: The Theory of Grammars

© 2018 Hans-Heinrich Lieb

1.  Introduction

The Integrational Theory of Grammars (ITG) was first developed in Lieb (1974a), (1976c). It was subsequently summarized and modified in Lieb (1983b: Part G) and Lieb (1993g: Chs. 20 and 21). It has recently been characterized and confronted with current approaches to grammar writing (compare Müller 2018) in Lieb (2018). (For some criticism of earlier versions of ITG, see Falkenberg 1996a.)

Integrational Linguistics (IL) appears to be unique in

i. sharply separating a theory of grammars (understood as texts) from a theory of language (understood as a general theory of languages, in a grammar-independent sense of “language”),
ii. developing a theory of grammars and a theory of language separately but in unison.

Concerning grammars of any kind and theories of grammars, it is assumed that both grammars and theories of grammars either do or should presuppose, explicitly or implicitly, theories of language, however informal and incomplete; it is argued that serious problems arise if presupposed theories of language are not admitted. (For a similar position, see Lehmann 2018.) The requirement of a presupposed theory of language is not met by standard grammar frameworks as discussed in Müller (2018) but is met indeed in Integrational Linguistics:

(1)    The Integrational Theory of Grammars and any grammar conforming to this theory presuppose the Integrational Theory of Language.

In Lieb (1974a), (1976c), the emphasis is on formulating a theory of grammars, and it is only within this context that a theory of language is also sketched. In Lieb’s later work, including most of Lieb (1983b) and (1993g), developing a theory of language – The Integrational Theory of Language (ITL) – has been the key concern; compare, in particular, Lieb ed. (2017). The parts of Lieb (1983b) and (1993g) that are devoted to presenting the Integrational Theory of Grammars presuppose the IL work on the Integrational Theory of Language.  

ITG attempts to cover the format, the interpretation, and the modification of existing grammars, which means that ITG falls within the domain of the ‘theory of linguistic description’ as one branch of ‘metalinguistics’. ITG does not include a part on how to arrive at grammars in the first place. This is considered as belonging in the domain of linguistic methodology construed as a different branch of ‘metalinguistics’. However, ITG is indirectly relevant to the methodology of grammar writing due to an emphasis put on the empirical nature of grammars.

The present brief outline of ITG will be based on the grammar part of Lieb (1983b) and, especially, on Lieb (2018). ITG presupposes the Integrational Theory of Language, but this theory will be mentioned only for a few features that are relevant to ITG; for details of ITL, the reader is referred to the relevant texts on this Homepage (“Integrational Linguistics: Development and topicality”, and “Integrational Linguistics: The Theory of Language”, Parts I and II).

In addition to the general work on ITG, there have been more specialized studies adopting ITG. Lieb (2013: Sec. 7) outlines how ITG may apply in the word-formation part of a grammar; similarly, outlining a related framework, Nolda (2018), with an application to German word formation and an implementation by a computer program ( ).  Lieb (2008), making use of Lieb (1998b), applies the axiomatic approach in contrasting German and French phonology. However, most axiomatic work in IL has been devoted so far to developing a theory of language rather than to formulating axiomatic grammars. Independently, a point for axiomatic grammar writing has recently been made in Neef (2014).

2.  General assumptions

Some essential assumptions of ITG, making use of ITL as the presupposed theory of language, are informally characterized in Lieb (2018: Sec. 4.2) (numbering in the quote is retained for the present text):

(2)    A grammar of a linguistic means of communication – i.e., of an idiolect – is a text that determines – either completely or in part – a system of the means of communication: a system such that something is a normal utterance by a speaker using the means of communication only if the utterance agrees with the system.

(3)    A grammar of a set of idiolects is a text that determines, or identifies, a system for the set of idiolects: a system construed, in the simplest case, as a set of component-specifying properties of idiolect systems such that for any system of any idiolect in set, the system has each one of the properties.

(4)    The object of a grammar is a pair consisting of either

i. an idiolect and a system of the idiolect, or
ii. a language construed as a set of idiolects (where a language is either a complete historical language or one of its periods – a certain subset) plus a system for the language, or
iii. a variety of a language (also construed as a subset) plus a system for the variety (stages of languages or varieties count as varieties of a language).

(5)    If a formal or semi-formal grammar has the same intended coverage as a part of an informal one, then any grammatical statement made in this part and any of its logical consequences or presuppositions can, in principle, have a semantic analogue in the formal or semi-formal grammar.

These assumptions can be modified to cover more complex grammars such as comparative ones, along the lines of Lieb (1993: Chs 20 and 21).

“Language” in (2) to (5) should be understood to allow for any realization mode, and for fully developed sign languages as languages. Lieb (1993) = Lieb (1993g).

Assumption (5) formulates an adequacy condition for formal and semi-formal grammars, giving priority to informal ones. This condition is to apply even outside an IL framework. In Lieb (2018) it is argued that the adequacy condition cannot be satisfied by generative grammars, be they generative-enumerative as in classical Generative Grammar or constraint-based, model-theoretic as in HPSG, whereas formal or semi-formal ITG grammars do satisfy the condition.

Generally, ITG is contrasted in Lieb (2018) with other theories of grammars and related to informal grammar-writing. The present characterization of ITG will be restricted to the conception of formal or semi-formal grammars, understood as texts.

3.  Axiomatic grammar format

Formal and semi-formal grammars as conceived in ITG satisfy the following basic condition:

(6)    Grammars are axiomatic theories, in a conservative sense of the term.

The presupposed conception of axiomatic theories is essentially the one characterized in Carnap (1958/2012: Sec. 42) as modified in Lieb (1983b: Part G) and Lieb (2018: Sec. 7).

Due to the axiomatic grammar format, a number of distinctions are rendered explicit in formal or semi-formal ITG grammars that may be hidden in informal ones. Due to its axiomatic format, we will be able to recognize, or partly recognize, even in a correlated informal grammar (Lieb 2018: Sec. 7.4, (9)): 

  1. The sentences that should be taken as definitions (including recursive definitions),or as formulating basic assumptions, or as formulating consequences of definitions and basic assumptions. [ … ]
  1. The ontological status that may be assumed for the linguistic objects described by the grammar. [ … ]
  1. The definitional status that may be assigned to the terms in the grammar (defined vs. undefined), their logical status (logical vs. non-logical, in particular,axiomatic), and their interpretation. [ … ]
  1. The relationship of the grammar to potential data. [ … ]

Important details for ITG grammars are given in Lieb (2018: Parts C and D), as follows.

4.  Axiomatic primitives: Language name and language-system name

Suppose that we are dealing with a grammar G of a language or language variety D (the ITG treatment of idiolect grammars will be disregarded here). There is at least one axiomatic primitive (undefined axiomatic constant) of the grammar, and this is a language name D*, say, “English”, which denotes the set D of idiolects. Taking the language name as an undefined constant of the grammar is to be preferred over defining it, and is supported by assuming a ‘language-name axiom’, see below (Sec. 8).

We also need a constant σ* for denoting a system σ for the language or language variety D, the language-system name.  If undefined, σ* is the only other axiomatic primitive. If defined, σ* is an axiomatic term in case the language name is used in its definition, but this need not be the case, for the following reason.

The language-system name may be defined using only terms from the ‘basic language’ of the grammar: the part of the grammar’s language that does not contain expressions in which axiomatic terms occur. The basic language contains any theory that is presupposed by the grammar, in particular, the presupposed theory of language; for an ITG grammar, this is the Integrational Theory of Language (ITL). The terms from the presupposed theory of language are available for defining the system name, and may be sufficient for this purpose. If so defined, σ* is not an axiomatic term of the grammar but is a term of its basic language. Even then, σ* is connected with the language name by the ‘language-system axiom’, see below (Sec. 8).

In addition to the language name and the language-system name, we need expressions in grammar G to refer to linguistic objects connected with the language or language variety D.

5.  The theory of language as a source of grammatical terms

Even in a typical informal grammar of a language, a general theory of language – however incomplete – that is implicitly presupposed in the grammar plays a basic part in creating the terminology needed to refer to entities belonging to the language. The following example is taken over, in a somewhat simplified form, from Lieb (2018).

Let us assume that ITL, the theory of language presupposed in a grammar G, contains the following definition (S is an arbitrary idiolect system in an arbitrary language):

(7)    W is an article of S if, and only if: S is an idiolect system, and W is a lexical word of S with an empty lexical meaning, and every form f of W is used in S as the auxiliary part of some (analytic) noun form of S.

We are assuming in ITL a conception that allows for analytic noun forms, not only analytic verb forms, such as the tree in English idiolect systems; moreover, grammatical terms like “article” are relativized to idiolect systems not language systems to account for language variability from the outset.

While I do subscribe to definition (7), other definitions may well be considered for “article” in a sense where it applies to the idiolect systems of arbitrary languages. (Also, “article” in an informal sense may be formally explicated not by a two-place predicate, as above, but by a one-place function term; see Lieb 2018: Sec. 6.5.) Even so, “article” can now be used in grammars of arbitrary languages as a grammatical term, or a term from which more complex grammatical terms are obtained by strictly logical operations.

6.  Grammatical terms in a grammar of a language

Suppose that G is a grammar of Standard British English that presupposes ITL and has “Standard British English” as the language name. The grammar may contain the following theorem:

(8)    For any system S of any idiolect that belongs to Standard British English and any W, W is an article of S if, and only if, W is a lexical word of S and a, the, sòme, àny, or is a form of W.

(sòme, àny, and have inherent secondary word accent, indicated by the accent sign, which distinguishes them from the corresponding forms of indefinite pronouns.) Equivalently:

(9)    For any system S of any idiolect that belongs to Standard British English and any W, W is an element of article (-, S) if, and only if, W is a lexical word of S and a, the, sòme, àny, or is a form of W.

Now consider the expression “article (-, S)”. By the definition of the hyphen notation in Carnap (1958/2012: § 33d), “article (-, S)” is a logically complex expression that denotes the set of lexical words of S of which a, the, sòme, àny, or are forms, for any idiolect system S of Standard British English  

Suppose that S* is a defined term of grammar G, whose definition may or may not involve the language name “Standard British English”, and assume that the sentence “S* is a system of an idiolect of Standard British English” is an axiom or theorem of G. We then obtain from (9) the following theorem:

(10)    article (-, S*) = the set of lexical words W of S* such that a, the, sòme, àny, or is a form of W.

The terms “article (of)”, “article (-, S)”, and “article (-, S*)” in (7) to (10) are related as follows.

7.  Grammatical terms: Analysis

The term “article (of)” is a grammatical constant, defined in (7), of ITL, the theory of language presupposed in the grammar. This constant is used in (8), and used in the complex open term “article (-, S)” in (9) and in the complex closed term “article (-, S*)” in (10): Each time it has exactly the sense in which it is defined in (7); there is no new definition of “article” in the grammar. The two logically complex terms cannot be defined, since neither is a constant; still, the two terms are available in the grammar for referring to a grammatical category either of any idiolect system S of Standard English British as in (9), or of a specific idiolect system as in (10). Neither of the two sentences (9) and (10) are definitions; they are theorems of the grammar making empirical claims that are possibly false (which they are if sòme, àny, or are not admitted as forms of lexical words).  

This analysis also applies to (8). True, (8) has a form that would be possible for a conditional definition of “article”, restricting application of the term to Standard British English, but this is not sufficient for being understood in this way. Anyway, as long as “article” from the theory of language is available in the grammar, no new definition is possible: Defining the same term twice in a single context is rightly forbidden in the theory of definitions.

But suppose ITL is not available in the grammar, or a formal difference is introduced, such as capitalization, to obtain two slightly different terms. Even so, if a sentence like (8) is construed as a definition (making “article” into a ‘descriptive category’, or a term for a ‘descriptive category’, in the sense of Haspelmath 2010, Haspelmath to appear), there are disastrous consequences: The term is logically unconnected with “article” in its general sense in a theory of language, as in (7); a new definition is needed for “article” in dealing with each new language; and sentence (7) ceases to make an empirical claim – definitions are either logically true, or are neither true nor false. In other words, interpreting sentences like (8) as definitions – as Haspelmath does – is simply inadequate. (For further discussion of the example and its generalizations, see Lieb 2018, where a different construal of functional terms like “subject” is also outlined, in Sec. 6.5.)

In summary, the three sentences (8) to (10) do not define expressions of the grammar but identify, by non-definitional properties, linguistic objects denoted by expressions whose meaning is directly or indirectly given by the definition in (7).

Some grammatical terms in a grammar may well be introduced as defined axiomatic constants, using the language name or the language-system name in their definition and thus restricting their use. However, most grammatical terms used in a grammar are either constants from a presupposed theory of language or are logically complex expressions obtained from such constants by logical means other than definitions. This is not only true of ITG grammars and ITL as the presupposed theory of language; it also holds of typical informal grammars, though not of formal ones written in standard frameworks, where theories of language are simply missing – to the detriment of the grammars.

8.  Axioms and theorems in a grammar of a language

The term “axiom” is ambiguous. When used informally, it may mean, roughly, “important self-evident truth for which no proof is needed”; in a formal context, its meaning is “non-definitional sentence in a deductive system that is not derived in this system from other sentences”. The sentences that are chosen in a theory as axioms in this second sense usually satisfy additional informal requirements of relevance to the theory, such as allowing for the derivation of a large number of ‘important’ theorems. Axioms in a theory of language or a grammar are axioms only in the second sense.

Adopting ITG, the role of axioms in a grammar can be characterized as follows (Lieb 2018: Sec. 10.6):

Two axioms are needed in any grammar G of D and σ: the language-name axiom, which uses the language name D* in stating that D is a language; and the language-system axiom, which uses D* and the language-system name in stating that σ is a system for D. There is at least one additional axiom, a language-determination sentence or, if the grammar is language complete, a language-identification sentence. If the grammar is system incomplete, at least one system-determination sentence is added as a further axiom.

Due to the language-name axiom, anything that can be said of languages in general can also be derived in the grammar in relation to D, to the extent that the presupposed theory of language has been taken over into the grammar. Analogously, for the language-system axiom: due to this axiom, anything that holds of all language systems may be derived for σ, to the extent that the theory of language has been presupposed. Each one of the additional axioms is a claim on something that is ‘specific to D’.

In a grammar that is language complete, there is a language-identification sentence, an additional language axiom or a theorem identifying D on the basis of σ. The identification sentence may well satisfy the form requirements for a definition of the language name, e.g., could have the following form: D* = the greatest set D´ such that σ* is a system for D´.

It would still be a fundamental mistake to construe the identification sentence as a definition, for the following reason. It must be possible for a language-identification sentence to be empirically false. However, definition sentences of a theory are either logically true or are neither true nor false, depending on one’s theory of definition. Thus, the empirical nature of the grammar evaporates if the language-identification sentence is misconceived as a (‘nominal’) definition.

Examples of theorems have been given above, in (8) to (10); for examples of proofs, see Lieb (2018: Sec. 10.6).

9.  Grammar application and modification

In an empirical theory such as a theory of language or a grammar, axioms and theorems are subject to revision or rejection once the theory is confronted with relevant data; even the system of definitions may have to be modified.  

On the ITG conception, grammars are confronted with data by grammar application. If G is a grammar of a language or language variety D, application consists in extending G into an applied grammar G´, in a number of steps.

First, constants are added as new axiomatic primitives of G´ that denote specific speakers, specific speech objects or speech events, and specific idiolects belonging to D; further constants, possibly defined, are added that denote systems of the idiolects. The added constants for speakers and speech objects or events are ‘observation terms’ relative to G´. The theory of language ITL, presupposed in ITG grammars, allows us to relate various entities in an idiolect system to speech objects or events. Most important in this respect is a concept of ‘normal utterance’ that is used in ITL to relate sentences of an idiolect system – at various levels of abstraction – to such objects or events.

Employing the new constants in G´, sentences of G´ (‘grammar sentences’) are formulated stating that certain speech objects or events by certain speakers are normal utterances of certain sentences of certain idiolect systems of D. The grammar sentences are now added in G´ to the axioms of G as new axioms (whose number may be quite large).

Consider one of the new axioms, and let V* be the name of the speech object or event V of which the axiom states that it is a normal utterance. The theory of language ITL that is presupposed in G and G´ will contain assumptions on the form and meaning of normal utterances of sentences of idiolect systems, assumptions that are available in G´. Using these assumptions plus the new axiom with V*, we derive a theorem by which V must have certain formal and semantic properties. If it does, the grammar sentences of G´ from which the theorem is derived are jointly confirmed to a certain extent; if it does not, at least one of these sentences is disconfirmed and may have to be revised or rejected, giving rise to a change – a modification – in G´ and, possibly, in G. (For details and an example, see Lieb 2018: Secs. 11.1 to 11.4.)  

10.  Grammar integration

A grammar G may presuppose a theory of language, which is then available in the basic (non-axiomatic) language of the grammar; an applied grammar G´ based on G is an extension of G. Presupposition and extension are two ways in which grammars may be integrated with other theories, both grammars and non-grammars.

Theory integration is characterized in Lieb (1983b: Ch. 29), and more briefly, in Lieb (2018: Sec. 11.4). Different kinds of theory integration are distinguished that allow us to combine grammars with grammars, with other linguistic theories, and even with non-linguistic ones; in addition, it is shown how different parts of grammars, such as the phonetic-phonological, the syntactic, and the semantic part, can be combined.

Integration in its various forms applies once theories fulfil at least some of the requirements satisfied by axiomatic theories. This is another argument in favour of adopting an axiomatic format for formal grammars. Axiomatic grammars can also be used as a counterfoil in analysing informal ones, which on the other hand provide a practical yardstick for evaluating formal grammars.


IL references

(Whenever possible, titles are quoted as they appear in the Bibliography that is part of this Homepage and should also be consulted for a complete view of the work done in Integrational Linguistics. In addition, you may check the Lists of References for other explanatory texts that are part of the Homepage, in particular, the List of References in “Integrational Linguistics: Development and topicality”.)  

Falkenberg, Thomas. 1996a. Grammatiken als empirische axiomatische Theorien. Tübingen: Niemeyer. (= Linguistische Arbeiten 346).

Lieb, Hans-Heinrich. 1974. "Grammars as theories: The case for axiomatic grammar (Part I)". Theoretical Linguistics 1, 39–115.

Lieb, Hans-Heinrich. 1976c. "Grammars as theories: The case for axiomatic grammar (Part II)". Theoretical Linguistics 3. 1–98.

Lieb, Hans-Heinrich. 1983b. Integrational Linguistics. Vol. I.: General Outline. (Current Issues in Linguistic Theory, 17). Amsterdam; Philadelphia: Benjamins.

Lieb, Hans-Heinrich. 1987. "Sprache und Intentionalität: der Zusammenbruch des Kognitivismus". In: Rainer Wimmer (ed.). Sprachtheorie: Der Sprachbegriff in Wissenschaft und Alltag. Jahrbuch 1986 des Instituts für deutsche Sprache. 76 (Sprache der Gegenwart 71.) Düsseldorf: Schwann, 11–76.

Lieb, Hans-Heinrich. 1993g. Linguistic variables: Towards a unified theory of linguistic variation. (Current Issues in Linguistic Theory, 108.) Amsterdam; Philadelphia: Benjamins.

Lieb, Hans-Heinrich. 1998b. "Morph, Wort, Silbe: Umrisse einer Integrativen Phonologie des Deutschen". In: Matthias Butt and Nanna Fuhrhop (eds). Variation und Stabilität in der Wortstruktur: Untersuchungen zu Entwicklung, Erwerb und Varietäten des Deutschen und anderer Sprachen . Hildesheim etc.: Olms. (= Germanistische Linguistik 141–142). 334–407. [Published in 1999]. 

Lieb, Hans-Heinrich. 2008. "The case for two-level phonology: German Obstruent Tensing and Nasal Alternation in French". In: Robin Sackmann (ed.). 2008b. 21–96. 

Lieb, Hans-Heinrich. 2013. Towards a general theory of word formation: the Process Model. Berlin: Freie Universität Ber lin. (An open access publication.). 101 pp.

Lieb, Hans-Heinrich. 2018. “Describing linguistic objects in a realist way.” In: Christina Behme and Martin Neef (eds.), Essays on linguistic realism. Amsterdam: Benjamins. (= Foundations of Language Companion Series 196). 19-138.

Lieb, Hans-Heinrich (ed.). 2017. Linguistic research in progress: Proceedings of the Berlin Research Colloquium on Integrational Linguistics 1992 – 2003 (Parts I to XXII) / Berliner Forschungskolloquium Integrative Sprachwissenschaft 1992-2003. Protokolle (Teil I bis XXII). Berlin: Freie Universität Berlin. [Ca. 2000 pp.] 

Nolda, Andreas. 2018. “Explaining linguistic facts in a realist theory of word formation”. In: Christina Behme and Martin Neef (eds.), Essays on linguistic realism. Amsterdam: Benjamins. 203-234.

Sackmann, Robin (ed.) 2008b. Explorations in Integrational Linguistics: four essays on German, French, and Guaraní. (Studies in Integrational Linguistics, 1). Amsterdam; Philadelphia: Benjamins. (= Current Issues in Linguistic Theory 285).

Other references

Carnap, Rudolf. 1958/2012. Introduction to symbolic logic and its applications. New York: Dover Publications. [Reprinted in 2012 by Courier Corp., North Chelmsford, MA.]

Haspelmath, Martin. 2010. “Comparative concepts and descriptive categories in crosslinguistic studies”. Language 86 (3), 663-687.

Haspelmath, Martin. to appear. "How comparative concepts and descriptive linguistic categories are different.” (revised version, January 2018). In: Brisard, Frank; Mortelmans, Tanja; and Daniel van Olmen (eds.), Festschrift for Johan Van der Auwera. Berlin and New York:  de Gruyter. [zenodo preprint]

Lehmann, Christian. 2018. “Linguistic concepts and categories in language description and comparison”. In: Marina Chini and Pierluigi Cuzzolin (eds.). Typology, acquisition, gramma grammaticalization studies. Milano: Franco Angeli. (in press) tion studies. Milano: Franco Angeli. (in press)

Müller, Stefan. 2018. Grammatical theory: From transformational grammar to constraint-based approaches. (Textbooks in Language Sciences 1). 2nd, rev. and ext. edition. Berlin: Language Science Press.