Montague semantics is a theory of natural language semantics and of its relation with syntax. It was originally developed by the logician Richard Montague (1930–1971) and subsequently modified and extended by linguists, philosophers, and logicians. The most important features of the theory are its use of model theoretic semantics which is nowadays commonly used for the semantics of logical languages and its adherence to the principle of compositionality—that is, the meaning of the whole is a function of the meanings of its parts and their mode of syntactic combination. This entry presents the origins of Montague Semantics, summarizes important aspects of the classical theory, and sketches more recent developments. We conclude with a small example, which illustrates some modern features.
- 1. Introduction
- 2. Components of Montague Semantics
- 3. Philosophical Aspects
- 4. Concluding Remarks
- Bibliography
- Academic Tools
- Other Internet Resources
- Related Entries
1. Introduction
1.1 Background
Montague semantics is the approach to the semantics of natural language introduced by Richard Montague in the 1970s. He described the aim of his enterprise as follows:The basic aim of semantics is to characterize the notion of a true sentence (under a given interpretation) and of entailment (Montague 1970c, 223 fn).The salient points of Montague's approach are a model theoretic semantics, a systematic relation between syntax and semantics, and a fully explicit description of a fragment of natural language. His approach constituted a revolution: after the Chomskyan revolution that brought mathematical methods into syntax, now such methods were introduced in semantics.
Montague's approach became influential, as many authors began to work in his framework and conferences were devoted to ‘Montague grammar’. Later on, certain aspects of his approach were adapted or changed, became generally accepted or were entirely abandoned. Nowadays not many authors would describe their own work as ‘Montague semantics’ given the many differences that have taken shape in semantics since Montague's own work, but his ideas have left important traces, and changed the semantic landscape forever. In our presentation of Montague semantics the focus will be on these developments.
Richard Montague was a mathematical logician who had specialized in set theory and modal logic. His views on natural language must be understood with his mathematical background in mind. Montague held the view that natural language was a formal language very much in the same sense as predicate logic was a formal language. As such, in Montague's view, the study of natural language belonged to mathematics, and not to psychology (Thomason 1974, 2). Montague formulated his views:
There is in my opinion no important theoretical difference between natural languages and the artificial languages of logicians; indeed I consider it possible to comprehend the syntax and semantics of both kinds of languages with a single natural and mathematically precise theory. (Montague 1970c, 222)
Sometimes only the first part of the quote is recalled, and that might raise the question whether he did not notice the great differences: for instance that natural languages develop without an a priori set of rules whereas artificial languages have an explicit syntax and are designed for a special purpose. But the quote as a whole expresses clearly what Montague meant by ‘no important theoretical difference’; the ‘single natural and mathematically precise theory’ which he aimed at, is presented in his paper ‘Universal Grammar’ (Montague 1970c). He became most well-known after the appearance of Montague 1973, in which the theory is applied to some phenomena which were discussed intensively in the philosophical literature of those days.
Montague's interest in the field arose while teaching introductory logic courses. Standard in such courses are exercises in which one is asked to translate natural language sentences into logic. To answer such exercises required a bilingual individual, understanding both the natural language and the logic. Montague provided, for the first time in history, a mechanical method to obtain these logical translations. About this, Montague said:
It should be emphasized that this is not a matter of vague intuition, as in elementary logic courses, but an assertion to which we have assigned exact significance. (Montague 1973, 266)
We next describe the basic ideas of Montague semantics. Section 2 presents several components of Montague semantics in more detail. Section 3 includes a discussion of philosophically interesting aspects, and Section 4 provides a detailed example and further reading.
1.2 Basic Aspects
To implement his objective, Montague applied the method which is standard for logical languages: model theoretic semantics. This means that, using constructions from set theory, a model is defined, and that natural language expressions are interpreted as elements (or sets, or functions) in this universe. Such a model should not be conceived of as a model of reality. On the one hand the model gives more than reality: natural language does not only speak about past, present and future of the real world, but also about situations that might be the case, or are imaginary, or cannot be the case at all. On the other hand, however, the model offers less: it merely specifies reality as conceived by language. An example: we speak about mass nouns such as water as if every part of water is water again, so as if it has no minimal parts, which physically is not correct. For more information on natural language metaphysics, see Bach 1986b.Montague semantics is not interested in a particular situation (e.g. the real world) but in semantical properties of language. When formalizing such properties, reference to a class of models has to be made, and therefore the interpretation of a language will be defined with respect to a set of (suitable) models. For example, in the introduction we mentioned that the characterization of entailment was a basic goal of semantics. That notion is defined as follows. Sentence A entails sentence B if in all models in which the interpretation of A is true, also the interpretation of B is true. Likewise a tautology is true in all models, and a contradiction is true in no model.
An essential feature of Montague semantics is the systematic relation between syntax and semantics. This relation is described by the Principle of Compositionality which reads, in a formulation that is standard nowadays:
The meaning of a compound expression is a function of the meanings of its parts and of the way they are syntactically combined. (Partee 1984, 281)
An example. Suppose that the meaning of walk, or sing is (for each model in the class) defined as the set of individuals who share respectively the property of walking or the property of singing. By appealing to the principle of compositionality, if there is a rule that combines these two expressions to the verb phrase walk and sing, there must be a corresponding rule that determines the meaning of that verb phrase. In this case, the resulting meaning will be the intersection of the two sets. Consequently, in all models the meaning of walk and sing is a subset of the meaning of walk. Furthermore we have a rule that combines the noun phrase John with a verb phrase. The resulting sentence John walks and sings means that John is an element of the set denoted by the verb phrase. Note that in any model in which John is element of the intersection of walkers and singers, he is an element of the set of walkers. So John walks and sings entails John walks.
An important consequence of the principle of compositionality is that all the parts which play a role in the syntactic composition of a sentence, must also have a meaning. And furthermore, each syntactic rule must be accompanied by a semantic rule which says how the meaning of the compound is obtained. Thus the meaning of an expression is determined by the way in which the expression is formed, and as such the derivational history plays a role in determining the meaning. For further discussion, see Section 2.5.
The formulation of the aim of Montague semantics mentioned in the introduction (‘to characterize truth and entailment of sentences’) suggests that the method is restricted to declarative sentences. But this is need not be the case. In Montague 1973 (248 fn) we already find suggestions for how to deal with imperatives and questions. Hamblin (1973) and Karttunen (1977) have given a semantics for questions by considering them with a meaning that is based upon sentences (viz. sets of propositions). Groenendijk and Stokhof (1989) consider questions as expressions with meanings of their own nature (namely partitions).
Since Montague only considered sentences in isolation, certain commentators pointed out that the sentence boundary was a serious limitation for the approach. But what about discourse? An obvious requirement is that the sentences from a discourse are interpreted one by one. How then to treat co-referentiality of anaphora over sentence boundaries? The solution which was proposed first, was Discourse Representation Theory (Kamp 1981). On the one hand that was an offspring of Montague's approach because it used model theoretic semantics, on the other hand it was a deviation because (discourse) representations were an essential ingredient. Nowadays there are several reformulations of DRT that fit into Montague's framework (see van Eijck and Kamp 1997). A later solution was based upon a change of the logic; dynamic Montague semantics was developed and that gave a procedure for binding free variables in logic which has an effect on subsequent formulas (Groenendijk and Stokhof 1991). Hence the sentence boundary is not a fundamental obstacle for Montague semantics.
