MontagueGrammar

Montague Grammar Amy H. Kao EECS 595 Fall 2004 “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 within a single natural and mathematically precise theory” (Montague 1970) 2 Abstract In 1970, Richard Montague’s radical theories of how formal logic could be used to describe natural language changed the way logicians and linguists perceived the connections between their fields. Using formal logic and a model-theoretic view, Montague creates a system where the syntactic structure and semantic structure of natural language are connected in a manner that allows for a better understanding of the semantic meanings of sentences. These theories are exemplified in what has been termed Montague Grammar. In this paper, we gain a better understanding of Montague Grammar’s impact on logicians and linguists by understanding the many components of Montague Grammar, including syntactic and semantic handling, intensional logic, general quantification, and model theory. History Richard Montague (1930 - 1971), both a logician and philosopher of language, was a student of Alfred Tarski, a logician often compared with the likes of Aristotle and Gödel. In 1993, Tarski’s paper The concept of truth in formalized language began the development of model theory, which provided the foundation for much of Montague’s work in semantics. Model theory describes the meanings of formal and natural languages by defining the classes of objects referred to by the expressions. In this manner, model theory examines the truth of an expression (using Tarski’s Truth definitions) by examining the truth of the expression’s corresponding class elements. Tarski’s influence may be seen in much of Montague’s approaches semantics, allowing for a logical model theoretic approach that was radical for Montague’s time. In Universal Grammar (1970), Montague introduced his theory of formal syntax and semantics as applied to both formal and natural language. Universal Grammar is significant because it was the first attempt at applying formal semantics to natural language. Logicians prior to Montague regarded natural language as too ambiguous and unstructured for formal logical analysis while linguists felt that formal languages were unable to capture the structures of natural languages. Montague more explicitly argues for the similarities of natural and formal language in English as a formal language (1970) where he writes “I reject the contention that an important theoretical difference exists between formal and natural languages.” Montague demonstrates the application of his theories in The Proper Treatment of Quantification in Ordinary English (1973), most commonly referred to as PTQ, where he defines the syntax and semantics for a large fragment of English. In the text, English phrases are translated into logical expressions based on intensional logic, which are then interpreted with Tarski’s model theory. The term Montague grammar generally refers to the theories outlined in Universal Grammar, English as a formal language, and PTQ, but because so much of Montague’s work is explained in PTQ, Montague grammar is often also referred to as PTQ. 3 Montague’s work came at a time when linguists were developing different approaches to semantics in relation to Chomsky’s generative grammar. The two main approaches, generative semantics (Lakoff, Ross, McCawley, Postal) and interpretive semantics (Jackendoff, Chomsky), were at odds with each other. Generative semantics did not separate semantic and syntactic rules while interpretive semantics contained a distinction between the two types of rules. Montague’s approach offered a perspective of how the structures of semantic and syntactic rules can be connected to each other but not required to be the same rules. Linguists hoped that Montague’s work would merge “the best aspects of both of the warring approaches, with some added advantages of its own” (Partee 2001). Montague was not the only linguist aware of the relevant advances in logic and philosophy of language but before Montague’s work became popular, linguist and logicians did not actively work together. Many attribute the lack of communication between the fields to the personalities of generative semanticists. The clash of these two fields was demonstrated in August of 1969 during a colloquium of logicians and linguists. The colloquium was aimed at fostering collaborations between the two fields but was deemed a failure by W.V. Quine, who declared it “a fiasco at bridge building” (Abbott, 1999). Though unsuccessful in generating new interactions between people, the papers from the colloquium, including articles by Montague, were read by many linguists and logicians. These articles were key in setting the grounds for future collaborations between the two fields. One of the important issues brought to the attention of linguists at the 1969 colloquium was the differentiation between semantic representation and semantic meaning. Philosopher David Lewis condemned the existing systems of semantic representation because they lacked the treatment of truth conditions. At the time, expressions were commonly given a corresponding marking to denote the semantic meaning. Lewis claimed that these markings simply translated sentences into an artificial language one could call Semantic Markerses without regard to the meaning of the sentence. Montague’s work in the years following that colloquium could be deemed an answer to Lewis’s criticisms with semantic analysis. His model theoretic analysis of semantics in natural language provided a treatment of meanings that Lewis had found lacking in semantic representations. Montague’s work in natural language is described in only three publications (Universal Grammar, English as a formal language, and The Proper Treatment of Quantification in Ordinary English) but most found his writing to be “highly formal and condensed, very difficult for ordinary humans (even logicians!) to read with comprehension.” (Abbott 1999) The proliferation of Montague’s work is therefore often attributed to Barbara Partee who presented Montague’s work in a more understandable fashion. Partee graduated MIT (1965) with a PhD in Linguistics under Chomsky and became familiar with Montague’s work when she began teaching at UCLA (1965-1972) where he was a professor. Partee’s Montague Grammar and Transformational Grammar 4 (Linguistic Inquiry 1975) was seen as the first introductory text on Montague Grammar. Dowty, Wall & Peters later published another comprehensive text called “Word Meaning and Montague Grammar” (1981). When Partee became a faculty member at the University of Massachusetts at Amherst, her classes on Montague Grammar became an attraction to many prominent linguists and logicians. Montague’s groundbreaking work in formal semantics created new areas of research that were of interest to both linguists and logicians, helping to proliferate interactions between the fields. Barbara Partee’s joint appointment in the linguistics and philosophy departments at University of Massachusetts is one of many signs that linguists and logicians are collaborating more closely. In 1977, the journal Linguistics and Philosophy published its first issue and to this day, continues to foster communications between the two communities. Montague Grammar Montague’s Universal Grammar (UG) is a general theory of language developed to encompass the syntax and semantics of known artificial languages as well as natural languages and “unnatural” languages. UG relates syntax and semantics by creating a formal interpretation of Frege’s philosophy that an expression’s meaning is a function of the meaning of its constituents and its syntax. The characteristics of UG are meant to be more general so as to serve as the reference framework for comparing formal and natural languages. The theories of Universal Grammar are applied in Montague Grammar as described in The Proper Treatment of Quantification in Ordinary English (PTQ). It is worth noting that the terms Universal Grammar (UG), Montague Grammar, and PTQ may be found interchangeably in various texts regarding Montague’s work. In general, UG refers to Montague’s theories of syntax and semantics found in Universal Grammar whereas PTQ is commonly used to denote the application and process of the theory. The term Montague Grammar is more general and is commonly used to refer to anything within Montague’s three main texts concerning syntax and semantics. The following sections of this paper will examine syntax and semantics as handled in Montague Grammar. Syntax Syntax in Montague grammar consists of syntactic rules and syntactic operations as described in Universal Grammar. The rules are composed of syntactic categories defined as basic or recursive clauses while syntactic operations are functions such as concatenation that define how categories form new phrases. These rules and operations may be applied to a lexicon to define the syntax of a ‘fragment’ of English. Syntactic categories are based upon two primitive categories: sentence and entity categories as denoted by t and e respectively. The t category does not contain lexical items. It is composed of sentences built by recursive rules. The label t represents the fact that all members of the category contain a truth-value. The primitive category e does not 5 actually contain any entities. Its usage lies not in categorization of phrases but in representing semantic information. For example, phrases categorized as t/e implies the usage of a function that takes the senses of entities (e) into truth-values (t). Using the primitive categories t and e, an indefinite number of categories may be constructed in the form X/Y where X and Y are categories and Y is a phrase that may be used to create an X phrase. To represent the fact that more than syntactic category may be represented by the same category type, the syntactic categories may be further divided by added more slashes. For example, two distinct categories that use an e phrase to create a t phrase may be represented by t/e and t//e. There is no limit to the number of slashes allowed but the original grammar defined in PTQ only used double slashes. Category Abbreviation PTQ Name Nearest linguistic equivalent t (primitive) Truth-value expression; or declarative sentence Sentence e (primitive) Entity expression; or individual expression (noun phrase) t/e IV Intransitive verb phrase transitive verb, transitive verb and its object, or other verb phrases t/IV T Term Noun phrase IV/T TV Transitive verb phrase Transitive verb IV/IV IAV IV-modifying adverb VP-adverb and prepositional phrases containing in and about. t//e CN Common noun phrase Noun or NOM t/t None Sentence-modifying adverb Sentence-modifying adverb IAV/T None IAV-making preposition Locative, etc., preposition IV/t None Sentence-taking verb phrase V which takes that-COMP IV/IV None IV-taking verb phrase V which takes infinitive COMP Table 1: Syntactic Categories (Partee 1976) For simplification, Montague created the abbreviations IV, T, TV, IAV, and CN for the first five derived categories. Without abbreviations, category labels would grow increasingly hard to read. For example, TV would need to be denoted as (t/e)/(t/(t/e)). These abbreviated categories contain both lexical and derived phrases while the last four unabbreviated categories (t/t, IAV/T, IV/t, and IV/IV) contain only lexical members. For large lexicons, it may be necessary to define further syntactic categories. 6 Rules Every non-primitive syntactic category in Montague grammar contains lexical rules and recursive rules. Lexical rules simply state the category of a lexical phrase. To generate a new grammar, one first categorizes the lexical terms in terms of syntactic categories. Each resulting set of lexical members is called a ‘basic expression’ of the category. Below is an example ‘fragment’ of English from Partee 1973 where BA represents ‘basic expression of category A’. The term ‘fragment’ refers to a subset of a language. When working with Montague grammar, one works with ‘fragments’ because the grammar was not intended to define syntax and semantics a whole language. Note that the example does not contain common categories such as determiners. This is because our example is only working with a very small ‘fragment’ of English. Fragments that require determiners would have a category for them. The ability to define new syntactic categories allows for extending the category set to accommodate larger and more complex ‘fragments’ of English. 1) BIV = {run, walk, talk, rise, change} 2) BT = {John, Mary, Bill, ninety, he0, he1, he2, …} 3) BTV = {find lose, eat, love, date, be, seek, conceive} 4) BIAV = {rapidly, slowly, voluntarily, allegedly} 5) BCN = {man, woman, park, fish, pen, unicorn, price, temperature} 6) Bt/t = {necessarily} 7) BIAV/T= {in, about} 8) BIV/t = {believe that, assert that} 9) BIV/IV= {try to , wish to} Figure 1: Example Expressions for a Fragment Every complex syntactic category of form X/Y has a corresponding recursive syntactic rule. The rule defines a function Fi(x,y) that creates a phrase in X from a phrase in X/Y and a phrase in Y. Abbreviations IV and CN represent categories t/e and t//e respectively. The category e contains no phrases, so a function on IV or CN would have no effect on the phrases. Therefore, IV and CN do not follow the recursive rule shown in Equation 1. If   X/Y and   Y then Fi(,)  X Equation 1: Syntactic Rule Most recursive rules are simply concatenations such that Fi(, ) =   but they may be as complicated as required. The following example rule for TV (from Thomason p.81) defines a more complex rule that checks if  is a TV or TV/T in order to determine the accurate position of . The rule also transforms hei to himi if  is hei. Note that the 7 Mary loves him, t, F1(love him, Mary) love him, IV, F3(love, he) Mary, T love, TV he, T subscript 3 from F3(, ) refers to the expression number of its corresponding category, BTV , from Figure 1. F3(, ) = If first word of  is a TV:   if  is not a variable  himi if  is hei If  is 1 2 where 1 is a TV/T: 1  2 if  is not a variable  himi 2 if  is hei Figure 2: Example Syntactic Rule Examples: F3(shave, a fish) = shave a fish F3(seek, he1) = seek him1 F3(read a large book, Mary) = read Mary a large book Figure 3: Examples of Rule F3 Syntactic recursive and lexical rules may be applied to generate basic expressions and sentences. A tree created from the rules display how the syntactic rules generate a sentence. Such a tree, known as an analysis trees, illustrates the syntactic structure of a sentence. The following analysis tree uses the lexicon from Figure 1 to generate the phrase Mary loves him from the lexical terms love he Mary. Note that the example function F3 is used to generate love him from love and he. Figure 4: Analysis Tree for "Mary loves him" 8 Semantics In PTQ, semantic interpretations are derived from the formal intentional logic translation of sentences. The process begins by translating sentences into a formal intentional logic that specifies a one-to-one correlation between syntactic and semantic rules. The formulas generated may then be analyzed to determine the sentence’s truth with respect to a given model. Intensional Logic and Types An intension is a function that generates extensions from a set of inputs. Extensions refer to things in one of the following three categories: the truth-value of a sentence, the thing named by a name, or the set of objects a common noun applies to. Inputs, known as an index, to an intension consist of variables about the context of the expression. Extension Category Appropriate Intension Function Truth-value of a sentence From indices to truth-values Thing named by a name From indices to things Set of objects a common noun or intransitive verb phrase applies to From indices to sets Table 2: Extension Categories and Intension Functions To understand extensions more thoroughly, we first examine each extension category. The first category defines extensions of names where an extension simply refers to the object being named. In this category, the extension of the name John would be the individual named John. In the next category, an extension refers to the set of objects represented by a common noun phrase or intransitive verb phrase. For example, the extension of common noun student is the set of all students whereas the extension of the intransitive verb phrase reads quickly is the set of all individuals who read quickly. The last category makes use of extensions of names and common nouns to generate extensions of sentences as a truth-value. The truth-value of a sentence refers to whether or not it is true in a particular possible world or worlds identified. In this context, the truth-value or extension is based upon the extensions within the sentence. A simple example would be John reads quickly where the sentence’s extension is based upon the extensions of John and reads quickly. This concept of the meaning of a sentence as a function of the meanings of its constituents is based upon Frege’s functionality principle. One of the problems with extensions is that alone, they are not able to accurately analyze sentences with intensional contexts. An example is the sentence Necessarily the University of Michigan is identical with the University of Michigan where “necessarily” is understood to mean “in all possible worlds”. The extension of this sentence is its truth-value, which is true. Replacing the second occurrence of University of Michigan with Columbia University would replace a constituent with a new 9 constituent containing the same extension but the resulting new sentence would no longer contain the same extension. Its truth-value would not be true because there exists a possible world in which the University of Michigan and Columbia University are not identical. A more complicated example of the failings of extensions is the phrase classmate of John where classmate is defined as a student at the same institution and level of study. If John is currently a graduate student at the University of Michigan, then the phrase has the same extension as the phrase graduate student at the University of Michigan. We now examine the compound phrase former classmate of John whose extension is the set of individuals who have at some point been at the same institution and level of study as John but are currently not a graduate student at the University of Michigan. If we replace classmate of John with the phrase graduate student at the University of Michigan, the resulting phrase is former graduate student at the University of Michigan. The extension of this new phrase is the set of individuals who were once a graduate student at the University of Michigan but currently are not. Though we replaced a phrase with a new phrase of the same extension, the resulting compound phrase does not contain the same extension as the original compound phrase. Because of these problems with extension, Frege suggested that in oblique contexts, an expression’s extension is the intension of the expression. The logic for determining intensions (or the sense) of an expression was partially developed by Church (1951) and based on Carnap’s suggestion that an intension is a function of the expression an