"There is, in my opinion, no important theoretical difference between natural languages and the artificial languages of logicians; indeed I consider it possible to integrate the syntax and semantics of both kinds of languages within a simple, natural, mathematically precise theory."(Richard Montague)
Montague's Grammar
Richard Montague, American logician, mathematician, and philosopher of language, was a disciple of Alfred Tarsky −one of the great logicians, sometimes compared to Aristotle, Frege, and Gödel− and professor of logic at UCLA.
Montague's grammar is a universal theory of language, in its two aspects of natural (or ordinary) language and formal (or logical) language. Montague was convinced that natural languages and formal languages shared the same principles and could therefore be formalized by a mathematical theory integrating the syntax and semantics of both types of languages.
Montague believed that these common principles of natural and artificial languages could be described by a universal grammar. In the late 1960s, Montague undertook the project of creating such a grammar: a syntactic-semantic-pragmatic system for describing expressions of natural and artificial languages. The result was later called "Montague grammar" or "Montague semantics", since Montague's main goal was the formalization of semantics.
Montague's grammar was a milestone that changed the view of the relationship between formal logic and natural language. Before Montague, logicians considered natural language too ambiguous and unstructured for formal logical analysis. And linguists believed that formal languages were too limited and rigid to capture the structures of natural languages. But a large segment of linguists sought an integration of the cognitive and formal perspectives of natural language. It was in this sense that Montague's work had an impact, since his system allowed syntactic (formal) and semantic (conceptual) structures to be closely connected. The idea that a natural language (such as English) could be formalized by means of logical techniques was very groundbreaking at the time.
Montague, in creating his grammar, had a philosophical motivation: the philosophical task is precisely the investigation of the logical-semantic structure of natural language. Montague's project can be considered as an attempt to create a formal philosophy.
For the development of his project, Montague integrated 5 tools to create the first "mathematically precise" semantic theory of natural language:
Set theory.
Categorial grammar.
A categorial grammar is a formalism for defining the syntax of natural language. It consists of a set of basic syntactic categories (associated with signs) and a formalism for combining those basic categories to define derived (also called "functional") categories. Normally, the formalism used are syntactic rules.
Church's lambda calculus.
The lambda calculus (or lambda abstraction) was created by Alonzo Church (1932-1933) as a general theoretical framework for functional expressions. A functional expression is a function defined from other functions. Kleene and Rosser (1936) showed that the lambda calculus was inconsistent. Church then developed (1940) a functional theory of explicit types − a simple, elegant and precise theory−, based on or inspired by Russell's type theory. Church's original lambda calculus had no types. Church's lambda calculus is the foundation of functional programming.
In Church's type theory, functional expressions are classified into types, and types restrict the combinatorial forms of those expressions. Types are categories of functions and play a role analogous to that of set types in set theory.
Intensional logic.
Intensional logic deals with the dual nature of all expression: the deep level of meaning (sense or intension) and the surface level of denotation (reference or extension), as well as the relations between the two. The intension (or sense) of an expression is a function of the expression and its "possible state of affairs" or context. In first-order predicate logic, intension plays no role; everything is superficial.
Although Frege provided a first approximation to the subject (through the concepts of "sense" and "reference"), it was Church who formalized it intensional logic in 1951.
Montague merged intensional logic and Church's type theory to create his own version of intensional logic, in which the possible state of affairs was represented by the context of the expression, defined as an "index" [see Addenda].
The version of intensional logic created by Montague has three salient features:
It allows higher-order logical expressions. In a first-order predicate logic, only variables for individual entities are allowed. In higher-order logic, properties can be variables.
Consider functions as generic expressions, where parameters play a role equivalent to quantifiers. It is precisely the operator that makes it possible to differentiate between the particular and the generic. If &lamda; appears in a formula, it is a generic expression (the intent). If it does not appear, we are dealing with a particular formula (the extension).
A theory of higher-order intensional types, based on Church's theory of types.
Tarski's model theory.
This theory was presented in his famous (and lengthy) 1933 paper "The Concept of Truth in Formalized Languages". In it he describes the meanings of formal and natural languages by defining classes of objects associated with linguistic expressions. Montague extended this theory so that it could be applied to formal semantics.
The essential ideas of Montague's grammar appeared in three seminal papers:
"Universal Grammar" (1970). Here he presents his theory of syntax and formal semantics applied to natural and artificial languages. It was the first attempt to formalize the semantics of natural language.
"English as a Formal Language" (1970). Here he makes more explicit the similarities between the two types of languages and where he argues that there is no important theoretical difference between the two.
"The Proper Treatment of Quantification in Ordinary English" (1973). This article is considered the most important, the one that had the greatest impact, and is often referred to in abbreviated form as "PTQ". In this text he demonstrates the application of his theory to a "fragment" (subset) of English. English sentences are translated into logical expressions (of intensional logic) which in turn are interpreted with Tarski's model theory, i.e. by assigning classes to the expressions of intensional logic. The formalization of the syntax and semantics of English did this by means of an intermediate intensional logic-formal language called L0, a combination of intensional logic of semantic types and lambda abstraction.
Foundations of Montague grammar
Montague grammar is rather cryptic and confusing. However, we can distinguish the following aspects:
A syntactic theory of language.
It is a theory aimed at describing the possible formal expressions of a language. For this purpose, it uses the concept of categorial grammar. In the case of Montague's grammar, an algebra is used to define new syntactic categories by combining some basic syntactic categories.
A semantic theory of language.
Aimed at describing the essential concepts of formal semantics (or meaning). The semantic description is carried out in the same way as the syntactic description of formal-logical languages, but using, not a categorial grammar, but an intensional logic of types (types are the semantic equivalents of syntactic categories). The formulas of intensional logic constitute the interpretative semantic structures of natural language. An algebra is also used as a combinatorial algebra of semantic types.
The syntax-semantics connection.
It is realized by means of the following correspondences, which make it possible to establish a close relationship between form and meaning:
Syntax is an algebra, semantics is an algebra, and there is a correspondence (homomorphism) that relates the elements of the syntactic algebra and the elements of the semantic algebra.
To each syntactic category corresponds a type (semantic category) of intensional logic.
To each syntactic rule corresponds a semantic rule (of meaning or interpretation). In the case of single-argument rules, it may happen that the semantic rule is equal to the syntactic rule.
The principle of compositionality.
This is the most important principle of Montague grammar, which is borrowed from Frege: the meaning of an expression is a function of the meanings of its component expressions and of its syntactic structure. The components of an expression always have the same meaning.
In natural languages, on the other hand, the components of expressions are not always interpreted in the same way, since they depend on the context. For example, the meaning of "Mary is the tallest girl here" depends on the context (specifically the meaning of "here").
Frege's strategy was "semantic glue": the application of functions to arguments to combine meanings and create new ones. Montague interprets compositionality as a mechanism (homomorphism) that relates syntactic and semantic structures. It is precisely this homomorphism between syntax and semantics that formalizes compositionality.
Generalized quantifiers.
Before Montague, natural language noun phrases had a compositionality problem. Montague solves this problem by means of generalized quantifiers, which are implemented by formulas from Church's lambda calculus.
A generalized quantifier denotes a set of elements or properties associated with a sentence (noun phrase) and expresses an intent or meaning. Examples:
"John" is the set of properties contained in the individual element "John". That is, a proper noun does not denote an individual or a basic syntactic category, but a set of properties.
"Every man" is the set of properties that every man has.
"Walking" is the set of individuals who share the property of walking. The same could be said of other properties such as "singing", "whistling", "jumping", etc.
"Walking and singing" is the intersection of the sets corresponding to "walking" and "singing".
"John walks and sings" means that John is an element of the set denoted by the verb phrase "walk and sing".
With generalized quantifiers you can specify concrete quantifiers such as: the man, a man, every man, no man, most men, not all men, etc., something impossible to do with the tools of predicate logic. It also allows specifying transitive intensional verbs, conjunction of phrases, adverbial modification, etc.
Generalized quantifiers have been compared to the concept of "continuation" in programming languages because successive generalizations are chained together.
The theoretical model.
The theoretical model makes it possible to model all kinds of situations (real or imaginary) in linguistic terms. The formalization of semantics refers to a class of models. For example:
The meaning of "walking and singing" is the intersection of the subsets of the meanings of "walking" and "singing" in all models.
The definition of "entailment" is: the sentence A entails the sentence B if in all models in which the interpretation of A is true, also the interpretation of B is true.
A statement is a tautology if it is true in all models.
A statement is a contradiction if it is not true in any model.
The meaning.
First, semantic values are assigned to natural language expressions. This assignment of semantic values is made on the basis of their syntactic structure, by virtue of the homomorphism defined between syntactic structure and semantic structure.
Secondly, once the semantic expression (an intensional language formula) has been obtained, its meaning is a "denotation" (an extensive or intensive expression), which is a function of the set of possible worlds and the set of contexts of use.
The meaning g of a sentence is always a function. It is formally defined as a function of two arguments: 1) a possible world i; 2) a context j. The result is a denotation d, which is a truth value (true or false). This function is called "intension".
g: I×J → D
(I is the set of possible worlds, J is the set of use contexts, D is the set of denotations)
Meaning postulates.
Montague's strategy in analyzing a sentence was to apply to all expressions the most general approach possible and to progressively reduce or particularize it by means of "postulates of meaning." Montague elaborated many postulates of meaning. Today they are mainly used to express relations between words and their meanings and to express structural properties of models (e.g., the structure of the time axis).
Representations of noun phrases in lambda abstraction:
"John" indicates or signifies all properties of John (a universal quantifier). It is represented as λP[P(John)], where P is variable over properties. Its meaning is a function that has as input a particular property and produces as result T if John has that property and F if he does not. This is called the "characteristic function" of the set of properties of John. For example, this function applied to the argument (property) "sings", i.e., λP[P(John)](sings), produces T or F, depending on whether John sings or not.
The expressions λP[P(John)](sings) and Canta(John) are equivalent. The latter is obtained by eliminating λP from the former and substituting P for its argument (canta). This is called "lambda conversion".
"Every man" indicates all the properties that every man has. In predicate logic it is expressed as ∀x[man(x) → Q(x)], where Q indicates a property of every man. And the intensional logic expression λQ∀x[man(x) → Q(x )] is a universal quantifier and denotes a meaning, which is a function that when applied to a particular property A produces T or F, depending on whether or not every man has that property.
"Every man sings" indicates all the properties of all men who sing. Its representation is λQ∀x[man(x) → sings(x )]](sings), denoting a function that when applied to a particular property A produces T or F, depending on whether or not every man who sings has that property.
MENTAL vs. Montague's Grammar
We can distinguish the following aspects:
Tools.
Montague's grammar uses 5 tools: set theory, categorial grammar, lambda abstraction, intensional logic, and model theory.
MENTAL allows to express these tools with a unified linguistic structure and in a simpler way. A single language that allows to specify directly (and at the same time) syntax and semantics.
Union of opposites.
Montague attempted to create a universalistic system based on the conjunction of three pairs of opposites: 1) the union of natural and artificial (or logico-formal) languages; 2) the union of syntax and semantics; 3) the union of theory and practice (pragmatics). MENTAL is the integral union of opposites or duals.
Principles.
Montague intuited that all languages (natural and artificial) shared the same principles. He intuited the essential union, deep down, of all languages and attempted to create a universal grammar. He did not make these principles explicit, but they are more or less implicit in all the techniques he used.
In MENTAL, those common principles are the universal semantic primitives, which constitute a universal grammar and a universal language. MENTAL provides all the necessary resources for the syntactic-semantic formalization of languages (natural or artificial) because of its universality, expressiveness and combinatorial flexibility.
Connection between syntax and semantics.
Montague's grammar is semantic. The syntactic aspect is secondary; it is only a means, a tool to achieve the ultimate goal, which is the formalization of semantics. "I see no interest in syntax except as preliminary to semantics" [Montague, 1970].
Montague had the intuition that it was necessary to establish a relation between syntax and semantics. He did this by means of a homomorphism that made each syntactic expression correspond to a semantic expression. Montague tried to formalize semantics by relying on syntax. Indeed, semantics, by itself, cannot be formalized because semantics resides in the deep and cannot be brought to the surface to formalize it. It is necessary to connect it with the superficial, with syntax.
In the case of MENTAL, this connection is made by establishing a biunivocal correspondence between syntax and semantics (isomorphism) and not a homomorphism from syntax to semantics. In MENTAL, syntax and semantics go together; they are two sides of the same coin. This solution is the simplest. Syntactic expression (the superficial and visible) refers to semantics (the deep and invisible), and semantics manifests itself as syntax.
MENTAL can be considered a simplification and generalization of Chomsky's and Montague's grammars, since it views syntax and semantics as an indivisible unit.
The domain of linguistics.
In Montague's time, there was an argument as to whether linguistics was a branch of psychology or of mathematics. According to Chomsky, all of linguistics, including semantics, is a branch of psychology. For Montague, linguistics is a branch of mathematics, but he recognized, however, that the principle of compositionality might have a psychological component that would explain why a person can understand sentences he has never heard before.
But linguistics, mathematics, and psychology are manifestations of the primary archetypes, which allow these sciences to be grounded.
Possible worlds.
Montague intuited that the semantics of possible worlds should play an essential role in the formalization of semantics. However, his solution was somewhat diffuse and complex, for the concept of "possible world" is not clearly defined. In Montague's grammar, possible worlds are elementary objects, without internal or external structure.
In MENTAL, possible worlds are grounded in the infinite combinatorial possibilities of universal semantic primitives. A possible world is an autonomous abstract space of interrelated expressions. MENTAL is the Magna Carta of possible worlds.
Formalization of semantics.
The semantics of a language cannot be formalized. The maximum possible formalization is based on basing it on primary concepts (archetypes). From these, secondary concepts can be specified.
Montague confuses semantics with a higher level of generalization of a syntactic expression. That explains why he found a mechanical procedure of transformation from syntax to semantics.
MENTAL is oriented to what is semantics in general, and not a particular aspect of semantics such as truth.
Meaning and function.
In Montague grammar, the meaning of an expression is a function, a function in which possible worlds and contexts of use are involved, and in which the result is a truth value.
When we have a noun phrase that specifies a set of properties, we have a problem if we apply a property to that expression that is neither true nor false. For example, "the set of all men" and the property "color".
But meaning is something deep, which is related to consciousness. It is not something expressible and visible, it is neither a function nor a truth value. Meaning is associated with the primary archetypes, the archetypes of consciousness.
In the MENTAL model, meaning arises (internally) by associating semantics with all syntactic (external) form. Meaning appears in the connection between the superficial and the deep, between the extensive and the intensive, between the internal and the external. Consciousness is the general mechanism that intervenes in all meaning, connecting these dual aspects. Consciousness, as a global and universal mechanism, is the mechanism that makes all meaning possible.
The "postulates of meaning" in MENTAL are the axioms that relate the primary archetypes.
Montague confuses "meaning" with "denotation". The meaning of an expression must always be the same. Denotation is a function of environment or context. Meaning refers to the deep, to the inexpressible and invisible, to the extralinguistic. Denotation is something superficial, tangible, visible.
In MENTAL these two concepts are differentiated. There are levels of signification governed by the evaluation processes leading to the final denotation (result of the evaluation process).
Degrees of freedom.
Montague's system does not clearly state degrees of freedom or semantic dimensions. The MENTAL primitives are the syntactic-semantic dimensions and define the boundaries of the expressible and the representable.
Algebra.
In Montague's grammar there are two algebras, one for syntax and one for semantics.
The "algebra" of MENTAL is the combinatorics of the primitives themselves with each other, which is realized with the primitives themselves.
Context.
Montague grammar considers only isolated sentences (they are interpreted one at a time), so they cannot refer to other elements of discourse. MENTAL allows the interrelation between all the expressions in a context.
Compositional.
In MENTAL the principle of compositionality applies, that is, the meaning of an expression depends on its components, and where syntax and semantics go together. But denotation is a function of context.
True.
Montague said that the goal of semantics is to characterize the notion of true sentence. Montague's grammar is aimed at describing concepts associated with truth. But semantics is not concerned with truth, but with the meaning of linguistic expressions, whether or not they have any relation to reality.
In Montague's grammar, truth values are T (true) or F (false). MENTAL allows truth to be treated as a qualitative magnitude, i.e., as f*T, with f between 0 and 1.
In Montague, all tautologies have the same meaning (they are synonymous) because they have the same truth value. For example, "John is either sick or he is not sick" and "white snow is white" are expressions whose meaning is T (true). In MENTAL, meaning is not linked to truth, so tautologies can be discriminated.
Generic expressions.
Montague discovered that in generic expressions parameters play a role equivalent to quantifiers. Exactly the same as in MENTAL.
In MENTAL we can specify generic expressions to represent semantics (as conceived by Montague), instead of the lambda calculus. Examples of noun phrases:
The semantics of "John" (J) is the set of properties of John.
( J = {⟨( p ← John/p )⟩} )
The semantics of "Everyman" (H) is the set of properties of everyman.
( H = {⟨( p ← ←( x/man → x/p )→ )⟩} )
In conclusion, the expression "formalizing semantics" is contradictory because semantics belongs to the internal (or deep) world and formalizing it would be to bring it to the surface, to the external world. Semantics cannot be formalized without relying on syntax and connecting the two in a biunivocal way. MENTAL is a universal language and a universal grammar. It is simpler than Montague's complex grammar. It represents the syntax-semantics connection in a simple, natural and effective way. It allows formalizing all kinds of expressions, since it is based on the primary archetypes common to mind and nature. MENTAL simplifies and clarifies things by using only the primary archetypes, which are the deepest syntax-semantic resources available.
Addenda
On Montague's grammar
Montague's interest in semantics arose when he taught logic at UCLA and set his students exercises in translating natural language sentences into formal logic. Montague managed to devise a mechanical method for performing these conversions. This method was based on the application of a theoretical model relating syntax and semantics and interpreting natural language sentences in terms of intensions.
The term "Montague grammar" was coined by Barbara Partee, his colleague at UCLA and the person most instrumental in spreading Montague's ideas in linguistics. The term "Montague grammar" first appeared published in [Rodman, 1972].
The importance of the lambda abstraction in Montague's grammar should be emphasized for its expressive capacity for generic, intensional, functional, and higher-order predicates. This importance was reflected by Barbara Partee with the statement "Lambdas truly changed my life" [Partee, 1996].
Montague's grammar constituted a milestone in the development of the formal semantics of natural languages. After Chomsky's revolution, which applied formal (mathematical) methods to syntax, Montague's grammar was a revolution in semantics, which also applied formal methods.
The impact of the publication of PTQ (1973) was very important for semantics, as Chomsky's "Syntactic Structures" (1957) was in its day for syntax. Montague's grammar was developed at about the same time as Chomsky's generative grammar, which used an exclusively syntactic approach. It also appeared at a time when several ways of formalizing semantics were being attempted. The two most important were: interpretive semantics (Jackendoff), which distinguished between syntactic and semantic rules; and generative semantics (Lakoff, Ross, McCawley, Postal), which did not distinguish between these two types of rules. Montague's system harmonized these two approaches, since it contemplated different rules for syntax and semantics, but connected to each other.
Montague's original grammar has been the subject of detailed study by linguists, logicians, and philosophers. Several aspects of his work have already been superseded, but, after his untimely death, his work has subsequently been continued and extended by other linguists. Montague's ideas changed the landscape of formal semantics forever, and he has served as an inspiration for new areas of research in the common ground between logic and linguistics.
Montague's three seminal publications were at the time considered rather cryptic, difficult to understand. Barbara Partee presented Montague's grammar in a more comprehensible way. Her work "Montague Grammar" [1976] is considered the introductory text to Montague's grammar. Another introductory text is that of [Dowty et al., 1981].
In [Thomason, 1979] all of Montague's essays on natural language semantics are collected. The most relevant essays are included in [Thomason, 1977].
Index
The meaning of some natural language expressions by themselves is incomplete, and can only be completed within a context. For example, expressions such as "this", "that", "that", "that", "here", "there", "I", "me", "your", "the", "people", etc. acquire full meaning if the context in which they appear is taken into account. They are context-dependent expressions.
The (triadic) equation "expression+context = meaning" must be considered in these cases. An index, indexical or indicating expression is one whose reference cannot be determined without knowing the context of use. An index "points to" or "indicates" a context, a state of affairs, and is part of the pragmatics of communication in general. There may be higher order indexes (meta-contexts, contexts of contexts).
The word "pragmatics" refers to the possible contexts of use of a language. In order to interpret a pragmatic language, one must determine the set of all possible contexts of use or all aggregates of relevant aspects of possible contexts of use.
The term "index" comes from Peirce, who considered that the index transcended language. For this author, a sign has two kinds of meaning: referential (context-independent) and indexical (context-dependent).
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