Can Formal Logic Explain Language?
Can Formal Logic Explain Language?
Discussing Richard Montague’s Formalised Approach to Natural Language Semantics and its Bifurcation with the Chomskyan Approach
Dear Thinkers,
We have spoken a great deal on this substack about the underlying structures of language, discussing the works of Chomsky, Fodor, Lenneberg, Searle, and Jackendoff. Building upon this conversation, today I bring to you another article in which we will discuss at length and in technical detail, the works of Richard Montague, a logician and philosopher, who revolutionised the field of formal semantics with his approach to natural language, arguing that the principles governing logical languages could be applied to our everyday speech.
I will cover the formal systems he developed, such as typed lambda calculus and intensional logic, and see how they provide a framework for analysing the semantics of natural language.
Of course, any discussion of linguistic theory would be incomplete without considering what Noam Chomsky had to say about it. While Montague and Chomsky operated in the same era, their approaches offer distinct yet complementary insights into the nature of language. I will discuss the points where their theories intersect and diverge.
Richard Montague
Richard Montague was born in Stockton, California, in 1930. His formal education began at the University of California, Berkeley, where he completed both his undergraduate and graduate studies. It was here that he came under the tutelage of Alfred Tarski, a prominent logician at the time.
Tarski's approach to formal logic and model theory provided Montague with a robust foundation in the rigorous methods that he would later apply to natural language. Montague earned his PhD in 1957, and his dissertation, "Contributions to the Axiomatic Foundations of Set Theory," already hinted at the formal precision that would characterise his later work in linguistics.
After completing his doctorate, Montague joined the faculty at UCLA, where he spent the entirety of his academic career. During his time at UCLA, Montague published a series of influential papers that laid the groundwork for what would come to be known as Montague grammar. His work sought to bridge the gap between formal logic and natural language.
Montague's notable publications include "English as a Formal Language" (1970) and "Universal Grammar" (1970), where he articulated his vision of a formal system that could capture the complexities of natural language semantics. These works set the stage for his most comprehensive treatment of the subject, "The Proper Treatment of Quantification in Ordinary English" (1973).
Montague's approach was grounded in the belief that there is no essential distinction between the formal languages used in logic and mathematics and the natural languages spoken by humans. He argued that the tools and methods developed for formal logic could be applied to natural languages with equal rigour and precision. This perspective was revolutionary at a time when many linguists and philosophers considered natural language to be too irregular and context-dependent to be captured by formal systems.
Fundamentals of Montague Grammar
Montague's philosophical stance on language was both bold and transformative. He challenged the prevailing notion that natural languages are fundamentally different from formal languages in logic and mathematics. Montague believed that the irregularities and complexities of natural language could be rigorously analysed using the same formal tools that logicians apply to mathematical systems. This perspective allowed him to build a formal framework for natural language semantics that was as precise and systematic as those used in formal logic.
Basic Components of Montague Grammar
Syntax
Montague grammar adopts a formal approach to syntax using context-free grammars. A context-free grammar consists of a set of production rules that specify how symbols can be combined to form valid sentences. This is clever because it allows for the systematic generation of syntactic structures, which can then be paired with their corresponding semantic interpretations.
For example, consider the simple sentence "The cat sleeps." In a context-free grammar, this sentence might be generated by the following rules.
1. S → NP VP
A sentence (S) consists of a noun phrase (NP) followed by a verb phrase (VP).
2. NP → Det N
A noun phrase consists of a determiner (Det) followed by a noun (N).
3. VP → V
A verb phrase consists of a verb (V).
4. Det → "The"
The determiner can be "The".
5. N → "cat"
- The noun can be "cat".
6. V → "sleeps"
The verb can be "sleeps".
These rules generate the syntactic structure of the sentence, which can be represented as a tree diagram.
S
/ \
NP VP
/ \ |
Det N V
| | |
The cat sleeps
Semantics
Montague deploys what is known as model theory for deriving the semantics of a sentence. Model theory deals with the relationships between formal languages and their interpretations, or models. Montague applied these principles to natural language, allowing for a precise representation of meaning.
In Montague's system, each syntactic category is associated with a corresponding semantic type. For example, nouns can be associated with individuals, verbs with functions from individuals to truth values, and sentences with truth values.
The Principle of Compositionality
According to this principle, the meaning of a complex expression is determined by the meanings of its parts and the rules used to combine them. This ensures that the semantics of natural language expressions can be systematically derived from their syntactic structure.
Consider our previous example, "The cat sleeps." To derive its meaning, we need to determine the meanings of its parts and how they combine.
1. "The cat": This noun phrase refers to a specific individual, the cat.
2. "sleeps": This verb phrase denotes a function that maps individuals to truth values, indicating whether they are sleeping.
The sentence "The cat sleeps" can be interpreted as applying the function denoted by "sleeps" to the individual denoted by "the cat," yielding a truth value. If the cat is indeed sleeping, the sentence is true; otherwise, it is false.
Typed Lambda Calculus
This is a formal system for representing the meanings of expressions. Typed lambda calculus extends the traditional lambda calculus by associating each expression with a type, ensuring that only meaningful combinations of expressions are allowed.
For example, the verb "sleeps" can be represented as a lambda expression that takes an individual and returns a truth value.
This expression denotes a function that maps an individual to the truth value of the proposition "x sleeps."
Intensional Logic
Montague also used intensional logic to handle modality and reference in natural language. Intensional logic allows for the representation of statements about possibilities, necessities, and other modal concepts, which are crucial for capturing the nuances of natural language meaning.
Consider the sentence "The cat might sleep." This sentence expresses a possibility rather than a certainty. In intensional logic, we can represent this modality by introducing possible worlds and stating that there is at least one possible world in which the cat sleeps.
Translation between Natural and Formal Languages
This translation process involves mapping syntactic structures to their corresponding semantic representations in a formal language. For instance, the sentence "The cat sleeps" can be translated into a logical formula that represents its meaning in the typed lambda calculus.
Here, denotes the individual referred to by "the cat," and denotes the truth value of the proposition that this individual is sleeping.
Montague Grammar in Action
Okay, now let us put these concepts to use with some more practical examples in order to understand Montague grammar in-depth.
Simple Declarative Sentence
Consider the sentence "Every dog barks." Using Montague grammar, we can break this sentence down into its syntactic components and translate it into a logical formula.
Syntactic Breakdown
S → NP VP
NP → Det N
VP → V
Det → "Every"
N → "dog"
V → "barks"
Translation to Logical Formula
Every dog:
Barks:
Every dog barks:
This formula states that for every individual , if is a dog, then barks. The translation captures the meaning of the sentence within the formal framework of Montague grammar.
Complex Sentence with Quantifiers
Now, let's consider a more complex sentence, "Some cat chases every mouse."
Syntactic Breakdown
S → NP VP
NP → Det N
VP → V NP
Det → "Some" | "Every"
N → "cat" | "mouse"
V → "chases"
Translation to Logical Formula
Some cat:
Every mouse:
Chases:
Some cat chases every mouse:
This formula states that there exists an individual such that is a cat and for every individual , if is a mouse, then chases . The translation handles the interaction between the quantifiers "some" and "every," preserving the meaning of the original sentence.
Ambiguous Sentence
Consider the sentence, "Every student read a book." This sentence is ambiguous because it can be interpreted in two ways.
Reading 1
Every student read the same book.
Logical Formula:
Reading 2
Each student read a different book.
Logical Formula:
Montague grammar allows us to formally represent both interpretations, providing clarity and precision in the analysis of ambiguous sentences.
Modal Sentence
Consider the sentence "The cat might sleep."
Syntactic Breakdown
S → NP VP
NP → Det N
VP → V
Det → "The"
N → "cat"
V → "might sleep"
Translation to Logical Formula Using Intensional Logic
The cat:
Might sleep:
The cat might sleep:
This formula uses the modal operator to represent possibility, indicating that there is a possible world in which the cat sleeps.
Montague’s Divergence from the Chomskyan Approach
Montague's integrated approach to syntax and semantics contrasts with the modular view of language proposed by Noam Chomsky. I have written at length about Chomsky’s Universal Grammar (UG). If you are not familiar with it, click the link above to read. Chomsky's theories emphasise the cognitive and biological underpinnings of linguistic competence, which Montague's formalism does not directly address.
Modularity
Chomsky posits that language consists of modular components, such as phonology, syntax, and semantics, each governed by its principles and constraints. Montague's approach, which integrates syntax and semantics within a single formal system, does not align with this modular perspective.
Cognitive Realism
Chomsky emphasises that linguistic theories should aim to reflect the cognitive realities of how humans process and produce language. Montague's work, while formally rigorous, is often viewed as abstract and disconnected from these cognitive processes. This disconnect raises questions about the psychological plausibility of Montague's theories.
Notes
Finally, I’d like to share a few thoughts on where I see the theories of Chomsky and Montague intersect. By integrating the formal rigour of Montague's grammar with the cognitive realism of Chomsky's generative grammar, we can develop a more comprehensive framework for linguistic analysis.
Formal Precision and Cognitive Realism
Montague's formal semantics provides a precise and systematic method for representing meaning, which can be particularly useful for detailed linguistic analysis and computational applications. Chomsky's emphasis on cognitive realism ensures that linguistic theories are grounded in the psychological and biological mechanisms underlying language acquisition and use.
Syntax-Semantics Interface
Montague's work on the syntax-semantics interface can be integrated with Chomsky's syntactic theory to develop a more robust understanding of how syntactic structures map to semantic interpretations. This integration can help resolve ambiguities and capture the nuances of meaning in complex sentences.
Compositional Semantics in Generative Grammar
Researchers are developing frameworks that incorporate Montague's compositional semantics into Chomskyan generative grammar. This allows for the systematic construction of meaning from syntactic structures while maintaining cognitive plausibility.
Computational Models
In natural language processing (NLP), integrating Montague's formal methods with Chomsky's syntactic theories can enhance the development of algorithms for parsing, interpreting, and generating natural language.
Cross-Linguistic Studies
Integrating Montague's and Chomsky's theories facilitates cross-linguistic studies that aim to uncover universal principles of language structure and meaning. By applying formal methods to diverse languages, researchers can test the universality of linguistic principles proposed by UG.