Montague Semantics - The Formal Foundation of Compositional Language Understanding

In the early 1970s, linguistics found itself in a peculiar state of imbalance. The syntactic revolution initiated by Noam Chomsky had transformed the field, giving linguists powerful formal tools for describing the structure of sentences. Phrase structure grammars, transformational rules, and hierarchical tree representations had brought mathematical rigor to the study of syntax. Yet semantics—the study of meaning—remained conspicuously informal, relying largely on intuitive descriptions, philosophical arguments, and informal paraphrases. Linguists could precisely specify why "colorless green ideas sleep furiously" was syntactically well-formed but semantically anomalous, but they lacked a formal calculus for computing what sentences actually meant.

Richard Montague, a mathematical logician at UCLA, found this situation both intellectually unsatisfying and theoretically unnecessary. Trained in formal logic and model theory rather than linguistics, Montague approached natural language with a logician's sensibility: meaning should be compositional, systematic, and mathematically tractable. If formal logic could assign precise truth conditions to statements about sets, numbers, and abstract entities, why couldn't the same framework handle statements about cats sitting on mats, politicians making promises, or anyone seeking a unicorn? The conventional wisdom held that natural language was too messy, too context-dependent, too riddled with ambiguity and vagueness for formal semantic treatment. Montague disagreed fundamentally. In his view, there was "no important theoretical difference between natural languages and the artificial languages of logicians."

Between 1970 and his untimely death in 1971, Montague developed what would become known as Montague Grammar or Montague Semantics, publishing three seminal papers that laid out a formal framework for compositional natural language semantics. His approach provided a rigorous method for mapping syntactic structures onto logical forms, specifying how the meanings of complex expressions derived systematically from their parts. The framework employed intensional logic, lambda calculus, and model-theoretic semantics—mathematical machinery far more sophisticated than anything previously applied to natural language. While Montague's work initially met skepticism from linguists, who found his methods forbiddingly technical and his linguistic judgments sometimes questionable, it would profoundly influence computational linguistics, formal semantics, and ultimately the development of language AI.

Montague's contribution was transformative not because it solved all problems of meaning—it decidedly did not—but because it demonstrated that formal semantic analysis of natural language was possible in principle. His work established compositionality as a central principle, showed how to handle intensional contexts (beliefs, possibilities, necessities), and provided techniques for dealing with quantification and scope that would influence generations of semantic theory. For computational linguistics, Montague Grammar offered something invaluable: a path from syntax to semantics, a systematic method for computing meaning representations from parsed sentences. Though modern language AI has largely moved away from explicit logical representations toward distributed neural semantics, the problems Montague grappled with—compositionally, quantifier scope, intensionality—remain central challenges for any system attempting to understand language.

Linguists and philosophers had long recognized that sentences have meanings—that "snow is white" means something different from "grass is green," and that both differ from "colorless green ideas sleep furiously." But what exactly did it mean for a sentence to "have a meaning"? Informal descriptions sufficed for many purposes: one could paraphrase, translate, or explain meanings without formalizing them. Yet this informality created problems. Without a precise notion of meaning, linguists couldn't rigorously test semantic theories, couldn't definitively determine when two sentences had the same meaning, and couldn't systematically predict the meanings of novel sentences from their parts.

The problem grew particularly acute when dealing with complex semantic phenomena. Consider quantification: "every student read some book" is ambiguous between two readings—one where there's a specific book all students read, another where each student read potentially different books. How could this ambiguity be formally represented and resolved? Consider intensional contexts: "John seeks a unicorn" doesn't require unicorns to exist, while "John finds a unicorn" does. What makes "seeks" different from "finds" in terms of existential commitment? Consider embedded clauses: "Mary believes that John left" has a different logical structure from "Mary believes John," involving a proposition rather than an individual. How could these structural differences be systematically captured?

Previous approaches to semantics had provided partial answers. The tradition of formal logic, descending from Frege, Russell, and Tarski, had developed sophisticated tools for analyzing mathematical and scientific language. First-order predicate logic could represent statements like "all men are mortal" as formal expressions whose truth could be evaluated in models. But logicians had largely restricted their attention to carefully controlled formal languages, avoiding the complications of natural language. When they did address natural language, they often dismissed problematic constructions as defective, needing reform rather than analysis.

Linguists, conversely, attended carefully to natural language data but lacked formal semantic tools. Generative semanticists in the late 1960s had attempted to collapse syntax and semantics, treating semantic representations as deep syntactic structures that transformed into surface forms. This approach generated productive research but faced conceptual difficulties: semantic representations looked suspiciously like logical formulas, yet the grammatical machinery of transformations seemed ill-suited to semantic composition. The relationship between syntactic structure and semantic interpretation remained unclear, more stipulated than explained.

What was needed was a bridge: a formal framework that could handle the full complexity of natural language meanings while maintaining mathematical rigor. The framework would need to specify exactly how complex meanings derived from simpler parts, handle ambiguity systematically, distinguish semantic from pragmatic phenomena, and provide a notion of meaning precise enough to support theoretical predictions and computational implementation. Montague set out to build precisely this bridge, bringing the tools of mathematical logic to bear on natural language semantics.

Montague's approach rested on several key principles, each reflecting deep commitments about the nature of meaning. The first and most fundamental was compositionality: the meaning of a complex expression is a function of the meanings of its parts and the way they're syntactically combined. This principle, often called the Frege principle after Gottlob Frege's earlier advocacy, entailed that one couldn't assign arbitrary meanings to sentences. If you knew what "cat," "the," "on," "mat," and "sat" meant, and you knew the syntactic rule that combined them, you could compute what "the cat sat on the mat" meant. This wasn't just a methodological preference—it was a necessary condition for semantic learnability and productivity. Humans understand novel sentences they've never encountered precisely because they compose meanings from known parts.

The second principle was that meanings should be defined relative to models—mathematical structures specifying possible worlds, individuals, properties, and relations. Following Tarski's model-theoretic semantics, Montague treated the meaning of a sentence as its truth conditions: the conditions under which it would be true in a given model. The sentence "snow is white" was true in models where the substance snow had the property white, false otherwise. This approach shifted focus from meanings as mysterious mental entities to meanings as systematic relationships between language and models of reality. It provided a clear criterion for semantic equivalence: two expressions had the same meaning if they were true in exactly the same models.

The third principle, more technically sophisticated, was intensionality. Montague recognized that natural language abounds with contexts where truth values alone don't determine meaning. Consider "the President of the United States" in 2024 versus 2020—same grammatical form, different referent. Consider "necessarily" versus "possibly": "2 + 2 = 4" is necessarily true, but "it's raining" is only possibly true. To handle these intensional phenomena, Montague employed possible worlds semantics. Rather than evaluating expressions in a single model, his framework considered sets of possible worlds—different ways reality might be. The meaning of "necessarily P" required P to be true in all possible worlds; "possibly P" required P to be true in at least one world. Intensional operators like "believe," "seek," and "want" similarly operated over propositions or properties rather than simple truth values.

To implement these principles, Montague developed a fragment of English with explicit syntactic rules and corresponding semantic rules. Each syntactic rule that combined expressions had a paired semantic rule specifying how meanings combined. Consider a simple example: combining a proper name like "John" with an intransitive verb phrase like "walks." The syntactic rule produced the sentence "John walks." The semantic rule took the meaning of "John" (an individual) and the meaning of "walks" (a property of individuals) and produced a truth value: true if the individual John has the property of walking, false otherwise.

The framework employed lambda calculus as its primary tool for representing compositional meanings. Lambda expressions allowed Montague to represent meanings as functions that could be applied to arguments. The intransitive verb "walks" might be represented as $\lambda x.\text{walk}(x)$—a function that takes an individual $x$ and returns the proposition that $x$ walks. When applied to "John," denoting individual $j$, the lambda expression $\beta$-reduces: $(\lambda x.\text{walk}(x))(j) \to \text{walk}(j)$. This functional application modeled semantic composition precisely, showing how subject-verb combination produced a complete proposition.

Montague's treatment of noun phrases demonstrated the framework's sophistication. Rather than treating noun phrases as denoting individuals, he treated them as denoting sets of properties—what logicians call generalized quantifiers. The proper name "John" denoted the set of all properties that John has. The quantified noun phrase "every student" denoted the set of properties that every student has. This uniform treatment allowed a simple semantic rule for combining noun phrases with verb phrases: a sentence was true if the property denoted by the verb phrase belonged to the set denoted by the noun phrase. For "every student walks," this meant: the property of walking belongs to the set of properties every student has, or equivalently, every student has the property of walking.

This generalized quantifier approach had remarkable benefits. It handled quantifier scope systematically: ambiguous sentences like "every student read some book" could be derived via different syntactic structures producing different semantic interpretations. It explained why certain inferences were valid: if "every student walks" is true and "John is a student" is true, then "John walks" must be true—a consequence of the underlying logic. It unified the treatment of different noun phrase types: proper names, definite descriptions, and quantified phrases all denoted sets of properties, differing only in which sets they denoted.

The most technically demanding aspect of Montague's framework involved handling intensionality—contexts where meaning couldn't be reduced to truth values or extensions (the sets of entities having properties). Consider the sentence "John seeks a unicorn." If we interpreted "a unicorn" extensionally, as denoting some particular unicorn, we'd face a problem: unicorns don't exist, so there's no unicorn for "a unicorn" to denote. Yet the sentence is perfectly meaningful and can be true—John might indeed be seeking a unicorn, futilely though his quest may be.

Montague solved this through intensions: functions from possible worlds to extensions. The intension of a property like "unicorn" was a function mapping each possible world to the set of unicorns in that world. In the actual world, this set is empty, but in other possible worlds—mythical worlds where unicorns exist—the set is non-empty. An intensional verb like "seek" operated on intensions rather than extensions: "John seeks a unicorn" meant that John stood in the seeking relation to the intension of unicorn, not to any particular unicorn. This explained the lack of existential commitment: seeking relates an individual to an intension, which can be meaningful even when its extension in the actual world is empty.

The same mechanism handled other intensional contexts. "John believes that Mary left" involved a belief relation between John and a proposition—the intension of "Mary left," understood as the set of possible worlds where Mary left. Whether Mary actually left in the real world was irrelevant to whether John believed she did. Modal operators like "necessarily" and "possibly" similarly operated on intensions: "necessarily P" was true if P's intension included all possible worlds; "possibly P" was true if P's intension included at least one possible world.

Temporal expressions required intensionality as well. The sentence "the President is tall" has different truth values at different times, depending on who holds office. Montague treated temporal expressions as involving implicit world-time indices: expressions were evaluated not just at possible worlds but at world-time pairs. The present tense "is" evaluated at the current world-time, while past and future tenses shifted the time index. This allowed systematic treatment of temporal reasoning: "John was a student" was true at time t if "John is a student" was true at some earlier time.

Implementing intensional logic required a sophisticated type system. Montague distinguished:

• Type e: entities (individuals like John, books, cities)
• Type t: truth values (true/false)
• Type $\langle s,\alpha \rangle$: functions from possible worlds (type $s$) to expressions of type $\alpha$ (intensions)
• Type $\langle \alpha,\beta \rangle$: functions from type $\alpha$ to type $\beta$

Common nouns like "student" had type $\langle s,\langle e,t \rangle \rangle$: functions from possible worlds to functions from entities to truth values (i.e., intensions of properties). Intransitive verbs had the same type. Transitive verbs had type $\langle s,\langle e,\langle e,t \rangle \rangle \rangle$: functions from worlds to two-place relations. Noun phrases had type $\langle s,\langle \langle s,\langle e,t \rangle \rangle,t \rangle \rangle$: functions from worlds to functions from properties to truth values (intensions of generalized quantifiers). These complex types ensured compositional coherence: only compatible types could combine, guaranteeing semantic well-formedness.

Montague's work appeared in philosophy and logic journals—venues unfamiliar to most linguists. His three key papers, "English as a Formal Language" (1970), "Universal Grammar" (1970), and "The Proper Treatment of Quantification in Ordinary English" (1973, published posthumously), employed dense logical notation, assumed familiarity with model theory, and rarely engaged with linguistic literature. The linguistic examples he analyzed—sentences like "every man loves a woman," "John seeks a unicorn," "necessarily nine is greater than seven"—were chosen to illustrate logical phenomena rather than to cover the empirical breadth linguists valued. His formal fragments covered only tiny subsets of English, and extensions to fuller coverage seemed daunting.

Linguists initially reacted with skepticism. Some dismissed the approach as irrelevant to empirical linguistic concerns, more interested in logical puzzles than in describing actual language use. Others found the formalism impenetrably technical, a barrier rather than a tool. The framework's treatment of syntax struck many as inadequate—Montague used a categorial grammar quite different from the phrase structure and transformational systems dominant in generative linguistics. His semantic judgments sometimes seemed questionable: he treated "the" as a quantifier similar to "every," yielding unintuitive semantic types for definite descriptions.

The transformational generative paradigm, still dominant in the early 1970s, viewed semantics as interpretive: syntactic structures generated by the grammar received semantic interpretations, but syntax was autonomous and prior. Montague's framework inverted this relationship in certain respects, with syntactic and semantic rules developed in tandem, neither fully autonomous. This philosophical difference created friction. Additionally, the emphasis on truth conditions and model-theoretic semantics seemed to exclude much of what interested linguists: pragmatic implicatures, context-dependency, metaphor, discourse structure. Critics argued that Montague had formalized a narrow slice of semantics while ignoring phenomena that made natural language natural.

Yet a subset of linguists and philosophers, recognizing the framework's power, began exploring its potential. Barbara Partee, who had trained in linguistics under Chomsky before encountering Montague's work, became a crucial bridge figure. She translated Montague's logical apparatus into more linguistically accessible terms, demonstrated how his approach could integrate with linguistic theory, and showed how the framework illuminated empirical problems linguists cared about. Partee's work made Montague Grammar intellectually accessible to linguists, spawning a research program that would dominate formal semantics for decades.

By the late 1970s and early 1980s, Montague semantics had become a major force in linguistic theory. Researchers extended the framework to handle wider ranges of linguistic phenomena: tense and aspect, plurals, mass terms, generics, anaphora, presupposition. They refined the syntactic component, developing versions that integrated with phrase structure grammar and later with categorial grammar. They addressed pragmatic phenomena within a broadly Montagovian framework, distinguishing semantic content from pragmatic implicatures while maintaining compositionality. The approach matured from a logical curiosity into a research program rich with empirical predictions and theoretical insights.

For computational linguistics, Montague's framework offered something transformative: a systematic method for semantic interpretation. If syntax could be parsed automatically—a capability that rule-based parsers of the 1970s and 1980s increasingly demonstrated—then Montague's compositional semantics provided a path from syntactic structures to logical forms. Each syntactic rule came paired with a semantic rule; implementing both allowed a system to compute meaning representations algorithmically. This promise of automated semantic interpretation energized early natural language understanding systems.

The approach meshed well with the knowledge representation paradigms dominant in artificial intelligence during the 1970s and 1980s. AI systems of that era typically represented knowledge in formal logic—predicate calculus, frame systems, semantic networks. Montague's framework delivered exactly this kind of representation: parsed sentences yielded logical formulas in intensional logic, which could be translated into whatever logical formalism the AI system employed. A question-answering system could parse "which students read every book?" into a logical query, match it against a knowledge base of facts, and retrieve answers through logical inference.

Several influential natural language understanding systems incorporated Montague-inspired semantics. The LUNAR system, developed at BBN in the early 1970s for analyzing moon rock samples, used a semantic grammar combining syntactic and semantic rules to map English questions into database queries. While not strictly Montagovian, LUNAR embodied the compositional philosophy: meanings built systematically from parts, with lambda calculus used for semantic composition. The SHRDLU system, Terry Winograd's famous blocks world system, employed a semantic framework with similar compositional structure, though implemented quite differently.

Through the 1980s and 1990s, more explicitly Montagovian systems emerged. The Core Language Engine, developed at SRI International, implemented a version of Montague Grammar with extensive coverage of English syntax and semantics. It translated English sentences into quasi-logical forms—semantic representations based on first-order logic augmented with intensional operators and other extensions. These logical forms could then drive reasoning systems, database queries, or translation into other languages. The Rosetta machine translation system, developed at Philips, used Montague Grammar as an interlingua: source language sentences mapped to logical forms, which then mapped to target language sentences, with the logical forms capturing language-independent meaning.

The framework also influenced the development of formal grammars for computational use. Categorial grammars, which Montague had employed, became a major research area in computational linguistics. These grammars elegantly unified syntax and semantics: syntactic categories corresponded to semantic types, and syntactic combination rules directly specified semantic composition. Combinatory Categorial Grammar (CCG), developed by Mark Steedman and others in the 1980s and 1990s, extended categorial approaches with more flexible composition rules while maintaining tight syntax-semantics coupling. CCG parsers could simultaneously build syntactic structures and semantic representations, implementing Montague's vision of compositional interpretation in computationally efficient ways.

Despite its theoretical elegance, Montague's framework faced substantial practical and theoretical limitations. The most obvious was coverage: Montague's fragments covered only tiny portions of English. Extending them to handle the full breadth of natural language—idioms, metaphors, ellipsis, discourse anaphora, intonation, word order variations—required enormous effort. Each new phenomenon demanded new syntactic rules, new semantic rules, new type assignments, new logical operators. The framework provided principles for how extensions should work, but actually implementing them consumed decades of research.

The intensional logic Montague employed, while powerful, was computationally expensive. Evaluating expressions in possible worlds semantics required considering multiple worlds, computing truth values in each, and combining results through modal operators. For real-time natural language understanding systems, this computational burden proved prohibitive. Even determining whether a simple inference was valid could require exploring exponentially many models. Researchers developed restricted logical formalisms—decidable fragments of intensional logic with better computational properties—but these restrictions limited expressiveness.

The framework's treatment of context and pragmatics remained underdeveloped. Montague focused on semantic content—truth conditions—but much of meaning in natural language derives from context and use. What "I" refers to depends on who's speaking; what "now" means depends on when it's uttered; what "here" picks out depends on spatial context. While Montague's possible worlds framework could theoretically handle these indexicals by including speaker-time-location parameters in models, the framework provided no systematic account of how context determined these parameters. Similarly, pragmatic phenomena like implicature, presupposition, and speech acts received minimal treatment.

Ambiguity, though Montague's framework could represent it formally, remained computationally problematic. Ambiguous sentences had multiple derivations, each producing a different semantic interpretation. But natural language sentences were often spectacularly ambiguous—structurally ambiguous at multiple levels, lexically ambiguous with multiple word senses, scope-ambiguous with different quantifier readings. A realistic parser might generate dozens or hundreds of interpretations for a single sentence. Determining which interpretation was intended required pragmatic reasoning, world knowledge, and context—resources beyond what the formal framework provided.

The relationship between logical form and conceptual structure raised deeper questions. Montague assumed that natural language meanings could be adequately captured by intensional logic—truth conditions, possible worlds, functions, and sets. But many cognitive linguists and psychologists argued that human conceptual representations were richer and more structured: concepts organized in frames, schemas, prototypes, mental models. Logical forms seemed too austere to capture the full richness of meaning. This tension between logical and cognitive approaches to semantics would drive much subsequent debate in linguistic theory.

Montague semantics occupies a curious position in the history of language AI: simultaneously foundational and superseded. Contemporary neural language models—BERT, GPT, and their successors—bear little surface resemblance to Montague's logical framework. These models don't construct explicit logical forms, don't employ possible worlds, don't use lambda calculus for composition. They learn distributed representations through gradient descent on massive corpora, capturing semantic regularities statistically rather than logically. In this sense, the field has moved decisively away from Montague's approach.

Yet Montague's influence persists in subtle but important ways. The principle of compositionality remains central to linguistic theory and computational semantics. Modern approaches to semantic parsing—mapping natural language to database queries, code, or logical forms—directly instantiate Montague's vision, even when implemented with neural networks. Systems like Sempre, developed at Stanford, combine neural components with compositional semantic parsing, learning to map questions to logical forms that can be executed against databases. The neural-symbolic integration in these systems reflects a synthesis: neural networks handle the ambiguity and variation of natural language, while logical representations provide interpretability and interface with structured knowledge.

Research in formal semantics continues to build on Montagovian foundations. Dynamic semantics, developed in the 1980s and 1990s, extended Montague's framework to handle discourse phenomena—anaphora, presupposition projection, temporal progression. These approaches treated meanings not as static truth conditions but as context-change potentials: the meaning of a sentence was its effect on the discourse context. Game-theoretic semantics applied game theory to semantic interpretation, treating understanding as a kind of game between speaker and hearer. These extensions preserved Montague's commitment to compositionality and formal rigor while addressing phenomena his framework had neglected.

In computational linguistics, the lambda calculus Montague employed remains a standard tool. Semantic parsing systems often target lambda-calculus expressions as intermediate representations. The programming languages ML and Haskell, influential in computational linguistics, are essentially typed lambda calculi, making Montague's compositional semantic framework implementable through functional programming. The unification-based formalisms common in computational linguistics—like Head-Driven Phrase Structure Grammar (HPSG)—incorporate Montague-style semantic composition alongside syntactic structure, maintaining the tight coupling between syntax and semantics that Montague advocated.

Perhaps most importantly, Montague established that natural language semantics could be studied with mathematical rigor. Before Montague, semantics was often dismissed as too vague, too context-dependent, too psychological for formal treatment. Montague demonstrated that this dismissal was premature: substantial aspects of meaning could be formalized, made precise, subjected to logical analysis. This demonstration opened space for formal semantics as a legitimate subdiscipline, spawning decades of productive research. Even researchers who reject Montague's specific proposals work within a space he helped create—a space where formal semantic theories are expected, tested against data, and refined through theoretical argumentation.

The tension between Montague-style symbolic compositionality and neural distributional semantics has sparked interesting recent research. One line of work attempts to inject compositional structure into neural models. Tree-structured neural networks, recursive neural networks, and neural module networks all try to build representations of complex expressions by explicitly combining representations of parts, mimicking the compositional structure of Montague's framework. These models often outperform flat neural models on tasks requiring systematic generalization—understanding novel combinations of known parts.

Another line of work studies whether standard neural models learn compositional representations implicitly. Recent analyses of transformer models suggest they capture compositional structure to some degree: representations of complex phrases reflect systematic combinations of their constituent representations. The attention mechanism in transformers can be viewed as learning soft compositional functions, attending to relevant parts and combining their representations. While this differs from Montague's hard, rule-based composition, it captures a similar intuition: meaning is built from parts, and the building process follows learnable patterns.

The debate over compositional versus holistic representation echoes older debates in cognitive science. Fodor and Pylyshyn's famous argument for compositional mental representations paralleled Montague's linguistic arguments: thought is productive and systematic, suggesting that complex concepts compose from simpler ones. Connectionists responded that neural networks could exhibit productivity through continuous, distributed representations without explicit compositional structure. This debate, initially about mental architecture, now plays out in language AI: do we need explicit symbolic composition (the Montague approach) or can statistical learning over distributed representations suffice (the neural approach)?

The practical answer seems to be: both have value. For tasks requiring precise semantic interpretation—translating natural language to code, answering complex logical questions, interfacing with structured databases—explicit compositional semantics remains valuable. For tasks requiring robustness to variation, learning from data, and handling the statistical regularities of natural language use—machine translation, sentiment analysis, text generation—neural distributional approaches excel. Hybrid systems that combine both, using neural networks to handle surface variation and ambiguity while outputting compositional semantic representations, show particular promise.

Richard Montague's work on natural language semantics represents a watershed in the history of language science. Before Montague, semantics was largely informal—intuitive, philosophical, descriptive. After Montague, formal semantics became a rigorous mathematical discipline, with precise theories, testable predictions, and systematic explanations of semantic phenomena. This transformation didn't happen instantly; it took a decade for linguists to fully appreciate and adopt Montague's methods. But once adopted, these methods reshaped linguistic theory and computational linguistics alike.

Montague demonstrated that natural language, despite its apparent messiness and complexity, could be treated with the same formal rigor as mathematical logic. The tools he introduced—possible worlds semantics, lambda calculus, type theory, model-theoretic interpretation—became standard equipment in semantic theory. His emphasis on compositionality established a guiding principle that remains central today. His treatment of quantification, intensionality, and scope influenced not only semantics but also philosophy of language, artificial intelligence, and cognitive science.

For language AI specifically, Montague provided a bridge from syntax to semantics, showing how parsed sentences could be systematically interpreted as logical forms. This enabled the first generation of natural language understanding systems, which mapped language onto knowledge representations through compositional semantic rules. While modern neural systems have largely moved away from explicit logical representations, the problems Montague addressed—compositionality, ambiguity resolution, systematic interpretation—remain central challenges. Contemporary work in semantic parsing, question answering, and neural-symbolic integration continues to grapple with these problems, often using tools Montague pioneered.

The story of Montague semantics also illustrates an important pattern in the history of language AI: formal precision often precedes broad coverage. Montague's fragments covered minimal subsets of English, but they covered those subsets with unprecedented precision. This precision enabled rigorous testing, theoretical refinement, and eventual extension. The lesson has recurred: formal, narrow approaches often generate insights that guide later broad-coverage work. Modern neural models achieve impressive coverage, but ongoing efforts to make them more systematic, more compositional, and more interpretable often return to principles Montague articulated.

Montague died tragically young in 1971, murdered during a robbery at age 40. He never witnessed the full impact of his work on linguistics, philosophy, or computer science. Yet his legacy endures: in the formal semantic theories that build on his foundations, in the computational systems that implement his compositional vision, and in the enduring questions about meaning, composition, and interpretation that he helped formalize. Natural language understanding remains an unsolved problem, but Montague showed us how to think about it rigorously—a contribution whose value has only grown with time.

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