Montague Grammar

Montague's program applies the same formal, model-theoretic methods to natural and formal languages. In a typed compositional grammar, each syntactic analysis assigns expressions semantic types, and composition rules constrain how their denotations combine (Montague 1970a, 1970b, 1973).

Before we can say what any word means, we need a type system to say what kind of thing its meaning is.

Atomic types

There are just two primitive types:

• $e$ — the type of entities (individuals in the world: John, the Eiffel Tower, your left shoe).
• $t$ — the type of truth values ($\top$ or $\bot$, i.e. True or False). A complete, declarative sentence denotes something of type $t$.

Functional types

Everything else is built by function formation: if $\alpha$ and $\beta$ are types, then $\alpha \to \beta$ is the type of a function from $\alpha$-things to $\beta$-things. This is the arrow type familiar from programming languages and typed lambda calculi. Montague-semantics texts more often write the same type as $\langle\alpha,\beta\rangle$.

The arrow is right-associative: $\alpha \to \beta \to \gamma$ means $\alpha \to (\beta \to \gamma)$, a function that takes an $\alpha$ and returns a function from $\beta$ to $\gamma$. This is just currying. Some important derived types:

Type	Reads as	Example
$e \to t$	function from entities to truth values	a one-place predicate like sings, is a man: maps an entity to True iff it sings/is a man
$e \to e \to t$	function from entities to $(e \to t)$ functions	a two-place predicate like loves: takes an object, returns a property of subjects
$(e \to t) \to t$	function from $(e \to t)$ to truth values	a generalized quantifier (GQ) like every man, some woman, John (lifted)
$(e \to t) \to (e \to t) \to t$	takes a noun property, returns a GQ	a determiner like every, some, no

The key rule — Function Application (called “Rule B” here as a local convention) — says: if $f$ has type $\alpha \to \beta$ and $x$ has type $\alpha$, then $f(x)$ has type $\beta$. Function application is one central mode of semantic composition.

Montague semantics uses the typed lambda calculus. If you've written any functional programming, this will feel familiar (Church 1940; Montague 1970b, 1973; Heim and Kratzer 1998).

Lambda abstraction

$\lambda x. \phi(x)$ denotes the function that takes an input $x$ and returns $\phi(x)$. The variable $x$ is bound by the $\lambda$. So $\lambda x. \text{Sings}(x)$ is the function that maps any entity to True iff it sings — this is the denotation of the predicate sings, of type $e \to t$.

You can abstract over any type. $\lambda P. P(\text{john})$, where $P$ ranges over $e \to t$ functions, is the function that takes a property and returns True iff John has it. This is of type $(e \to t) \to t$.

Beta-reduction ($\beta$-reduction)

When you apply a lambda function to an argument, you substitute the argument for the bound variable everywhere in the body. This substitution is called $\beta$-reduction, written with the $\xrightarrow{\beta}$ arrow:

$$(\lambda x. \phi(x))(a) \xrightarrow{\beta} \phi(a)$$

Two concrete examples:

Beta-reduction is the principal term-reduction operation used to simplify compositions of lambda-denotations. A full Montague grammar also contains translation rules, intensional operators, model-theoretic interpretation, quantification, connectives, and meaning postulates.

Alpha-equivalence

The name of a bound variable is irrelevant. $\lambda x. \text{Sings}(x)$ and $\lambda y. \text{Sings}(y)$ are the same function ($\alpha$-equivalent). We'll silently rename variables to avoid clashes during substitution.

We now assign denotations to a small vocabulary, annotated with types.

Word	Type	Denotation	Gloss
john	$e$	$j$	a specific entity
sings	$e \to t$	$\lambda x.\, \text{Sings}(x)$	the property of singing
man	$e \to t$	$\lambda x.\, \text{Man}(x)$	the property of being a man
every	$(e \to t) \to (e \to t) \to t$	$\lambda P.\, \lambda Q.\, \forall x\,(P(x) \to Q(x))$	universal quantification

Deriving "every man sings"

We build meaning bottom-up from the syntax tree:

The problem with "John sings"

Now try "John sings." Syntactically, it has the same structure: NP + VP. But john has type $e$, while "every man" has type $(e \to t) \to t$. The VP combinator can't accept both — we have a type mismatch.

Type lifting (called “Rule A” here as a local convention) resolves this. We lift the proper noun john from an entity $j : e$ to a generalized quantifier of type $(e \to t) \to t$:

$$\text{john}_{\text{lifted}} = \lambda P.\, P(j)$$

This reads: "John is the set of all properties that John has." Now the derivation works uniformly:

Both "John sings" and "Every man sings" are now derived by the same rule (GQ + VP). This uniformity is the whole point: the grammar has a single, general composition rule rather than special cases for proper nouns versus quantifiers.

The framework so far treats noun phrases as denoting things (or sets of properties of things) that actually exist. But natural language can refer to non-existent objects:

John seeks a unicorn.

This sentence is perfectly grammatical and can be true — even though there are no unicorns. A naive analysis would assign "a unicorn" an empty extension and force the sentence to be trivially false (or undefined). That's clearly wrong.

Compare it with:

John finds a unicorn.

Here, if the sentence is true, a unicorn must exist. Finding requires a real thing to be found. Seek does not. The contrast is called the distinction between intensional and extensional verb positions.

Denotation of "a unicorn" as a GQ

Using our GQ machinery, the determiner a has the denotation:

$$\text{"a"} = \lambda P.\, \lambda Q.\, \exists x\,(P(x) \land Q(x))$$

So "a unicorn" (type $(e \to t) \to t$) is:

$$\text{"a unicorn"} = \lambda Q.\, \exists x\,(\text{Unicorn}(x) \land Q(x))$$

This is the set of properties $Q$ such that at least one unicorn has $Q$. If unicorns do not exist, it is the empty generalized quantifier. In an extensional model it is therefore indistinguishable from every other empty existential NP, so this representation alone cannot explain why seeking a unicorn differs from seeking a centaur.

Intensional verbs: seek

Montague instead analyzes an intensional transitive such as seek as taking an intension of an NP denotation. In simplified notation, with $s$ the type of indices or worlds:

$$\llbracket\text{seek}\rrbracket : \langle\langle s,\langle\langle e,t\rangle,t\rangle\rangle,\langle e,t\rangle\rangle$$

The object thus supplies different generalized-quantifier extensions at different indices, preserving the conceptual difference between unicorns and centaurs even when neither exists in the actual world. The exact PTQ types are more elaborate because intensional structure is represented systematically (Montague 1973; Zimmermann 1993).

Extensional verbs: find

Find is different. An extensional classroom analysis combines the existential object with the property of being found:

The existential quantifier $\exists x(\text{Unicorn}(x) \land \ldots)$ is now explicit in the logical form. The math enforces what our intuition demands: if John finds a unicorn, there exists a unicorn that John found.

Consider the sentence:

Every man loves a woman.

This sentence is genuinely ambiguous — not vague, but structurally ambiguous between two distinct logical readings:

• Weak (surface) scope: For every man, there exists (possibly a different) woman he loves. $\forall z(\text{Man}(z) \to \exists y(\text{Woman}(y) \land \text{Love}(z,y)))$
• Strong (inverse) scope: There is one particular woman that every man loves. $\exists y(\text{Woman}(y) \land \forall z(\text{Man}(z) \to \text{Love}(z,y)))$

These have different truth conditions. The weak reading is much weaker — it's true in almost any world with romantic activity. The strong reading makes a specific claim about one woman. Montague's account derives both readings from the same surface string by tracking derivational history — the order in which the syntax tree is assembled.

Derivation 1: weak scope (direct assembly)

Derivation 2: strong scope ("quantifying in")

To get the inverse scope reading, Montague uses a quantifying-in rule: we first insert a pronoun placeholder $\text{him}_1$ in the object position, build the sentence with this placeholder, and then bind it by the object GQ from outside.

The two derivations use the same lexical entries but different syntactic derivations. Montague's PTQ obtains inverse scope with a quantifying-in rule (Montague 1973). This differs from Quantifier Raising (QR), a later syntactic movement analysis (May 1985), and from continuation-based accounts that derive scope without movement (Barker 2002).

Everything so far evaluates meaning at a single fixed world and time. But natural language routinely talks about possibilities, necessities, beliefs, and counterfactuals — contexts where truth can vary across worlds and times.

Montague's Intensional Logic (IL) adds a type $s$ for indices and forms an intension of an expression of type $a$ as a function of type $\langle s,a\rangle$. An index may encode a world–time pair $\langle w,t\rangle$, though treating worlds $W$ and times $T$ as separate primitive sets is only one implementation (Montague 1970c, 1973; Gallin 1975).

Evaluating expressions relative to an index gives us the key distinction:

• Extension of $\alpha$ at $\langle w, t \rangle$: the value $\alpha$ actually takes in world $w$ at time $t$. Written $\vee\alpha$ (the "down" operator, also ${}^\vee$ or "ext"). For an individual concept like "the temperature," the extension at $\langle w, t \rangle$ is the actual temperature reading in that world at that moment.
• Intension of $\alpha$: a function from indices to extensions, mapping every index to what $\alpha$ denotes there. Written $\wedge\alpha$ (the "up" operator).

The temperature paradox

Consider the sentence pair:

1. "The temperature is 90."
2. "The temperature is rising."

If we naïvely substitute the number 90 for "the temperature" in (2), we get "90 is rising" — nonsensical. The problem is that "is 90" and "is rising" make claims at different logical levels:

• "is 90" evaluates the extension at the current index: ${}^\vee({}^\wedge\text{temperature}) = 90$.
• "is rising" predicates change over time of the individual concept: $\text{Rising}({}^\wedge\text{temperature})$.

You cannot substitute the current numerical value into a context that requires an individual concept. A calculus derivative is a useful illustration, but natural-language time need not be continuous and rising need not mean mathematical differentiability.

The Cain/Adam paradox and modal contexts

Consider: "Necessarily, the father of Cain is Adam."

In the actual world, “the father of Cain” and “Adam” may have the same extension while differing in modal profile: a name is often modeled as a rigid designator, whereas a description may vary across worlds. Whether Cain could have had a different father depends on further assumptions about parentage, essence, and existence, so the example illustrates rigid versus non-rigid designation rather than establishing that the sentence is false.

The following is a major extension of the Montague tradition, not part of Montague's original grammar. Our earlier lexicon treats sings as a simple predicate of entities, while event semantics adds explicit event variables (Davidson 1967; Parsons 1990; Kratzer 1996).

• "John sings" (habitual: John regularly sings) vs. "John is singing" (progressive: a singing event is happening now).
• "John walked quickly" — where should quickly attach? It seems to modify the verb, but the standard type of walks ($e \to t$) has no "slot" for manner adverbs.
• "John buttered the toast in the kitchen with a knife" — do we want a separate predicate for every combination of location and instrument? That scales catastrophically.

Neo-Davidsonian Event Semantics (building on Donald Davidson's 1967 proposal) solves all three problems by introducing a new atomic type $v$ for events, alongside $e$ and $t$.

Verbs as event predicates

In a simple neo-Davidsonian entry, a verb denotes a predicate of events:

$$\text{"sings"} = \lambda e.\, \text{Singing}(e) \quad [\text{type } v \to t]$$

A complete clause also needs compositional rules that introduce participants and existentially close the event variable. Participants are linked to events via thematic roles — predicates relating an event to a participant:

• $\text{Agent}(e, x)$ — $x$ is the intentional instigator of event $e$
• $\text{Theme}(e, x)$ — $x$ is the thing affected or moved by $e$
• $\text{Instrument}(e, x)$, $\text{Location}(e, x)$, etc.

Progressive vs. habitual

• Progressive — asserts the existence of a current, ongoing event: $$\text{"John is singing"}: \quad \exists e\,(\text{Singing}(e) \land \text{Agent}(e, j) \land \tau(e) \ni \text{now})$$ where $\tau(e)$ is the run-time of event $e$.
• Habitual — uses a generic operator GEN over a regular pattern of events: $$\text{"John sings"}: \quad \text{GEN}[e]\,(\text{Singing}(e) \land \text{Agent}(e, j))$$ This does not assert that any particular singing event is happening now.

Manner adverbs and the conjunction property

Adverbs attach as conjuncts on the event variable. This is the key insight — it requires no new composition rules:

This elegantly predicts the entailment patterns we observe. If “John walked quickly in the park” is true, then “John walked” follows by dropping conjuncts. Event semantics offers a uniform account of such intersective modification and argument–adjunct entailments.

The liar's paradox and Tarski's undefinability theorem

Consider: "This sentence is false."

Compositionality by itself does not block this sentence. Demonstratives, quotation, and other context-sensitive devices can support self-reference; the paradox arises from the interaction of self-reference, negation, and a truth predicate, not from the order in which a parse tree is assembled.

But we can engineer the paradox with Gödel numbering: encode sentences as natural numbers, find the number $n$ that encodes the sentence "the sentence with Gödel number $n$ is false," and write:

$$\text{"The sentence encoded as } n \text{ is false."}$$

Now the self-reference is indirect, going through arithmetic rather than syntax — and the machinery of Montague grammar cannot rule it out a priori.

Tarski's Undefinability Theorem identifies a precise limit under standard assumptions (Tarski 1936/1983):

An arithmetically sufficiently expressive consistent theory cannot define a truth predicate for its own sentences that satisfies all expected Tarskian biconditionals.

Tarski's object-language/metalanguage hierarchy is one influential response. Montague's original fragment lacks much of the machinery required to formulate semantic paradoxes, but richer extensions can reintroduce them. The theorem is closely related to incompleteness results, but it is not simply a semantic corollary of Gödel's 1931 First Incompleteness Theorem.

Hyperintensionality: the problem with logical equivalence

Possible worlds semantics has a structural flaw. Consider two tautologies:

• $A$: "Green grass is green."
• $B$: "White snow is white."

Both are true in every possible world. In standard intensional logic, two expressions that agree on all possible world/time indices are the same intension — the same function. So $A$ and $B$ have identical meanings. But they clearly don't mean the same thing! Similarly, "groundhog" and "woodchuck" corefer in every world, but "Phil believes groundhogs are cute" and "Phil believes woodchucks are cute" can have different truth values if Phil doesn't know they're the same animal.

This is the problem of hyperintensionality: some semantic contexts are too fine-grained for possible-worlds intensions to distinguish. Two main solutions have been proposed:

1. Structured meanings / structured propositions. Instead of treating a proposition only as a set of possible worlds, give it constituent structure. Different theories take the constituents to be objects, properties, senses, words, or other semantic values—not simply the sentence's parse tree (Cresswell 1985; King 2007). “Green(grass)” and “White(snow)” can therefore differ in structure even when they agree at every world.
2. Impossible worlds. Expand the semantic model with points at which logical or necessary truths may fail. These are commonly formal representational devices, not necessarily concrete worlds that literally exist. They can distinguish attitudes toward necessarily equivalent contents (Priest 1997; Jago 2014).

Looking forward: Dynamic semantics and discourse

Montague grammar is largely sentence-level, but natural language also has cross-sentence anaphora, presupposition, and temporal dependencies. Discourse Representation Theory (DRT) and Dynamic Predicate Logic (DPL) address these by reconceiving meanings as context-change potentials or relations between assignments (Kamp 1981; Groenendijk and Stokhof 1991). Some dynamic systems can be embedded in typed lambda calculi, but their architectures differ; dynamics is not merely an extra layer placed unchanged on Montague grammar.

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