So far, we have looked at types from several angles. Types describe which inputs a program accepts, reveal algebraic structure, and even express propositions whose values are proofs. We will now use types to look at something less obviously mathematical: natural language.
Richard Montague's central idea was that ordinary language could be studied with the same precision as a formal language. We can give words and phrases mathematical meanings, give those meanings types, and specify rules for combining them. The meaning of a sentence is then determined by the meanings of its parts and the way those parts are put together. This is the principle of compositionality.
The comparison with typed programming will often help us: expressions have types, and they can combine only when those types fit. Still, English is not a programming language. The analogy is a way to understand the machinery, not Montague's claim.
We begin with two basic semantic types:
- $e$: entities — Samson, Doherty, your left shoe.
- $t$: truth values — True or False.
A name such as Samson seems, at first, to pick out an entity. A complete declarative sentence such as Samson sings is either true or false. These observations motivate the types $e$ and $t$.
As with function types in programming, if $a$ and $b$ are types, then $a\to b$ is the type of functions that take an $a$ and return a $b$. Consider the verb phrase sings:
$$\llbracket\text{sings}\rrbracket=\lambda x.\operatorname{Sing}(x):e\to t$$The brackets $\llbracket\cdot\rrbracket$ mean “the semantic value of.” This function takes an entity $x$ and returns True exactly when $x$ sings. In the same way, the common noun student denotes a function that tests whether an entity is a student:
$$\llbracket\text{student}\rrbracket=\lambda x.\operatorname{Student}(x):e\to t$$Both meanings have type $e\to t$. We call values of this type properties. A property can also be viewed as a set: the property $\lambda x.\operatorname{Sing}(x)$ corresponds to the set of all entities that sing.
Combining a name and a property
Let $\mathsf{samson}:e$ be the entity Samson. Applying the meaning of sings to $\mathsf{samson}$ gives:
$$(\lambda x.\operatorname{Sing}(x))(\mathsf{samson})\xrightarrow{\beta}\operatorname{Sing}(\mathsf{samson}).$$The result has type $t$. We have turned the sentence into a truth condition: it is true exactly when Samson sings. This is our first example of compositionality.
Our first analysis of Samson sings treated the subject as an entity. That strategy cannot handle Every student sings: the phrase every student does not name one particular entity. Instead, it says something about how two properties are related.
The determiner every first takes the property of being a student and then the property of singing. It returns True when everything with the first property also has the second. More generally, a determiner relates a restrictor property $P$ to a scope property $Q$:
| Expression | Denotation | Type |
|---|---|---|
| every | $\lambda P.\lambda Q.\forall x(P(x)\to Q(x))$ | $(e\to t)\to((e\to t)\to t)$ |
| a | $\lambda P.\lambda Q.\exists x(P(x)\land Q(x))$ | $(e\to t)\to((e\to t)\to t)$ |
Applying a determiner to a common noun produces a generalized quantifier (GQ). A generalized quantifier has type $G=(e\to t)\to t$: it takes a property and returns a truth value. For example, every student takes the property of singing and says whether every student has it.
Worked derivation: “Every student sings”
The semantic applications mirror the syntactic combination of determiner with noun, and then noun phrase with verb phrase.
Making all noun phrases behave uniformly
We now have two apparent kinds of subject: Samson has type $e$, while every student has type $G$. Montague's PTQ avoids needing two different subject–predicate rules. It assigns a proper name a generalized-quantifier meaning directly:
$$\llbracket\text{Samson}\rrbracket_{\mathrm{PTQ}}=\lambda P.P(\mathsf{samson}):G.$$This function takes a property $P$ and returns True when Samson has that property. Applying it to sings gives:
$$(\lambda P.P(\mathsf{samson}))(\lambda x.\operatorname{Sing}(x))\xrightarrow{\beta}\operatorname{Sing}(\mathsf{samson}).$$Now Samson and every student have the same semantic type, so the same composition rule handles both sentences.
Optional note: deriving the same name meaning by type lifting
It is pedagogically natural to begin with $\mathsf{samson}:e$ and convert it into a generalized quantifier:
$$\operatorname{LIFT}=\lambda x.\lambda P.P(x):e\to G,$$ $$\operatorname{LIFT}(\mathsf{samson})\xrightarrow{\beta}\lambda P.P(\mathsf{samson}).$$This later type-shifting analysis reaches the same denotation, but PTQ itself assigns the higher noun-phrase denotation directly.
Samson finds a unicorn.
Samson seeks a unicorn.
If the first sentence is true, then a unicorn must exist: Samson cannot find something that does not exist. The second sentence is different. Samson may search for a unicorn despite there being no unicorns at all. Yet both sentences have the same visible subject–verb–object shape.
The phrase a unicorn is another generalized quantifier:
$$A\text{-}\operatorname{UNICORN}=\lambda P.\exists u(\operatorname{Unicorn}(u)\land P(u)):G.$$It takes a property $P$ and says that some unicorn has that property.
Why the extensional analysis works for “find” but fails for de dicto “seek”
Combining the GQ with the property of being found by Samson gives:
$$A\text{-}\operatorname{UNICORN}(\lambda u.\operatorname{Find}(\mathsf{samson},u))\xrightarrow{\beta}\exists u(\operatorname{Unicorn}(u)\land\operatorname{Find}(\mathsf{samson},u)).$$This correctly entails existence. Replacing Find with Seek would incorrectly force the same commitment on the de dicto reading. The extensional fragment is too coarse.
The reading on which no actual unicorn need exist is called the de dicto, or notional, reading. There is also a de re, or objectual, reading on which there is a particular unicorn that Samson seeks. We will first explain the de dicto reading, because it exposes exactly what our current system cannot represent.
Montague's solution lets seek relate Samson to the intension of the entire generalized quantifier “a unicorn,” rather than to an actual unicorn. To understand that sentence, we first need to understand what an intension is.
Our first model describes only how things actually are. In that model, if there are no unicorns, then the phrase a unicorn has an empty extension. But seeking is concerned not only with what exists here and now; it can be directed toward how the seeker wants, imagines, or tries for things to be.
To model this, semantics considers different possible worlds: complete ways the world might have been. This does not require believing that these worlds physically exist. They are points in a mathematical model that let us ask how the value of an expression changes as circumstances change. Montague's full system also tracks time, so we will use index as a general word for a world-and-time point.
- The extension of an expression is its value at one index.
- The intension of an expression tells us its extension at every index. Mathematically, it is a function from indices to extensions.
We use $s$ for the type of indices. If an expression $\alpha$ has type $a$, then its intension, written $\mathord{\wedge}\alpha$, has type $s\to a$.
A first intuition: temperature
The temperature is 90, but it is rising.
At the current time, the extension of the temperature may be the number 90. But 90 itself is not rising. What rises is the temperature considered across times: its intension. “Is 90” uses the present value, while “is rising” needs access to how that value changes.
Returning to the unicorn
Let $G=(e\to t)\to t$. The phrase a unicorn has type $G$, so its intension has type $s\to G$:
$$\mathord{\wedge}A\text{-}\operatorname{UNICORN}:s\to G.$$This intension tells us, at each index, which properties at least one unicorn has there. It preserves the difference between unicorns and centaurs even if neither happens to exist in the actual world.
To keep the main idea visible, we use a simplified type that suppresses some of PTQ's additional formal machinery. Here seek accepts Samson's individual concept and the intension of the object generalized quantifier:
$$\operatorname{Seek}:(s\to e)\to((s\to G)\to t),$$ $$\operatorname{Seek}(\mathord{\wedge}\mathsf{samson})(\mathord{\wedge}A\text{-}\operatorname{UNICORN}):t.$$The second argument is not an “intensional unicorn.” It is the intension of the whole phrase a unicorn.
Type-check the de dicto reading and locate the existential
The second argument is a complete function from indices to generalized quantifiers:
$$\mathord{\wedge}A\text{-}\operatorname{UNICORN}=\lambda w.[\lambda P.\exists u(\operatorname{Unicorn}_w(u)\land P(u))].$$The existential is inside the value delivered at each index. Exposing it would require applying the intension to a world and the resulting GQ to a property: $$(\mathord{\wedge}A\text{-}\operatorname{UNICORN})(w)(P)\to\exists u(\operatorname{Unicorn}_w(u)\land P(u)).$$ But seek receives the whole $s\to G$ object and returns a truth value; it performs neither application. No actual-world existential follows.
Why find still entails existence
PTQ allows transitive verbs to combine uniformly with the same abstract kind of object. It then uses a meaning postulate to constrain the meaning of find. A meaning postulate is a condition that acceptable interpretations of a word must satisfy. For find, the condition connects the abstract object back to an actual individual, producing the familiar result:
$$\exists u(\operatorname{Unicorn}^*(u)\land\operatorname{Find}^*(\mathsf{samson},u)).$$The stars indicate extensions at the current index. The important lesson is that existence follows from the particular meaning of find, not just from its syntactic category.
Every student has a teacher.
This sentence can describe two different situations. Perhaps each student has a teacher, but different students have different teachers. Or perhaps there is one teacher shared by every student. We record the two readings by writing which quantifier has wider scope:
- $\forall>\exists$: each student may have a different teacher.
- $\exists>\forall$: one teacher is shared by every student.
Compare the two derivations
Build has a teacher first, then apply every student:
$$\forall x(\operatorname{Student}(x)\to\exists y(\operatorname{Teacher}(y)\land\operatorname{Has}(x,y))).$$For inverse scope, begin with “Every student has him$_1$” and quantify in “a teacher”:
$$\exists y(\operatorname{Teacher}(y)\land\forall x(\operatorname{Student}(x)\to\operatorname{Has}(x,y))).$$PTQ's distinctive claim is that these readings arise from different derivations. The first is built in place. For the second, the object quantifier is introduced from outside by a special rule called quantifying in. The history of how the sentence is assembled therefore helps determine its meaning.
Consider the inference from “Samson sang loudly” to “Samson sang.” Donald Davidson proposed that action sentences implicitly quantify over events:
$$\text{Samson sang loudly: }\exists e(\operatorname{Sang}(\mathsf{samson},e)\land\operatorname{Loud}(e)).$$The adverb adds a condition on the same singing event. Dropping the $\operatorname{Loud}(e)$ conjunct immediately gives the weaker claim that Samson sang.
Later neo-Davidsonian analyses separate the kind of event from the roles played by its participants:
$$\exists e(\operatorname{Singing}(e)\land\operatorname{Agent}(e,\mathsf{samson})\land\operatorname{Loud}(e)).$$Here $\operatorname{Agent}(e,\mathsf{samson})$ says that Samson is the agent of the singing event. This decomposed formula comes from later work rather than Davidson's original proposal. Event quantification alone also does not explain the difference between habitual “Samson sings” and progressive “Samson is singing”; temporal and aspectual theories supply additional machinery.
The basic typed derivations follow the accessible reconstruction in the Stanford Encyclopedia of Philosophy, especially §§2 and 4. The treatment of proper names, find, seek, intensions, and quantifying in is based on Montague's PTQ, with Dowty, Wall, and Peters used for exposition. The optional lifting discussion comes from later work by Partee and Rooth. The final section draws separately on Davidson and Parsons.
- Davidson, Donald. 1967. “The Logical Form of Action Sentences.” In The Logic of Decision and Action, 81–95.
- Dowty, David R., Robert E. Wall, and Stanley Peters. 1981. Introduction to Montague Semantics. Reidel.
- Montague, Richard. 1970a. “English as a Formal Language.”
- Montague, Richard. 1970b. “Universal Grammar.” Theoria 36: 373–398.
- Montague, Richard. 1973. “The Proper Treatment of Quantification in Ordinary English.” doi.
- Parsons, Terence. 1990. Events in the Semantics of English. MIT Press.
- Partee, Barbara H. 1987. “Noun Phrase Interpretation and Type-Shifting Principles.”
- Partee, Barbara H., and Mats Rooth. 1983. “Generalized Conjunction and Type Ambiguity.”
- Stanford Encyclopedia of Philosophy. “Montague Semantics.” plato.stanford.edu.