Formal Semantics 1: Historical Prelude and Compositionality

English is really complicated. For a long time, logicians looking at natural languages believed that there could be no formal system detailing their grammar and semantics. They resigned themselves to extremely simple idealized fragments of English, like propositional logic (formalizing “and”, “not”, and “or”) and first-order logic (formalizing “every”, “some”, and “is”).

The slogan of the time was “ordinary language has no logic” (Bertrand Russell and Peter Strawson). Chomsky famously argued that the languages invented by logicians were too artificial and entirely unlike natural languages, and that therefore the methods of logicians simply couldn’t be applied to this more complex realm.

This attitude has changed over time. Perhaps the most important figure in the “logic of natural language” movement is Richard Montague, a student of the giant of logic Alfred Tarski. The first line of his paper English as a Formal Language reads “I reject the contention that an important theoretical difference exists between formal and natural languages”, and he follows this up by more or less single-handedly invented formal semantics, now a thriving field. Hilariously, Montague apparently saw this work as child’s play, writing:

I (…) sat down one day and proceeded to do something that I previously regarded, and continue to regard, as both rather easy and not very important — that is, to analyze ordinary language.

(This had to hit hard for linguists of his time.)

Alright, enough prologue. In the next few posts I want to describe a naive first pass at formalizing a fairly substantial fragment of English, modeled off of Montague semantics. The key concept throughout will be the notion of compositionality, which I’ll briefly describe now.

Compositionality

Compositionality is all about how to construct the meaning of phrases from their smaller components. Take a sentence like “The cat sat on the mat.” The meaning of this sentence clearly has something to do with the meanings of “the cat” and “sat on the mat”. Similarly, the meaning of “sat on the mat” must have something to do with the meanings of “sat”, “on”, “the”, and “mat”.

The compositionality thesis says that this is all that determines the meaning of “the cat sat on the mat.” In other words, the meaning of any phrase is a function of the meanings of the individual words within it. These meanings are composed together in some way to form the meaning of the sentence as a whole.

The natural question that arises now is, what is the nature of this composition? Take a very simple example: “Epstein died.” According to compositionality, the meaning of “Epstein died” depends only on the meanings of “Epstein” and “died”. That seems pretty reasonable. What about: “Epstein died suspiciously”? How do we compose the meanings of the individual words when there are three?

One proposal is to compose all three simultaneously. That’s possible, but a simpler framework would have us build up the meanings of our sentences iteratively, composing two units of meaning at a time until we’ve generated the entire sentence’s meaning.

Let me now introduce some notation that allows us to say this compactly. If X is some word, phrase, or sentence, we’ll denote the meaning of X as ⟦X⟧. Then the principle of binary compositionality is just that there’s some function F such that ⟦X Y⟧ = F(⟦X⟧, ⟦Y⟧).

There’s two major questions that arise at this point.

First, in which order should we compose our units of meaning? Should we combine “Epstein” with “died” first, and then combine that with “suspiciously”? Or should it be “Epstein” and “suspiciously” first, then that with “died”? Or should we combine “Epstein” with the combination of “suspiciously” and “died”?

One might suggest here that the order actually doesn’t matter; no matter what order we combine the meanings in, we should still get the same meaning. The problem with this is that “The Clintons killed Epstein” has a different meaning than “Epstein killed the Clintons.” If order of composition didn’t matter, then we’d expect these to mean the same thing.

Second, how exactly does composing two meanings work? Is there a single rule for composition, or are there multiple different rules that apply in different contexts? It would be most elegant if we could find a single universal rule for generating meanings of complicated phrases from simple ones, but maybe that’s overambitious.

For instance, you might model the meaning of “died” as a set of objects, namely all those objects that died at some moment in the past, and the meaning of “Epstein” as one particular object in the universe. Then we might have our composition rule be the following: ⟦Epstein died⟧ will be a truth value, and it will be True if and only if the object denoted by “Epstein” is within the set of objects denoted by “died”. So in this framework, ⟦X Y⟧ = True if and only if ⟦X⟧ ∈ ⟦Y⟧.

This works nicely for “Epstein died”. But what about “Epstein died suspiciously”? Now we have two compositions to do, and the order of composition will matter. The problem is that no matter how we compose things, it seems not to work. Suppose that we combine “died” and “suspiciously” first, then combine “Epstein” with that. Using our model, ⟦died suspiciously⟧ will be True if and only if ⟦died⟧ ∈ ⟦suspiciously⟧, which is already a little bit weird. But even worse, ⟦Epstein died suspiciously⟧ will be True if and only if ⟦Epstein⟧ ∈ ⟦died suspiciously⟧. But what would it mean for the object denoted by “Epstein” to be an element of a truth value? It looks like in this framework, most three-word sentences end up becoming vacuously false.

Anyway, the last two paragraphs only show us that one particular attempt to formalize composition fails to be universal. It doesn’t show that it’s impossible in general. In fact, we’ll end up doing pretty well with a small set of composition rules centered around function application. The idea can be very simply phrased as: ⟦X Y⟧ = ⟦X⟧(⟦Y⟧). And in particular, the meaning of “Epstein died suspiciously” will be ⟦suspiciously⟧(⟦died⟧)(⟦Epstein⟧). And that’s enough warm-up! Next we’ll explore this idea further and dive into our Montague-style system.