In the previous research-related post, I’ve introduced the idea that natural language determiners (these little words or phrases that you can put in front of a noun, like a, the, every, three, most, all but one, several, infinitely many…) can be analysed as relations between sets. The reason that I love this intuitively simple idea so much is that it basically turns meaning into a kind of puzzle that doesn’t require a lot of background knowledge – just some basic math skills. So, as a kind of intermezzo, let’s look at the interesting case of every and only, whose meanings don’t seem obviously related in actual language, but turn out to be each other’s exact mirror images when you look at the math behind them. Then, next time you’re at a party and the topic of formal semantics comes up (as it is wont to do) you will be able to dazzle everyone with this super-interesting case study in applied set theory!
To make things more easy, let’s introduce some basic formal notation. (No! you might object, formal notation makes things hard! I think this is a sad misconception rooted in the allergy-inducing way that math is generally taught in secondary school (an allergy that took me years to get over, especially since I was used to people telling me that I wasn’t very good at math… but that’s a different story). Don’t worry, though – I will try to paraphrase all the formal stuff in actual English as well. And if you really don’t like math, just skip the bits in green – they are not essential to the story.)
We will use D as a placeholder for determiners, and A and B as placeholders for sets. In this way, we can represent every English sentence according to the template D(A)(B), where A refers to the set corresponding to the subject noun, and B refers to the set corresponding to the predicate. For example, we can represent the sentence Every student smiles as:
every(the set of students)(the set of smilers)
or, for short:
This representation doesn’t yet tell us anything about the meaning of the sentence: in order to know the meaning, we need to know what kind of relation between sets every denotes. So let’s define this relation:
every(A)(B) = 1 iff A is a subset of B
“The sentence Every A Bs is true if and only if every member of the set corresponding to A is also a member of the set corresponding to B“
So, Every student smiles is true if every member of the set of students is also a member of the set of smilers. (And you can describe the meaning of any determiner in this way, for example:
less than three(A)(B) = 1 iff |A ∩ B| < 3
“The sentence Less than three As B is true if and only if the number of entities that are a member of both A and B is smaller than 3″
half of the(A)(B) = 1 iff |A ∩ B| = |A – B|
“The sentence Half of the As B is true if and only if there are as many members of A that are also members of B, as there are members of A that are not members of B“
This is what I meant when I wrote that this way of looking at determiners turns language meaning into a kind of puzzle.)
Another nice feature of set theory is that it’s very easy to illustrate visually. Like this:
The rectangle represents the universe (the set of all existing entities), the circles represent various sets of such entities. As you can see, in this particular universe, the frogs are a subset of the princes (every entity that’s a member of the set of frogs is also a member of the set of princes), so the sentence Every frog is a prince is true.
Now, on to only. What does the visual representation of Only frogs are princes look like? (Think about this for a second.)
Right. It looks like this:
In order for the sentence Only frogs are princes to be true, every prince must be a frog: the meaning of the sentence rules out the existence of non-frog princes. (It does not care about the existence of non-prince frogs: only frogs are princes does not mean that all frogs are princes.)
So now you see why I claimed that every and only are each other’s ‘mirror images’ in terms of meaning. Here is another way to represent this idea:
Every A is B ↔ Only Bs are A
“The sentence Every A is B is equivalent in meaning to the sentence Only Bs are A“
Which means that, in our official notation:
only(A)(B) = 1 iff B is a subset of A (the reverse of our definition for the meaning of every, above)
Let’s summarise all this. Departing from the idea that the meaning of a determiner can be described as a relation between sets, we looked at the meanings of every and only and discovered that they represent the same relation between sets, only mirrored. And this fun little case study is a real-life example of the way formal semanticists look at language – I could do this stuff all day!
Still hadn’t enough? Read more about the linguistic and set-theoretic properties of only below the cut.
In actual linguistic theory, only isn’t generally analysed as a determiner, because it actually has a much broader use: unlike real determiners like every, it can do things like:
Left-wing politicians only drink tea. (modifying a verb or verb phrase)
Only a left-wing politician could drink this much tea. (modifying a complete noun phrase, i.e. a noun that already has a determiner)
For theoreticians and other people who like neat generalisations, this is actually pretty good news, because if we analyse only as a determiner (like we did in this post), it lacks a particular set-theoretical property that all other natural language determiners do have: conservativity. Formally, we call a determiner conservative if the following equivalence holds:
D(A)(B) ↔ D(A)(A ∩ B)
“If D is a relation that holds between two sets A and B, then D also holds between the set A and the intersection of A and B, and vice versa”
Say that D is the relation every, A is the set of frogs, and B is the set of princes. The above definition tells you that every is conservative if:
every frog is a prince ↔ every frog is both a frog and a prince
Or, if D is exactly two, A is the set of semanticists and B is the set of sleepers:
exactly two semanticists sleep ↔ exactly two semanticists are semanticists that sleep
It isn’t hard to see that the equivalence holds in both cases – in fact it seems rather trivial. (So trivial, in fact, that it took me a while before I understood why people even bothered talking about it.) So let’s try a slightly different way to put it into words. The main thing that the property of conservativity tells us is that in order to determine whether a relation D holds between two sets A and B, we only have to look at the elements that are in A – the non-A’s are irrelevant. So in order to determine whether Every frog is a prince is true, we don’t have to consider anything that isn’t a frog.
By now you’ve probably already realised that only does not have this property. In order to determine whether Only frogs are princes is true, we do have to look at the non-frogs. If there exists even one non-frog that is a prince, the sentence is false. Note also that the following two sentences are not equivalent:
Only frogs are princes <-/-> Only frogs are both frogs and princes
The statement on the right follows from the one on the left, but not vice versa: only frogs are both frogs and princes does not preclude the existence of non-frog princes.
To sum up this part: only isn’t conservative, but it also isn’t a real determiner. This allows us to maintain the generalisation that all determiners in natural language have the property of conservativity. Why is this the case? That’s one of the remaining mysteries of natural language semantics – if you have a suggestion, let me know.
(Want to know more? This textbook chapter by Edward Keenan contains a full introduction to what is called generalised quantifier theory.)