Philosophy Of Language in 9 Sentences

I was recently reading something on Newtonian mechanics and was impressed by how much information about the world could be presented so concisely: in three laws by Newton, and in several pages in the pop science book I was reading. I got to thinking whether the same could be done for my field, analytic philosophy of language, and the below is the result.

There’s a lot of material here. It would be optimistic to think I’ve described it all suitably well that someone without background could understand it all. But I hope even if some of the details elude you, the style of argument, the sort of theoretical considerations, and the basic shape of the discipline will be clear to you if you give this article the fifteen minutes medium estimates it’ll take to read.


One way to cash this out is by means of referential semantics. We say that names(\pronouns) refer to objects, and that verbs(\adjectives\adjectival phrases) refer to sets of objects, namely the set of objects the verb (\etc.) correctly describes (‘run[s]’ refers to the set of things which run). We can then say that a sentence like ‘Obama runs’ is true provided the object is a member of the set referred to by the verb, that is provided Obama is a member of the set of running things.

We can then add a layer of fanciness. Consider ‘it’s not the case that Obama runs’. If we could view ‘Obama runs’ as referring to an object, then we could treat ‘it’s not the case that’ as kind of like a verb(\etc.) in that it describes an object. But what sort of object does ‘Obama runs’ stand for?

Well, we know this: a sentence is true or false. Moreover, at least for the case we’re considering, that seems to be the only relevant thing about it: if we know merely whether or not a sentence s is true or false, we also immediately know whether ‘it’s not the case that s’ is true or false.

So maybe we could say this: a sentence refers to its truth value, which is a weird new sort of object out there in the world. So ‘grass is green’ refers to True, and ‘grass is black’ refers to False. We could then say that ‘it’s not the case that’ refers to the set of sentences refer to False. We can say the same thing with ‘and’ and ‘or’: they describe pairs of sentences, just as ‘loves’ describes pairs of objects.

That already gets us quite far. Having spelled out the details about how transitive verbs like ‘loves’ works we will be able to give a referential semantics for all sentences of the form: ‘Obama runs’, ‘Obama doesn’t run’, ‘Obama loves Michelle’, ‘Obama runs and Michelle doesn’t dance or Malia sings’, and so on.

We’re going to take this model and run with it, by considering how in the course of the 20th century it has been modified and improved to deal with ever greater parts of natural languages.

9 Tricky Sentences

1. Everybody loves somebody (Frege, Begriffsschrift)

Now it’s not massively hard to think about how to generalize that idea to account for words like ‘everybody’ or ‘somebody’. We simply say that something like ‘Everybody golfs’ is true provided every object golfs. Think of it like this: take everything, and go through them one by one. If every object we look at is in the set of golfers, our sentence is true; if not, it is false. The same goes for ‘somebody’: take all the objects, look through them one byone, and if we find one that’s in the set of golfers, our sentence is true.

And it’s not so hard to generalize the account we gave of verbs to account for transitive verbs. We just need to work out what object the sentence is talking about, and how it is describing it. So consider ‘Obama loves Michelle’.

There are two things we could say: we could say that the bit that’s doing the describing is ‘loves Michelle’ and the object described is Obama. Or we could say that the bit doing the describing is ‘Obama loves’ and the object described is Michelle. Either works: they’ll give the same truth condition (think about that for a second to convince yourself).

But when we turn to 1, this ceases to be true. If we say that what’s doing the describing is ‘loves somebody’ and the thing described is ‘everybody’, we’ll be okay. But we could also take the object described as provided by ‘somebody’ and ‘everybody loves’ as doing the describing. But that yields a different reading: it yields one on which somebody is such that everybody loves them. That’s a very popular person.

The crucial point is that depending on how you break the sentence up, you get different readings. Frege introduced a way to talk about such sentences to account for this, the quantifier variable notion. It was part of his project to introduce a scientifically precise language free from the ambiguities of natural language, a project that was very influential at the start of the 20th century. For now, let’s just introduce a bit of the language itself. Consider:

yx Loves(y,x)
yx Loves(y,x)

We read ∃y as saying ‘there exists an x such that’, and we read ∀x as saying ‘for all y’. These two formula correspond to different ways the sentence can be broken up: the first corresponds to the reading on which ‘everybody loves’ is doing the describing bit, while the second corresponds to the other reading.

In doing so, Frege was able to make sense of the fact that our sentence doesn’t have a unique breakdown into object described and description. For now, the key point is simply that Frege realized, and gave a way to deal with, this problem: by associating a sentence with more than one logical formula. We’ll explore what the symbols mean more below, but first let’s consider our second sentence.

(Let me just make something clear: expressions like the above, despite the presence of the English-looking ‘Loves’, are not English. They are a separate, formal language that we’re trying to use to regiment English and reveal its underlying structure. I merely use English-looking expressions like ‘Loves’ to make them easier to read.)

Frege’s 1879 Begriffsschrift introduced many of the themes and problems that remain important today in the philosophy of logic and language. Thankfully the notation has improved since then.

2. The King of Ireland doesn’t exist (Russell, ‘On Denoting’)

But that causes problems. Ireland doesn’t have a king, but the above sentence is true (it sounds a bit unnatural, but you could imagine an ill-informed diplomat saying that he met the kings of Norway, Holland, and Ireland, only to be told ‘you couldn’t have — the king of Ireland doesn’t exist’).

What that means is that we can’t treat ‘the king of Ireland’ as a name, as simply standing for an object, because otherwise the sentence should be meaningless. Russell’s thought was that should treat such sentences as general sentences. In particular, he thought it meant the same as ‘it’s not the case that there is one and only one King of France’. Translated, this looks like (where ¬ translates ‘not’, A→B translates ‘if A then B’ and & translates … well you can probably guess):

¬∃x KingOfIreland(x) & ∀y (KingOfIreland(y) → x=y).

Again, the exact detail of the translation doesn’t matter (though feel free to try to figure out what that means and why). The key point is that by using the sort of logical analysis Frege used, by looking at the sentence and trying to clarify its underlying structure, Russell was able to use the philosophy of language to guard himself against claims that referential semantics led one to postulate all sorts of weird objects, like, for example, round squares (which would be referred to by ‘the round square’). And he extended his theory to account for names that cause similar problems by saying that a name like, for example, ‘Pegasus’ is in fact a disguised definite description.

3. This sentence is false (from antiquity)

The liar sentence, of which 3 is an example, is a good test of philosophical sensibility. You might think it’s a silly trick not worthy of attention, or you might think it’s the sort of thing a life would be well-spent trying to resolve. Personally, it took me about 7–8 years of studying philosophy before I moved from the former to the latter camp.

Whatever your take, its importance in at least mathematical logic can’t be denied: versions of something like it play a role in many of that discipline’s greatest hits, like Goedel’s incompleteness theorem, Tarksi’s indefinability of truth view, and it was Russell’s version of it for sets that put paid to Frege’s goal of showing that mathematics was nothing over and above logic. Moreover, work continues to this day to try to figure out what we should say about it: is the sentence meaningless? Is it both true and false maybe? Or neither? There are many options out there.

4. ∃xGolfs(x) (Tarski, ‘The Concept of Truth In Formalized Languages’)

We said that we could translate a sentence like ‘everybody golfs’ into ∀x Golfs(x). A natural way to understand that is that we translate the ‘everybody’ bit into ∀x and the ‘golfs’ bit into ‘Golfs(x)’.

And that in turn would mean that ‘Golfs(x)’ in our artificial language functions like ‘golfs’ in natural languages, as what I’ve called describing.

But how can this be? Because in our formal language ‘Golfs’ by itself functions as a describing term. Are we to say that both ‘Golfs’ and ‘Golfs(x)’ have the same function despite their different form?

That would be awkward. Its awkwardness can be magnified when we note that expressions containing ‘x’s can be as complicated as sentences. Just as we can have sentences like the below (where v translates ‘or’):

Golf(Obama) & Sings(Obama) v ¬Dance(Obama)

So we can have

x Golfs(x) & Sings(x) v ¬Dance(x)

A moment’s reflection should reveal that expressions-containing-’x’s can have all the same forms that sentences do. That suggests we’re missing an important generalization by treating them as different types of thing.

So can we say they are sentences? Well, that seems difficult, for several reasons. As already noted, we seem to want ‘Golfs(x)’ to function as a describing expression, in particular as referring to the set of golfers. But sentences aren’t describing expressions. Moreover, if it’s a sentence, just what does the x stand for?

So have a problem: we want expressions like Golfs(x) to be both sentence-like and description-like.

Tarski showed how to simultaneously satisfy these desiderata: how a formula like Golfs(x) could be both sentence-like and verb-like. The basic thought is that such formula are sentences, but only one you’ve specified a value of the ‘x’ . They are sentences relative to such a specification. For example, a specification is x=Beto; another is x=Cruz.

We can then say how quantifiers behave: a sentence like ∃x Golfs(x) is true provided provided there is some specification of a value to x that makes Golfs(x) come out true. A sentence like ∀x Golfs(x) is true provided for every specification of a value to x, Golfs(x) comes out true.

To repeat, the clever idea is to introduce a parameter to make formulas like Golfs(x) sentence-like enough to account for their distribution, but property-like enough to reflect how they contribute to the meaning of sentences in which they occur.

(The same thing applies in natural language. The key move here is to treat pronouns like variables, and then roughly the same argument goes through. In something like

Every man believes he is great

We treat — ignoring a few questions — ‘he is great’ as standing for a sentence only relative to a choice of ‘x’.)

5. Everybody golfs (Montague, ‘Proper Treatment of Quantification In Ordinary English’)

Now we know how to interpret ‘John golfs’. It’s true provided the reference of John belongs in the reference of ‘golfs’. Could we possibly say the same thing here?

Well, that seems difficult, because, as we’ve already noted, ‘everybody’ doesn’t stand for any one object, but rather we use it when we want to check whether a bunch of objects — all of them — are as the rest of the sentence describes them to be.

So it seems hard to say that in 5 ‘everybody’ stands for something that ‘golfs’ describes. And so it seems like we’re going to have to give up the prima facie attractive idea that sameness in subject-predicate form results in same analysis.

But, in fact, we don’t. The insight of Frege that was first properly formalised by Montague was that we can treat ‘everybody’ as if it describes the object referred to by ‘golfs’. In particular, we can treat ‘everybody’ as describing descriptions: it truly describes ‘golfs’ provided the property of golfing has the property of being possessed by everyone (just as ‘runs’ truly describes ‘Obama’ provided Obama has the property of running). In set talk, we say that ‘everybody’ stands for a set of sets. A set is in this set provided all men are members of it: that is to say, if it’s a property all men have.

This simple and clever idea enables us to treat, in a sense, ‘everybody golfs’ and ‘Obama golfs’ as involving the same semantic operations, a desirable result in light of their similar form. And it generalizes to more complicated construction, like ‘everybody loves someone’ and ‘every man thinks he is great’ but, with some regret, I will not describe how it does so here.


6. I thee wed (J.L. Austin, How To Do Things With Words)

7. She was rich but nice. (Grice ‘Logic and Conversation’)

8. It’s necessarily the case that 2+2=4 (Kripke, Naming and Necessity)

Well, its use is limited. To see this, consider the above sentence. It’s true, right? If a sentence stands for a truth value, then any two sentences which stand for the same truth value stand for the same thing. Now ‘2+2=4’ is true, as is ‘grass is green’. So they stand for the same truth value. That means that anything that truly describes what ‘2+2=4’ stands for truly describes what ‘grass is green’ stands for, because they stand for the same thing, and so that means that because ‘it is necessarily the case that’ truly describes what ‘2+2=4’ stands for, it also truly describes what ‘grass is green’ stands for.

But it doesn’t, and what that suggests is that the account of sentence meaning according to which sentences just mean truth values is no good.

Now, something like this observation dates back to Frege, and indeed you might have thought even without any fancy philosophizing the meaning as truth value stuff was weird and wrong.

But no one had a really formally good idea to deal with it before Kripke. On Kripke’s view, a sentence has different truth values relative to different ways the world could be (these ways are called ‘possible worlds’). ‘Grass is green’ stands for True relative to our world, but false relative to a possible world in which grass is coal black. ‘2+2=4’, on the other hand, stands for true relative to every world.

The useful idea is that we can treat ‘it’s necessary that’ as, in essence, a universal quantifier that ranges over worlds, and so we can make use of our logical understanding of quantifictional logic to account for things like necessity and possibility and in general all environments like them (which includes our talk about belief, knowledge, duty, and much else).

9. I am here now (Kaplan, ‘Demonstratives’)

Now consider 9. There’s something kind of necessary-truth-like about it. Just as ‘2+2=4’ is always true in the sense that it’s true no matter what world you consider it relative to, so ‘I am here now’ is always true in the sense that, whenever uttered, it’s true.

So is it a necessary truth? No, obviously not. For me to say ‘I am here now’ is to say Matt is in Northern Ireland on 10/11/2018. But that’s not necessarily true. I could easily have been in France, for example.

In order, in part, to account for this notion of being always true when uttered but not necessary, David Kaplan further complicated the Kripkean story. The interesting thing about our sentence is that it contains context-sensitive elements. That is, it contains elements whose reference differs depending on where when and by whom they are uttered. The reference of ‘Obama’ doesn’t depend on where, when, or by whom it is uttered. But ‘I’ does: in my mouth it means me, Matt, in Obama’s mouth it means him, Obama. ‘Saturday 10th November 2018’ doesn’t depend on where, when, or by whom it is uttered, but ‘now’ does. Uttered now, it stands for that Saturday. Uttered next Saturday, it stands for 17th November 2018.

Now recall again Kripke: a sentence doesn’t have a truth value once and for all, but rather has different truth values relative to different worlds. Call an assignment of truth values to worlds like this a Kripke-meaning.

Kaplan’s thought is that an expression has a Kripke-meaning only relative to a given context of utterance. A sentence like 9 is special because, no matter what context it is uttered in, it expresses a sentence whose Kripke-meaning is true relative to the world it is uttered in. This is complicated, and there’s no real way to make it less so — you just have to think about it. But once you’ve read it a few times you might be able to see it, and anyway the key point is that we move from a picture first according to which sentences just mean truth values, to one on which they truth values relative to worlds, to one on which they mean truth values relative to worlds and contexts. The story keeps on getting more complicated, but by doing so it becomes appeal to capture more data about how language works.


I would reiterate that understanding every detail is not completely necessary to benefit from this post: if you understand the basic structure of the theory, and the problems it should be sensitive to, then you now know roughly the aims and methodology of much work in analytic philosophy of language.



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