What’s In A Name?

Most of us have this intuitive idea that two equal things can substitute each other without consequence.¹ To show what I mean, consider Vladimir Ulyanov, better known to the word simply as “Lenin”.

1. true Vladimir Ulyanov = Lenin.
2. true Ulyanov [ was the first Premier of the Soviet Union ].
3. true Lenin [ was the first Premier of the Soviet Union ].

Ulyanov and Lenin are the same person, therefore I can always swap them out. And actually, we just found a new “name” for Lenin — “the first Premier of the Soviet Union”.

4. true Lenin [ founded Iskra ].
5. true The first Premier of the Soviet Union [ founded Iskra ].

Of course, Lenin might not have been the leader he was — perhaps Julius Martov could have become the leader of the Soviets.

6. true Ulyanov might not have been the first Premier of the Soviet Union.
7. false Ulyanov might not have been Ulyanov.

Awesome… we broke it…

In this post, let’s look at exactly how (7) just broke; was “the first Premier” just a bad name, or do our problems go deeper? It might seem like a pedantic question, but if we want to understand or model language, equality would be a pretty important thing to nail down!

References

The technical term we’re discussing is reference — a word (or words) that “point” to something (if you have a CS background, your intuitions might be useful here). Broadly speaking we could break references into two categories: names and descriptions.

The important note here is that references are not the things themselves. If the word “rose” were a rose, then a rose by any other name would not be a rose. And that’s the root of our problems — the space between our words, and the world we’re describing.

Linguists, philosophers, and logicians have had a knot in their stomachs about reference for ages, but their conversations are not always the most accessible. It doesn’t need to be that way, though; a few good examples can demonstrate just how sticky the topic is, and I think Quine (1956) did a great job providing some.

He breaks down the usage of references into two categories: well behaved transparent references, and the more problematic opaque references.

So, when do opaque references occure?

Modal

Sentences with worlds like “must”, “may”, “could”, and “might” are called modal sentences. The modern way to think about modality is with two operators: necessity, and possibility.²

Necessity \(\square P\)

In all possible words, \(P\) must be true.

Possibility \(\lozenge P\)

There exists a word in which \(P\) is true.

We actually already saw an example of a modal reference with (7), and could express it using “\(\lozenge\)”.

\[ \begin{align*} &\text{Ulyanov} = \text{First Premier} & \text{true}\\ \lozenge & \text{Ulyanov} \not= \text{First Premier} & \text{true}\\ \lozenge & \text{Ulyanov} \not= \text{Ulynov} & \text{false}\\ \end{align*} \]

Written out explicitly like this, it’s more clear exactly how (7) failed. References seem to operate slightly differently within the scope of “\(\lozenge\)”.³

Here’s another example, this time using necessity:

8. true The number of English letters is 26.
9. true Necessarily 26 should be 26.
10. false Necessarily the number of English letters should be 26.

Now, written with our \(\square\) operator:

\[ \begin{align*} n \, \text{letters} &= 26 &\text{true}\\ \square 26 &= 26 &\text{true}\\ \square n \,\text{letters} &= 26 &\text{false} \end{align*} \]

Again we see that “the number of English letters” under the scope of “\(\square\)” can refer to different numbers, whereas the name “26”, in all possible words, can only refer to the number 26.

Propositional Attitudes

Propositional attitudes get at the relationship between some proposition \(P\) and another person. For example:

11. Li Na claims [ she doesn’t know him ].
12. Dolores believes [ they are seeing each other ].
13. He hopes [ they will get a divorce ].

None of these sentences say that the proposition in brackets is true or false — they only say what the members of this love triangle say, think, and want.

Right off the bat, we can see how such sentences have an issue with references:

14. true Jozefína thinks the number of English letters is 46.⁴
15. false Jozefína thinks 26 is 46.

In (14), Jozefína simply doesn’t know how may letters English speakers use. That certainly doesn’t imply that she is totally ignorant of numbers!

Names Fails As Well

Propositional attitudes work just as bad with names as they do with definite descriptions. Consider Omar, who has a test in his Russian history class. He knows there will be an essay question, and he hopes it will be about Lenin. He does not know, however, that Lenin’s birth name was Ulyanov.

16. true Omar hopes he will write about Lenin.
17. ??? Omar hopes he will write about Ulyanov.

(17) is not true, in the sense that, when Omar sees an essay question about Ulyanov, he will probably panic, walk out of the room, and change majors.

It gets more complicated, though. Let’s say Omar’s parents (both history professors) are talking over breakfast. Omar’s mother says “Omar hopes the essay is about Ulyanov”. Omar’s father, knowing who Ulyanov is, would declare this sentence to be true!

Not Just Noun-Phrases

Everything we’ve done so far has been a noun-phrase, but consider my baby cousin Grace, age two.

18. true Grace knows people have two legs.
19. true “Being bipedal” means “having two legs”.
20. ??? Grace knows people are bipedal.

Grace may know people have two legs, so in some sense (20) is a true statements. That being said, grace is a smart kid, but I wouldn’t put a lot of money on her coming up to me any time soon and saying “people are bipedal”. So, in another sense, (20) is false.

Possible Worlds

Quine didn’t have the tools he needed to understand these sentences, but we’re a luckier bunch. Here’s my claim about the difference between transparent and opaque references:

That might be a little abstract, so here are some examples of what I mean:

Once we make this revelation, it starts to become clear what was happening all along: references resolve within the context of a particular world \(w\), and that world is not always the one in which we are living; it can be the one about which we are speaking.

A Computational Connection

References are essentially a kind of program — call it ref. If I say the name “Le Guin”, I’m asking you the listener to bring to mind the person who has that name. (ref "Le Guin"). Similarly, if I say “the author of The Dispossessed”, I’m asking you to bring to mind the person who fits that description. (ref "The author of The Dispossessed"). I can only assume that, in your mind, those should be the same individuals.

But the result depends which “machine” runs ref! Remember Jozefína?

You-Machine> (ref "Num English Letters")
  26
Jozefína-Machína> (ref "Num English Letters")
  46

The programing language Lisp is famously able to “talk about programs”.

In natural language, we also can easily switch between talking with words (known as de re statements) and talking about words (known as de dicto statements). If I want to say “Jozefína thinks the number of English letters is 26”, there are two interpretations

;; De Re Interpretation
;; (no normal speaker would parse this way).
(eq (eval-jozefína (ref "English Letters") 26)
;; De Dicto Interpretation
;; (the intended meaning).
(eq (eval-jozefína '(ref "English Letters") 26)

De dicto literally means “about words”, so a de dicto reading means “this person thinks/wants/believes these exact words”.

De re means “about the thing”, so a de re reading would be “this person thinks/wants/believes the things these words mean”.

Scope Is Always Tricky

Natural language, unlike Lisp, doesn’t have clear scope rules. If I say “all students talked about a move”, you could take that in one of two different ways:

• \(\forall k\), \(\exists m\), \(k\) talked about \(m\). (Each kid talked about a movie.)
• \(\exists m\), \(\forall k\), \(k\) talked about \(m\). (There is a movie that all kids talked about.)

The way we manage this kind of ambiguity is ultimately a question of psychology — it’s contextual⁵. And, as we’ve described it, we’ve made reference a problem of scope ambiguity. When you hear a reference, you mind must decided, “was that reference de dicto, or de re?” For example:

21. Gloria thinks someone here is a murderer.

When you hear this sentence, there are two totally legitimate ways to interpret it.

Let’s drive this home with one last example. Consider Antonio and Yolanda, who are visiting an art show. Yolanda points to one painting in particular, and says:

22. That painter is amazing!

Later that night, Antonio says to his sister:

23. Yolanda thinks the girl down the street is an amazing painter.

Antonio “translated” the reference “that painter” (which Antonio’s sister wouldn’t understand) to “the girl down the street” (which Yolanda wouldn’t understand). Antonio’s sister hears the sentence as:

(think yolanda (is (ref "the girl down the street") amazing-painter))

And because Yolanda doesn’t know who the girl down the street is, she would not likely think:⁶⁷

(think yolanda '(is (ref "the girl down the street") amazing-painter))

We could also think back to (17) — the de dicto reading is false, because Omar didn’t know who Ulyanov was. However, when his mother made the same statement to a different audience, it was received with a de re meaning, and was therefore true.

Parting Notes

I hope you enjoyed following along with me on my topic of the week. Referential Opacity can seem a little abstract; historically the main problem had always been that modeling language would never be possible without understanding reference. Quine (1956) is a remarkably entertaining paper, and I would highly recommend it (it’s a short read, full of great examples)! It and Hall-Partee (1979) do a good job laying out why the problem matters.

One critical piece of required reading here that has been left unmentioned is Kripke (1980) — “Naming and Necessity” is the book on reference, which servers as the modern foundation of the problem in Philosophy, Language, Logic, and anyone else who may be interested. Before Kripke, this was considered a massive open problem. It’s considered to be required reading in these fields. The computation connections are my own intuition, but they largely shadow the thinking of Sual Kripke.

References