Ray: How many planets are there?
Dick: I think its ten.
Ray: The number of planets is ten?
Dick: Right. I think. But it might not have been, you know. It might have been almost anything.
Ray: You mean, there might have been some other number of planets?
Dick. Right. The number of planets isn't necessarily ten.
Ray: No, wait. The number of planets is necessarily ten.
Dick. You crazy SOB, you just agreed there could have been any number of planets.
Ray: You should never call your brother a SOB. It doesn't matter what could have been. Fact is, the number of planets is ten, and so it follows that the number of planets is necessarily equal to ten.
Dick: You just contradicted yourself.
Ray: No, I didn't. You just don't understand logic, is the problem.
Dick. Logic shmogic. Look, you just said that it was necessarily true that the number of planets was ten...
Ray: No, I didn't say that.
Dick: Yes you did.
Ray: Didn't.
Dick: Did.
Ray: Didn't.
Dick: Well, what did you say, then?
Ray: I said that the number of planets was necessarily equal to ten.
Dick: That's exactly what I said, and you said you didn't say it !!
Ray: No, it's not the same at all.
Dick: What's the difference?
Ray: It might be clearer if I were to say it formally.
Dick: I knew this was going to happen.
Ray. Take off your shades for a minute. Now, look. You said this wasn't true, and I agree its not:
Necessarily(number-of-planets = 10).
What I said was this:
(lambda x (Necessarily(x= 10)))(number-of-planets).
Dick. How do you make that out? I didn't hear you say 'lambda' anywhere.
Ray: What I said was, that a number had a property. If I say that Mother is a lady, I'm saying that a thing, called Mother ...
Dick: Watch your mouth.
Ray: ... a person, called Mother, has the property of being a lady. Ok, so, listen carefully. The number of planets is... I havn't used the 'necessary' word yet, right? So I'm talking about a number, which is the actual number of planets, which is... ten. You said that yourself, the number of planets is ten, right?
Dick: Right, but it might not have been.
Ray: Never mind what might not have been, I'm talking about what is true. It is ten, as a matter of fact, which is all that I'm saying. And ten couldn't possibly be any other number than ten, right?
Dick: Right, ten couldn't be ...
Ray: Well, there you go, that's what I said. The number of planets - that is, ten - is necessarily ten. Ten is necessarily ten.
Dick. Ten is necessarily ten, but the number of planets isn't necessarily ten.
Ray: You just contradicted yourself.
Dick: Didn't.
Ray: Did.
Dick: Didn't.
Ray: Enough with the dids. Look, ten is the number of planets. So when you say 'ten', and when you say 'the number of planets', you are referring to the same thing. There is only one number we're talking about, however we refer to it. Roman X, decimal 10, binary 1010, English ten, the number of planets: these are all just different ways of talking about the same number. And what I said about that number, was that it is necessarily ten. Ten is necessarily ten, or the number of planets is necessarily ten, are just two ways of saying the same thing. You can't say one is true and the other not.
Dick: Look, ten is necessarily ten, I'll grant you that; but that doesn't mean that the number of planets is necessarily ten.
Ray: Sure it does. If being-necessarily-ten is a property, and its true of ten, then its true of the number of planets, because the number of planets is ten. There's only one number we are talking about here, so if it has that property then it has it. It can't both have and not have the same property.
Dick. Hmm. Seems to me that you are cheating by sneaking the 'necessarily' into a property. When I said that the number of planets wasn't necessarily ten, what I meant was that it wasn't necessarily true that the number of planets was equal to ten...
Ray: Well then you should say what you mean, because those are very different things to be saying.
Dick: Seems to me they are the same. In fact, you want formal? I can show it formal. Suppose that
not Necessarily(number-of-planets = 10)
then use lambda-abstraction:
(lambda x (not Necessarily(x = 10)))(number-of-planets)
and you can go the other way by using lambda conversion.
Ray: But there is obviously something wrong here, because the first one is true but the second one is false. Just go on with the derivation:
(lambda x (not Necessarily(x = 10)))(10)
because number-of-planets is ten, after all, and then do the application, and you get
not Necessarily (10 = 10)
which is obviously not right. You made a mistake back there when you did the abstraction, because you moved the phrase 'number of planets' from inside a Necessarily to outside it. That changes the meaning.
Dick: It was you that started talking about 'necessary properties', and that seems to be where the problem is coming from. If I just stick to using 'necessarily' in sentences, it all makes perfect sense. If that thing is a property, why doesn't lambda abstraction work on it?
Ray: It was you that said the number of planets is necessarily equal to ten. That's a statement about a property. You didn't say "Necessarily, the number of planets equals ten".
Dick: So now I can't even talk English when I speak to you? Necessarily equals, equals necessarily, what kind of hairs do you split on Saturdays? Look, go back to the example and do the abstraction this way:
not Necessarily(number-of-planets = 10)
(lambda x (not Necessarily(number-of-planets = x)))(10)
then there's no problem. According to you, it shouldn't matter what name we use. But it does.
Ray: It doesn't matter what name you use outside the scope of Necessarily. Of course it matters inside the scope. Ten is always ten, but the number of planets might not have been ten. So that phrase "the number of planets" is sensitive. The properties of necessarily-being-ten, and necessarily-being-the-number-of-planets, aren't the same property.
Dick: Sensitive? What are you, some kind of poet now?
Ray: How did I get you for a brother? What I mean is, phrases like that have to be treated differently inside the scope of Necessarily from the way they are outside the scope. You agree with that, surely? That's why it isn't necessarily true that the number of planets is ten, even though the number of planets is in fact ten. It's because that phrase, "The number of planets", might have meant a different number. Like, if there were two more planets than there are, it would have meant twelve. But that's not a statement about the actual numbers. The numbers are necessarily what they are.
Dick: Sure sounds to me like "the number of planets" refers to a number. Just look at it, it actually says "the number of..". If its not about a number, what else is it about?
Ray: The cardinality of the set of planets, maybe? Anyway, its at least partly about the planets, and its the planets that might have been different, not the numbers.
Dick: Well, you are the hairsplitter. So lets say it is about a relationship between the planets and a number, is that OK with Mr. Pedantic?
Ray: Good enough for a conversation with you, yes.
Dick: I should have let you drown in that pond when you were three.
Ray: Tell you what, let's list the things we agree about. We agree that
number-of-planets = 10
not Necessarily(number-of-planets = 10)
Necessarily(10 = 10)
Necessarily(number-of-planets = number-of-planets)
Right?
Dick: Right, but I'm not sure I trust you. Speak very slowly.
Ray: Now I'm going to mention prop-er-ties.
Dick: Already I'm suspicious.
Ray: Just listen carefully. Now, it is OK to use lambda abstraction provided that we don't move any phrases whose reference might change when they go in or out of a modality. So these are right:
(lambda x (x = 10))(number-of-planets)
(lambda x (not Necessarily(number-of-planets = x)))(10)
(lambda x (Necessarily(x = 10)))(10)
but this isn't:
(lambda x (not Necessarily(x = 10)))(number-of-planets)
Dick: So, according to you, it is OK to say that ten is not necessarily the number of planets, but not to say that the number of planets is not necessarily ten?
Ray: Oh, well done. I knew I'd get it into your thick skull eventually. Being necessarily ten is one thing, being necessarily the number of planets is something else.
Dick: Sheesh. Seems like quartering hairs now, not just splitting them. Why bother? What use is all this care about what is necessarily what? Take that last example, is there anything that is necessarily the number of planets?
Ray: Well, er... no. There isn't.
Dick: I mean, according to you, even the number of planets wouldn't hack it, right? Because it - that number, ten - might not have been the number of planets.
Ray: Right. So that is an empty property that is always false.
Dick: Oh, whoopee. That is really a lot of use, that is. I've always wanted to have a good stock of empty properties, it reminds me of uncle Hiram's rental business.
Ray: OK, OK, I'm not saying it's useful, I'm saying it's right. They are different properties, if only because the first one is true of ten and the second one isn't true of anything. And at least this keeps all the rules straight and meaningful. Whereas with your sloppy mouth, all you have to do is say 'necessarily' anywhere in a sentence and all the logic is shot.
Dick: Nonsense. All we have to do is stop trying to talk Greek. Properties are for talking about things in the world, and necessity is for talking about different ways the world might have been. "Necessarily" belongs in sentences, and stop saying 'lambda'. Seems to me that if you just stuck to sentences and asked if they were true or false, instead of thinking so much about what things phrases referred to, you wouldn't get into such a muddle.
Ray: But the only way to make sense of sentences is to ask what they say about the relationships between the things referred to by phrases. Things are what the world is made of, after all. And it doesn't seem like a muddle to me, it seems elegant and clear.
Dick: Maybe we should just agree to disagree. You focus on the things sentences talk about, and I'll focus on how things are said in language. We might come to different conclusions at times, but at least each of us won't get so muddled.
Ray: Until we try to talk to one another.
Dick: Well, lets try not talking to one another for a while. Then maybe I can get some extra sleep.
Ray: OK, but I tell you what, let's name the styles after ourselves. I'm going to talk de Ray from now on.
Dick: And I'm going to talk de Dickto.
Ray: Why not de Dick?
Dick: Who wants to sound like a Dick?

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