Monty's Unicorns, Fermat version

[Nick Nicholas <opoudjis@optushome.com.au>]

This is rushed, because I do have work to do. I may come back with a 
fuller version later --- or I may just abandon Lojban. We'll see.

Property is defined here Monty style, as an intension: a function 
mapping indices (world + time) to things. This allows its extension 
to vary according to what planet you're on. (Today's Prez is not the 
same as the Prez in 1880 --- or in the Uchronia where the South won.) 
Property is the mapping of indices to one-place preds, which are 
taken to denote sets of individuals.

So the property {ce'u merjatna} is actually \l<w, t>.\lx.Prez(x)<w,t>.
In 2000, {ce'u merjatna} denotes {W}
In 1996, it denotes {Big Dog}
In 2000 in the world The South Rose Again, it denotes {Jefferson Davis VI}
In 2000 in the world Dummvirates Are Allowed, it denotes {Gore, Bush}

The intension of a is ^a. The reverse, the extension of a, is \va. 
So, [[merjatna]] (the current denotation of "merjatna"} is {W} in 
2000, and {Big Dog} in 1996. ^[[merjatna]] is the function which, 
given world and time, gives you the current denotation: it is the 
property, rather than the one-place predicate.

You seek a president.
Do you seek the property {ce'u merjatna}?

No you don't. Lojban, welcome to the world We Fucked Up Because We 
Didn't Know Enough Logic.

The reason you don't is that you can quantify what you seek. Let's go 
to senators.

I seek a senator: mi sisku leka ce'u turnrsenatore
I seek two senators: mi sisku leka ce'u turnrsenatore
I seek every senator: mi sisku leka ce'u turnrsenatore

... Oh shit, you might well say. Well, obviously you gotta quantify 
the senators, right? OK, let's see you try.

mi sisku ro ce'u turnrse--

"What the hell are you doing?"

I'm, uh, quantifying ce'u?

"Let's see you do that in lambda, with the quantifier.

OK, mi sisku \lx.senator(x)...

mi sisku \lx\Ax.senator(x)

Oh. I see. If x is a variable, I can't go lambdaing it: it's already 
bound. \lx picks out individuals meeting a criterion; it's nonsense 
to turn around and say, all of them meet the criterion. I might be 
able to massify them into a bundle and start speaking of that...

"And you're not going to. Concentrate. You seek all senators. \Ax is 
non-negotiably part of that expression."

Right.

"And you need something to attach \Ax to, right?"

Eh, Ax.senator(x)...

"You're back to de dicto. You still need an intension."

Here, Monty pulls a swifty.

A one-place pred is something that needs an entity to yield a true or 
false value: [[merjatna]] is \lx.merjatna(x) . Satisfy the x, and you 
have a truth claim: merjatna(W), merjatna(Big Dog), merjatna(Fred).

Monty make a noun be not an entity, but something that needs a 
one-place predicate to yield a truth value. John is \lP.P(j).

In simple cases that's fine. "John walks" is

{\lP.P(j) } {\lx.walks'(x)}
= {\lx.walks'(x)}(j)
= walks'(j)

Quantified common nouns follow the same template, with the 
quantification wedged in further in than the predicates applying to 
it:

some man = \lP.\Ex.man(x) & P(x)
every man = \lP.\Ax.man(x) -> P(x)

In first-order contexts, these too combine fine:

"some man walks"
{\lP.\Ex.man(x) & P(x)} {\ly.walks'(y)}
= \Ex.man(x) & {\ly.walks'(y)} (x)
= \Ex.man(x) & walks'(x)

"every man walks"
{\lP.\Ax.man(x) -> P(x)} {\ly.walks'(y)}
= \Ax.man(x) -> {\ly.walks'(y)} (x)
= \Ex.man(x) -> walks'(x)

The problem with that is, it's tenseless and worldless. But the 
expression means different things in 1950 and now, because there are 
different men in those two times in the world. In order to get the 
intensional contexts to work later, Monty makes his predicates 
properties: time contingent. So John is something that needs a 
property, not just a predicate: \lP.\vP(x) (because once you've got 
the timeless property, you need to convert it to a predicate for a 
particular time and world. Monty does this all the time, and writes 
it as \lP.\vP(x).

Now, all well and good when the \lP property is plugged in. What 
happens when it's left dangling?

That's precisely what Monty does with his de dicto unicorn:

John seeks a unicorn:

seeks'(j, ^\lP\Ex.unicorn(x) & P{x})

John seeks every unicorn:

seeks'(j, ^\lP\Ax.unicorn(x) -> P{x})

Oh, with one more thing. The lambda expression itself becomes 
intensional. Because it is an intensional thing we're looking for: on 
this planet there are no unicorns, but on the planet D&D there are; 
so the denotation of what you seek depends on your imaginary or real 
world.

In fact, because there's already intension going on up the front, 
Dowty allows you to turn the property (an intension) back into a 
predicate (extension) --- because it will hold for the particular 
time and world of your intension. So:

seeks'(j, ^\lP\Ax.unicorn(x) -> P(x))

which is really

seeks'(j, \l<w,t>\lP\Ax.[unicorn(x) -> P(x)]<w,t>)

So. Current Lojban-cum-Monty take: John is not looking for something 
such that it is a unicorn in this world. If he was, he'd be looking 
for the empty set. He's looking for the concept of unicorn, formally 
defined as a mapping from worlds to unicorns. And if you plug in the 
given time and world (D&D game #3, 1457 AD), you get an actual 
denotation of unicorns being searched for: Mr Uni, Mr Corn, Ms 
Licorn, Ms Einhorn... If you plug in the real time and world (this 
planet, now), you get zilch.

World and intension is how you distinguish centaurs from unicorns. 
Sure, in the here and now unicorns and centaurs are the same thing: 
nada. But in the D&D world, unicorn denotes a distinct set from 
centaurs (Chiron, Ixion, Frisky...) You use that distinction in 
mapping to differentiate centaurs from unicorns.

So what's the P(x) doing there? Well, you wanted an \A(x) and a \E(x) 
in there, and you couldn't quite attach them to unicorn on its own, 
right? So they're attaching to something else. In fact, this is kinda 
sorta propositionalism through the back door: there's still a 
predicate for the embedded quantification to hang off of.

What Monty does with this quantification is actually quite icky. 
\lP.P(john) isn't just "John needs a predicate to make sense as a 
truth claim". \lP.P(john) is the set of all preds such that they 
hold of John: that bunch of preds identifies John as distinct from 
any other entity, and make up his haeccity. So that's what the P(x) 
is doing there for the unicorns as well: they are the sum total of 
all the preds applying to at least one unicorn, or two unicorns, or 
every unicorn: their own haeccity. And those haeccities distinguish 
"one unicorn" from "all unicorns". That's the individual sublimation 
and the existential sublimation (defining individuals and populations 
in terms of their properties), and I'll leave that for another 
lifetime.

And it seems to me that when you imagine something (that problem 
propositionalism couldn't crack because it couldn't find a predicate 
for the something to be), the haeccity is your pred. Actually, your 
conglomerate of preds.

But what Monty goes on to do with the preds, and how he forces "find 
a unicorn" to have only a de re interpretation, is pretty vile. So 
I'll get off the bus now.

Does this help us? Yes, in that it shows {sisku leka ce'u broda} not 
to be tenable, if we insist that ce'u is \lx. Because that x *is not 
quantifiable*. There's just one of them. If we quantify it, then it 
is no longer \lx, it is something else. Which may turn out to be 
acceptable after all.

And the usual brickbat: Dowty says matter of factly, in a 1980 
textbook, "of course, you can't seek a property, because properties 
don't admit quantification." So why the hell did we go with seeking 
properties? Because we're bozos. None of us bothered looking up any 
1980 textbooks. And I hereby officially proclaim sisku broken.

xod had his "Lojban is fucked" moment with quantifier ordering. I've 
resisted that long and hard, with all my compromise and 
fundamentalism and kludges. But this is my Rubicon: the current 
understanding of sisku must be destroyed. I don't know what will 
replace it, and I still don't like Mr Shark replacing it, but we were 
bozos to put in the lambda. Some stuff should be kept behind the 
scenes in semantics. We're not going to start quantifying over 
avatars; we're going to gloss over quantifying worlds and times by 
using tense and CAhA (and yes, the notion of countable times and 
worlds *is* deeply bogus); and we don't need a ce'u exposed in sisku 
and unquantifiable. We do need a ce'u, true. But I don't see now how 
sisku isn't a mistake.

Sorry if this was too dense and made no sense. It's pretty new to me too.
--
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* Dr Nick Nicholas, French & Italian Studies nickn@unimelb.edu.au *
University of Melbourne, Australia http://www.opoudjis.net
* "Eschewing obfuscatory verbosity of locutional rendering, the *
circumscriptional appelations are excised." --- W. Mann & S. Thompson,
* _Rhetorical Structure Theory: A Theory of Text Organisation_, 1987. *
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