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August 22, 2005
Indexicals in the Wilder
Now that I'm paying attention, I'm noticing odd or playful uses of indexicals all over the place. Here's one from the programme of the Edmonton Fringe Festival:
Magic. Mirth. Monkeys. Dave Cox is hilarious. And I'm not just saying that because I'm his hastily thrown together program blurb. He's very, very funny. More than that, he'll fool the pants of you. And then he'll politely give them back, unlike some of those weaselly trouser stealing magicians I'm sure you've heard so much about.
For a second I wondered if this was an example in which 'I' fails to refer to the agent of the context, and refers instead to the text itself. But actually, I think the conceit is that the text is the agent of the context - it's as if the text itself is speaking to us.
Perhaps I should take a leaf out of Mr Cox' book next time I write the blurb for one of my courses...
Posted by logican at 11:30 AM | Comments (0) | TrackBack
August 18, 2005
Routes to Montague
Here's an update on the old "getting to know Montague" post. From Rich Thomason's 70-page introduction to Formal Philosophy (1974):
A paper by Barbara Partee, Partee [11], contains detailed comparisons of the grammar of Chapter 8 [Montague's "The Proper Treatment of Quantification in Ordinary English] with transformational grammar, and her work in Rodman [13] contains, as well as syntactic extensions of Montague's fragments, further material helping to make Montague's work accessible to linguists. In this introduction we will therefore try to illuminate Montague's work on English in a different way, which may help to make it understandable to those familiar with logic. (16)
(Partee [11] is a reference to "Comments on Richard Montague's "Quantification in Ordinary English" in Hintikka, Moravcsik and Suppes (eds.) Approaches to Natural Language, (Dortrecht, 1973) and Rodman [13] is Papers in Montague Grammar, (LA, 1972) and (charmingly) $3 per copy to Mr Robert Rodan, Linguistics Department, UCLA, Los Angeles, Calif. 90024)
I don't know whether that address will still work. I'm definitely on the logic track though - Thomason's introduction is super-clear.
UPDATE: If I were in Melbourne on Friday, I'd be going to this:
Melbourne Logic Seminar
Allen Hazen (Melbourne)
English as a Formal Language: Montague for Beginners and Others.
11am, B07, Old Quad.
Posted by logican at 10:00 AM | TrackBack
August 17, 2005
Indexicals in the Wild
This morning I noticed an interesting use of "I" in an email from Amnesty:
There isn't much I can add to the details shared here. I was horribly beaten and abused. Police. Hospitals. Restraining orders. Leaving and returning. Fear (of him) and loathing (of myself). I am your wife, sister, mother, daughter, niece and neighbor. I am the same story told in a different voice hundreds of thousands of times.
I doubt Amnesty sent this to me so that I can comment on the language, so if you want to know more you can check out the site here.
But what's interesting about the use of "I" in "I am your wife, sister, mother, daughter, niece and neighbor" is that i) this seems like a legitimate use of language and ii) it is plausible that the sentence is analytically false, since, for example, no-one can be both your mother and your daughter. (Putting worries about time travel aside for now.)
If this is right, then it seems to mitigate the damage done by answer-machine examples ("I'm not here right now, please leave a message") to theories of indexicals on which the sentence 'I am here now' comes out as analytic. It's just not that unusual for us to say things that are, not only false, but, analytically false - false in virtue of what they mean and transparently so.
Posted by logican at 12:35 PM | Comments (5) | TrackBack
August 10, 2005
A note on Kit Fine's Rigidity Axiom
I've been reading Kit Fine's "The Logic of Essence" (Journal of Philosophical Logic, vol. 24 1995 pages 241-273) and just wanted to record a puzzle I had about one of the axioms. It's the axiom of rigidity that I am puzzling over, and it looks like this:
There are a two of bits of unusual notation in there. The first is Fine's essentiality operator. If you want to formalize the sentence
It is a property of singleton Socrates that it has Socrates as a member.
one way to do it is as Rab, where a refers to the set, b to Socrates and R expresses the set-membership relation. But suppose you want to formalise something like
It is an essential property of the singleton set of Socrates that it contain Socrates as a member.and you want to do it in such a way that the logical force of "essential" isn't lost.
Well, you might try making "essentially" a kind of modal operator (represented here by the box) and write:
Fine argues (quite convincingly) that this would be wrong, since this next sentence would be translated the same way and is false, where the above sentence is true:
It is an essential property of Socrates that he is a member of the set singleton Socrates.
(Having trouble getting the right intuitions about these sentences? Fine thinks of essential properties of an object as being those that are had by virtue of its nature. The thought is that it is part of the nature of singleton Socrates that it contain Socrates, though it is not part of Socrates' nature that he be a member of any set---it doesn't even follow from his nature that sets exist. Fine's project in this paper is that of "developing a logic of essence, not now as a fragment of a modal system, but as a system in its own right" (241))
Instead we have, for each predicate F, an operator:
and the predicate F picks out the subjects of the essentialist claim - that is, the objects (if any) whose natures underwrite the truth of the claim. We call predicates playing this role "delimiters."
To formalise "Singleton Socrates essentially has Socrates as a member", we need a predicate which applies only to singleton Socrates (singleton-Socratises? lx(x=Socrates)? (l is for "lambda"), say, F, and then we use the associated operator:
This can be read as "Rab is true in virtue of the nature of the objects that F". So where a refers to singleton Socrates, R expresses the membership relation, b refers to Socrates himself and F expresses the property of singleton-Socratising, this formula says that Socrates is a member of singleton-Socrates in virtue of the nature of singleton Socrates.
There is another piece of unusal notation that you need to understand the rigidity axiom. Fine doesn't only represent predicates using single letters. He also uses lambda abstraction and some abbreviations of expressions so formed. Most relevant here, he writes (y) for lx(x=y), where x is the first variable distinct from y (i.e. an predicate for the property of being identical to y) and in using such abbreviations in delimiters, he sometimes abbreviates them even further by removing the parentheses. Hence
can be read as "A is true in virtue of the nature of objects which are identical to y" (that is, of course, in virtue of the nature of y.)
Finally, it seems that if we have more than one predicate in the delimiter space, separated by a comma, that means that the inside formula is true in virtue of the objects which fall under the disjunction of the predicates. (I say "it seems" because I can't find a place in the text where this is stated, but I think it makes sense of what comes later.)
So now we have the resources to understand the rigidity axiom:
Roughly what this says is that if Px is true, then it is true in virtue of the natures of x and the objects which fall under P. Even more roughly, Px is true (if it is) in virtue of the natures of the things which it is about. Fine says that the axiom is "clearly correct" and gives a proof to demonstrate this correctness:
For if x is one of the objects x1, x2..., say xi, then it is true in virtue of the nature of x that it is xi, and hence true in virtue of the nature of x1, x2...that x is one of x1, x2....
What I'm puzzling over is this. I'm not sure that sentences are always true in virtue of the natures of the objects that they are "about," that is, the objects which they refer to or which fall under the predicates they contain. Take a sentence like "fred is a frog or it is not the case that fred is a frog." Is that true in virtue of the nature of Fred, or froggy things? One reason why you might think it isn't is that it would be true regardless of Fred's nature, or the nature of Froggy things. The expressions could even apply to things entirely unfroggy objects which are entirely unlike Fred, and the sentence would still be true.
So does that mean that there's something wrong with Fine's proof - er, no, though in the spirit of the intuition I've just drawn on I might deny that it is true in virtue of the nature of x that x is identical to xi. (After all, things are self-identical regardless of their natures, aren't they?)
But what I suspect is really going on here is that there are several senses of "in virtue of". Channeling the Quine of "Carnap and Logical Truth" for a second, we might say that there's a sense of "in virtue of" on which "Fred is a frog or it is not the case that Fred is a frog" is true in virtue of Fred's nature, and a sense in which it is not. It really doesn't make any difference which we choose to use (and so Fine is perfectly justified in using his sense.)
Here's another place where the assumption comes in - in the following rule of Fine's system:
This says that if you've proved A as a theorem, then it is also a theorem that A is true in virtue of the natures of the objects contained in the proposition expressed by A (Fine is working with a very Russellian conception of propositions) which seems like a fairly explicit committment exactly the claim I was puzzled about, namely that sentences are always true in virtue of the natures of the objects the refer to or which satisfy their predicates.
"in virtue of" is puzzling...
Posted by logican at 09:12 AM | Comments (10) | TrackBack