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Sunday, October 22, 2006

Hyperintensional Operators

The latest issue of Philosophical Issues features a very intersting article called Indicative versus Subjunctive Conditionals, Congruential versus Non-Hyperintensional Contexts by Timothy Williamson. As the title indicates, the articles is mostly about indicative and subjunctive conditionals. However, the article got me thinking about hyperintensional operators again.

Uncontroversially, Williamson takes an operator O to be intensional iff it is not extensional, and O is extensional iff the following condition is satisfied:

If p is materially equivalent to q, then Op is materially equivalent to Oq.

An operator O is hyperintensional iff it is not non-hyperintensional, and O is non-hyperintensional iff the following condition is satisfied:

If p is strictly equivalent to q, then Op is strictly equivalent to Oq.

Hyperintensionality, then, is a special case of intensionality. As I believe some but not all truths, 'Brit believes that' is intensional. Moreover, as I believe some but not all necessary truths, 'Brit believes that' is hyperintensional. If S is a perfectly rational and omniscient being, then 'S believes that' is extensional, and non-hyperintensional. However, as a matter of fact, most propositional attitude operators will be hyperintensional.

Many story prefixes are also hyperintensional. 'Andrew Shepherd is identical to the actual current president' is strictly equivalent to '2 + 2 = 5', but 'in the movie the American President, Andrew Shepherd is the actual current president' is not strictly equivalent to 'in the movie the American President, 2 + 2 = 5'. Or if you dislike examples with 'actual', consider this example instead: 'Andrew Shepherd is not identical to the President' is strictly equivalent to '2 + 2 = 4', as 'Andrew Sherpherd' is empty, but 'in the movie the American President, Andrew Sherpherd is not identical to the president' is not strictly equivalent to 'in the movie the American President, 2 + 2 = 4', as the fomer is false but the latter true.

According to David Lewis ["Tensed Quantifiers" in Zimmerman (ed) Oxford Studies in. Metaphysics 2005], span operators -- operators that shift the time feature of the index of evaluation from the time of utterance to some past or future time span -- are hyperintensional. According to Lewis, 'it is raining, and it is not raining' is strictly equivalent to '2 + 2 = 5', but 'it WAS the case that it is raining, and it is not raining' is not strictly equivalent to 'it WAS the case that 2 + 2 =5', as the former has a true reading, whereas the latter is necessarily false. According to David Lewis, the reason that the former has a true reading is that it could be true at one time but false at another time during the time span in question that it is raining. Lewis takes this to constitute a serious problem for presentism. If presentism is true, then span operators are required to capture certain truths (e.g. 'there have been several kings of England named Charles'). But span operators are hyperintensional. Yet modal operators (including tense operators) are thought to be intensional, not hyperintensional.

However, I think a presentist could reply as follows: a proposition is true at a span circumstance only if it holds at some, most or all times during the time span in question. But 'it is raining, and it is not raining' does not hold at any time during the time span in question. For there is no time at which it is raining and not raining. In other words, the result of embedding a contradiction under a span operator does not yield a truth. So, span operators are not hyperintensional.

6 comments:

Mike said...

I think a presentist could reply as follows: a proposition is true at a span circumstance only if it holds at some, most or all times during the time span in question. But 'it is raining, and it is not raining' does not hold at any time during the time span in question. For there is no time at which it is raining and not raining.

We want to be able to say that, for a sufficiently large region (maybe Earth-sized) it holds in the entire spatial region that it is both mountainous and not, even if we deny that it holds at each (or any) sufficiently small sub-region. So maybe there's an ambiguity between a proposition holding during the entire time span and the proposition holding at each time in the span. Let the span be t-t'. It might hold during the entire span that p and ~p, even if it does not hold at any instant that p & ~p. So the your truth-conditions seem to be these,

T. A proposition is true at a span circumstance only if it holds at some instant, or each instant of time during the time span in question.

But given (T) I can't say, for a sufficiently large time span, that it was raining and not raining in that span. Seems like I should be able to say that truthfully, i.e., without uttering a contradiction.

Brit Brogaard said...

Hi Mike
Here is another reply on behalf of the presentist (perhaps more to your liking :-) Treating span operators as hyperintensional is unproblematic for the presentist. For the presentist already thinks time spans are illusions (so span operators are fictional operators or at least very much like them, and fictional operators are hyperintensional).

But I guess I doubt that 'it WAS the case from 1010 to 1011 that it was raining and not raining' has the mentioned ambiguity in ordinary language. The reason we read it that way, I think, is that it seems true that it was raining at some time and not raining at another. But actually 'it is raining and not raining' must be true AT the span. So, it cannot be split up that way and be evaluated at different time during the span.

Mike said...

But actually 'it is raining and not raining' must be true AT the span

But isn't it true at the span? Suppose we change the example to baseball. Did your team win yesterday? Yes. Did your team lose yesterday? Yes. The team won and failed to win yesterday? Yes, that's true. Didn't that happen AT that time span?

Brit Brogaard said...

But then a part of the proposition is true at one part of the span and another part of the proposition is true at another part of the span. I find that odd.

An analogy: "John is hungry at t1 and John is not hungry at t2" can be true when evaluated with respect to a possible world.

But when the time adverbials are left out, as in "John is hungry and John is not hungry", then we have a contradiction regardless of whether you are a Fregean who thinks there are times in the propositions or a Kaplanian who thinks otherwise.

But spans are no different. To evaluate a proposition at a span you must evaluate the WHOLE proposition at the span, not one part of the proposition at one end of the span and another part of the proposition at the other end.

Mike said...

But when the time adverbials are left out, as in "John is hungry and John is not hungry", then we have a contradiction . . .

But isn't that because you are reading it stating "at the same time" rather than "at the same span"? I guess intuitions just vary, but I can't see why "I was both cold and not cold" is not a coherent answer to "were you cold in that room yesterday?" Of course, not at the same time, but at the same span.

Brit Brogaard said...

But "I was both cold and not cold" is at best ambiguous between I was both cold and not cold throughout the time span and the reading you suggest. But I would add that we cannot rule out that "I was both cold and not cold" is to be understood as "I was cold yesterday, and I wasn't cold yesterday"

Anyway, I am open to the span operator interpretation you are suggesting. I just don't think this is how operators normally function in ordinary language.