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Anonym sa...

a. Paris Hilton is Paris Hilton.

b. All nontrivial zeros of ζ have real part 1/2.


1. b is the Riemann Hypothesis, the most famous unresolved conjecture in all mathematics.

2. For a declarative sentence R, to know whether R is to know that R (if R is true) or to know the denial of R (if R is false).

3. There is only one necessary truth, so whichever of b and its denial is true expresses the same proposition as a.

4. Paris Hilton knows that Paris Hilton is Paris Hilton.

5. So if b is true, then Paris Hilton knows that all nontrivial zeros of ζ have real part 1/2.

6. And if b is false, then Paris Hilton knows that not all nontrivial zerosof ζ have real part 1/2.

7. Hence, Paris Hilton knows whether all nontrivial zeros of ζ have real part1/2.

The source of the granularity problem is the antisymmetry of entailment asmodelled in the standard theory, which arises directly from the modelling ofpropositions as sets of worlds (which themselves are ontological primitives) and of entailment as the subset inclusion relation on the powerset of the set of worlds. There is simply no getting around the fact that the inclusion relation on a powerset is an order,and therefore antisymmetric.

By way of background, there is another tradition, one which is bothintuitively satisfying and of long standing of constructing worlds (or equivalently, maximal consistent sets of propositions) as ultrafilters (i.e. meet-closed and upper-closed proper subsets which for each proposition p contain either p or its complement) over the boolean algebra of propositions.That is, one takes the propositions as primitive and constructs worlds out of them rather than the other way around as in the standard theory. Now if the boolean algebra is finite, the two approaches coincide. This is because in a finite boolean algebra, a subset is an ultrafilter if it is a principal filter generated by an atom; in the special case where the boolean algebra is the power set of the set of worlds, the atoms, namely singleton sets of worlds, (and thefore the “constructed worlds” themselves) are in one-to-one correspondence with the primitive worlds.However, if the ambient set theory has Choice, it is well known that any infinite boolean algebra can be shown to have a nonprincipal ultrafilter. Since there are uncontroversially infinitely many propositions, it follows (assuming Choice) that there are some ways things might be (maximal consistent sets of propositions) which are not being taken into account by limiting the domains of intensions (as functions) to the set of primitive worlds.

(There are Enough Ultrafilters)

Now that we have worlds, we can say what what it means for something to be the extension of a given meaning (hyperintension) at a given world. The obvious move here is to treat the notion of extension as a family of functions (parametrized by the set of hyperintensional types) from hyperintension-worldpairs to other things. But we want to take into consideration the possibility that some meanings (e.g. meanings of names of fictional beings) may lack extensionsat some worlds. To this end, we introduce instead a similarly parametrized family of realization functions from hyperintension-extension pairs to propositions, and then say that a meaning a has extension e at world w just in case the proposition that e realizes a is true at w. (Remember that we can’t just“evaluate” a at w, because our meanings are hyperintensions, not intensions!)

Two closed hyperintensional terms a, b are equivalent iff Ext maps them to the same intension (or, equivalently, if they have the same extension at every world), i.e. ext(a) = ext(b).

(c) Paris Hilton is Paris Hilton and whichever is true, the RiemannHypothesis or its denial.

29 mai, 2008 16:05

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