In 'Assertion', Robert Stalnaker distinguishes between the sema

1.79m, 1.8m, and 1.81m. Suppose the context set prior to my assertion is one where
Bill’s height is unknown. I discover he is 1.82m and exclaim ‘Bill is tall’. If the content
of my assertion is the semantic content then Weak Uniformity is violated. So, by
Stalnaker’s lights, I assert a diagonal proposition and given Don’t Care, it doesn’t matter
which.
Now consider a world in the context set where the sharp cut-off for ‘tall’ is 1.79m. Given
our model, epistemically accessible worlds from that world will have a different profile to
the actual case: they will include worlds where the cut-off is restricted to the values
1.78m, 1.79m, and 1.8m. At that world, for familiar reasons, asserted content will not be
semantic content, but rather a diagonal content. However given that the set of worlds in
the context set at that world do not include all of the worlds in the actual context set, the
worlds that fall within the ambit of Don’t Care for inhabitants of that world will include
some worlds (namely those in which the cut-off is 1.81m) that fall within the actual
context set. It is now an easy matter to construct a diagonal proposition for members of
that world that differs from the diagonal proposition actually asserted (for example, given
Don’t Care, we can assert a diagonal at the 1.79 cut-off world that it is true if and only if
Bill is above 1.9m at the 1.81m worlds. Our actual assertion of ‘Bill is tall’ is true at the
1.81m cut-off worlds, but that 1.79m-world assertion is false at the 1.81m cut-off
worlds). The upshot is that Weak Uniformity will be violated.
A similar story will apply to supervaluationism. Without going into details, that
framework posits a range of acceptable precisifcations for a predicate at a world, without
privileging any one of them semantically. In this setting, let us think of the meaning of a
predicate as given by its range of acceptable precifications. Our use somehow determines
a range of acceptable precisifications. It has to be allowed, course, that small,
imperceptible differences in use sometimes makes for small differences in the range of
acceptable precifications (After all, one can construct a series of cases, with each adjacent
case imperceptibly different but where the meanings of the cases at each end are wildly
different. So small differences have to make a difference in meaning). The upshot is that,
within the context set, the meaning of a vague predicate will not be constant. Moreover,
there will be worlds that are just far enough away from the actual world to be outside the
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