Central Problems of Philosophy

Prof. Stephen Schiffer

Fall `98

Handout 7

- A person with $50,000,000 is rich.
- A person with $0 is not rich.
- An argument for the sorites premise:
- If the sorites premise isn’t true, then it’s false.
- If it’s false, then there is an
*n*such that having $*n*is sufficient for being rich but having $*n*- $1 isn’t sufficient for being rich. - If there is such an
*n*, then there is a*sharp one-dollar cutoff*between what suffices to make a person rich and what fails to suffice to make a person rich. That is, there is some particular number such that the proposition that*that number*is the cutoff is true, which, in this case, is equivalent to saying that there is some numeral a such that the proposition expressed by ‘Having $a is sufficient for being true but having $a - $1 isn’t’ is true. - But there is no such sharp one-dollar cutoff.
- Therefore, the sorites premise is true.
*Bivalence*: Every proposition is true or false (a proposition p is false iff not-p is true).- Reconstruction of the student’s argument:
- The exam won’t be held unless it’s unexpected, and I’ve already shown that I’d expect it on Thursday if it wasn’t on Tuesday.
- There will be a surprise exam on Thursday.

For any *n*, if a person with $*n* is rich, then so is a person with $*n* -
$1.

Therefore, a person with only $3 is rich.

For any *n*, if a person with $*n* isn’t rich, then neither is a person with $*n* +
1.

Therefore, a person with $50,000,000 isn’t rich.

*Excluded middle*: Every proposition of the form *p or not-p* is true.

If the exam were on Thursday, it wouldn’t be a surprise, for, since it wasn’t on Tuesday, I’d expect it to be on Thursday.

But if the exam were on Tuesday, it also wouldn’t be a surprise, for, since Thursday was ruled out, I’d expect it to be on Tuesday.

Therefore, there can be no surprise exam.