Central Problems of Philosophy

Prof. Stephen Schiffer

Fall `98


Handout 7



  1. A person with $50,000,000 is rich.
  2. 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.

  3. A person with $0 is not rich.
  4. For any n, if a person with $n isnít rich, then neither is a person with $n + 1.

    Therefore, a person with $50,000,000 isnít rich.

  5. An argument for the sorites premise:
    1. If the sorites premise isnít true, then itís false.
    2. If itís false, then there is an n such that having $n is sufficient for being rich but having $n - $1 isnít sufficient for being rich.
    3. 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 ĎHaving $a is sufficient for being true but having $a - $1 isnítí is true.
    4. But there is no such sharp one-dollar cutoff.
    5. Therefore, the sorites premise is true.
  6. Bivalence: Every proposition is true or false (a proposition p is false iff not-p is true).
  7. Excluded middle: Every proposition of the form p or not-p is true.

  8. Reconstruction of the studentís argument:
  9. If the exam were on Thursday, it wouldnít be a surprise, for, since it wasnít on Tuesday, Iíd expect it to be on Thursday.

    But if the exam were on Tuesday, it also wouldnít be a surprise, for, since Thursday was ruled out, Iíd expect it to be on Tuesday.

    Therefore, there can be no surprise exam.

  10. The exam wonít be held unless itís unexpected, and Iíve already shown that Iíd expect it on Thursday if it wasnít on Tuesday.
  11. There will be a surprise exam on Thursday.