What follows is a discussion of knowledge. If method M is “Given a valid deduction from the premise that someone believes P to the conclusion that P, believe that P.”
“Williamson points out that one can, using M, come to believe that there is at least one fallible believer… since the proposition that there is at least one fallible believer follows from the proposition that someone believes that there is at least one fallible believer” (Hawthorne, Deeply Contingent A Priori Knowledge 259).
“…he asks the reader to consider the situation in which there are (up to now) no believers except for a necessarily not fallible one, who is now in the act of coming to know (1′) a priori, where (1′) is the belief that there is at least one fallible believer.” (ibid)
Hawthorne goes on to raise doubts:
“Suppose there is a single believer in a world. That believer has three true beliefs before noon and it is 90 percent likely that he will form only true beliefs after noon. At noon he deduces ‘There is at least one fallible believer’ using M. The unlikely thing happens: he forms a false belief after noon” (Hawthorne 260).
This is perhaps the clearest example I’ve seen in a long time of where I think philosophical method goes wrong. Concepts get pushed past their breaking point–applied to circumstances distant from the ones where they play an important role in our thought. In the case at hand, it is more obvious, since we appeal to a believer who has a mere three beliefs, or necessarily non-fallible believers, and so on. But it goes on all the time.
There are deep roots to this impulse. For one, we have few ideas for how to do (philosophical) semantics without some version of this notion. For another, philosophy is manifestly concerned with an expansive sphere of possibilities–typical philosophical examples do not necessarily need to be actual or even nomically possible to be relevant. The temptation to jump straight from that conclusion to examining a completely unlimited space of possibilities is easy, especially without a well-grounded story of why we’re interested in the possibilities we are.