An argument against shaving

The argument is a reductio.

1. Shaving is sometimes permissible.
2. There is a relation I such that IxP holds if and only if P is a property which x has.
3. Define the property N by NP=~IPP.
4. Either N has N, N does not have N.
5. If N has N then ~INN by definition of N, and so N does not have N, which is absurd.
6. If N does not have N then by definition of N we do not have ~INN, and so by double negation, we do have INN, and so N has N, which is absurd.
7. Thus absurdity ensues on both horns of the dilemma in (4).

The reader will, of course, notice that (1) is not used anywhere in the argument. Instead, the argument gives (1), and then launches into a standard variant of Russell's paradox. It's obvious, thus, that we learn nothing from the argument about the permissibility of shaving.

But one can dress up the argument if one so desires, so that (1) gets used further on down. For instance, instead of working with I, one can work with I_s where I_sxP holds if and only if P is a property which x has and shaving is sometimes permissible. If N_sP=~I_sPP, then absurdity ensues from assuming that N_sN_s only assuming that shaving is sometimes permissible. (If shaving is not permissible, then N_sN_s holds because I_sN_sN_s unproblematically fails as the second conjunct in its definition fails.) Thus, we can take the paradox and dress it up into an argument against the permissibility of shaving. But of course we still learn little about shaving from it.

I claim that Patrick Grim's arguments against omniscience are another such dressing up of this paradox, and hence we learn little about omniscience from them. But I am not going to argue for this here, since I am still working on the paper where I show this.