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On Thu, Sep 13, 2012 at 1:46 AM, Martin Bays <mbays@hidden.email> wrote: > > The following looks like an error, though: > >> One other thing to note about the M function is that the following >> transformations are valid: >> >> o sV[2] R MV[1] P <=> MV[1] sV[2] R P >> >> where V[1] is different from V[2] and V[1] does not occur free in R. > > That's surely false for (some) natural interpretations of M. For > example, taking R to be trivial and P(x,y) to be "x=y", and making the > reasonable-seeming assumption that if we apply M to a unary formula with > singleton extension the result has the same singleton extension, then > left hand side is: > Ey. Mx. x=y <=> Ey. x=y > whose extension is the entire domain; but the right hand side > Mx. Ey. x=y > has singleton extension. It seems you have shown that when you singularize the identity relation, you end up with the entire domain being a singleton, which sounds right. ma'a xrxe