Propositions are nullary relations, and properties are unary relations. Suppose the Fs are a plurality.
- For any object x, there is the property Dx of being distinct from x.
- For any n-ary relation P, there is an (n+1)-ary property P+ such that x1,...,xn,xn+1 stand in P+ if and only if x1,...,xn stand in P.
- For any pair of properties P and Q, there is the property P∨Q of having at least one of P and Q.
- If for every x among the Fs there is a property Px, then there is a property that is the conjunction of the properties Px as x ranges over the Fs, a property that is had by y provided that y has Px for every F x.
But we cheated in 2. For suppose P is nullary (i.e., a proposition). Then the second part of the biconditional says that an empty list of objects stands in P, which is just a fancy way of saying P is true.
Still, it is interesting that we can define a property of truth as long as we're given the arity-raising operator (·)+, plus some plausible (abundant) property formation rules.