How many properties?

To count the number of contingent properties (monadic predicates) predicable of an object x, we start with the number of subsets of states of affairs (assuming theWorld is finite, of course):

#SSA = 2^#S

From this, we exclude {S} and {O}, since, as no object has either S or O as states, no property could hold of an object by virtue of the inclusion of {S} or {O} in the property's extension.

From the resulting 2^{#SSA - 2} properties, we exclude the 2^{#SSA - 2 - #x} properties which contain all elements of x, since these are always true of x, hence not contingently predicable of x. An equal number of properties are disjoint from x, so are never a property of x.

This leaves 2^{#SSA - 2} (1 - 1/2^{#x - 1}) or

2^#SSA (1/4 - 1/2^#x+1)

We note that, as we might expect, the larger x is, the more contingent properties it has.