According to strong Platonism about properties, if F is a fundamental predicate, then F expresses the property Fness and the sentence "x is F" should be analyzed as saying that x instantiates Fness.
Weak Platonism makes no such analysis claim. It merely claims that to each fundamental predicate F there corresponds a property Fness such that, necessarily, x is F if and only if x instantiates Fness. Weak Platonism thus drops two claims that strong Platonism makes: (a) that the predicate expresses the property; and (b) that we should analyze predications in terms of instantiation.
Here is a problem with strong Platonism.
- If strong Platonism is true, then in an ideal logic—one that follows the metaphysics as closely as possible—there is one and only one (multigrade) predicate: Instantiates.
- In an ideal logic there would be more than one predicate.
- So, strong Platonism is false.
If strong Platonism were true, then in some sense there would only be one thing one would ever be doing—instantiating (of course sometimes one would be instantiating together with others, and what property one would be instantiating would vary).
I think weak Platonism is much more attractive. An example of weak Platonism is Lewis's account of properties as cross-world sets of objects. This is merely weak Platonism. We don't want to say that "is a circle" expresses the set of all of circles. Nor do we want to say that "Socrates is a circle" holds because of Socrates' membership in the set of all circles. But Lewis does not intend it as more than a weak Platonism.
I think that on some readings, divine conceptualist views on which properties are divine ideas are a weak Platonism.