A common interpretation of the relation of Substance to it's modes in Spinozas states that modes are properties of the Substance [1]. Let's assume this for now: Modes are properties of Substance. I think I see a problem here. Consider the identity conditions for predicates, i.e. in which circumstances two predicates are considered identical. Swoyer and Orilia list four different identity conditions for predicates [2]. In this framework, two properties are identical iff...
a) ... they have the same extension (S&O call this an "infra-coarse" identity condition).
b) ... they necessarily have the same extension (S&O call this a "medium-coarse" identity condition).
c) ...they confer the same causal powers on their instances (a "medium-fine" identity condition).
d) ...they are (nearly) the same linguistic expressions ("ultra-fine").
It seems that every case leads to a contradiction. If a) is the case, then every property (mode) would be the same, because there is only one object, namely Substance, that has properties. Thus every mode would be identical to the set, which has Substance as its only member. The same holds for b), because of E1P29: "Nothing in the universe is contingent, but all things are conditioned to exist and operate in a particular manner by the necessity of the divine nature." So there are no properties which are non-necessary. If c) holds, then two modes are identical iff they confer the same causal powers on their instances. But there is only one instance, namely Substance. But as the modes would confer causal power on its instance, Substance, the modes would determine Substance. But then Substance would not be in itself anymore, because it would be determined by something else. So c) doesn't make sense as well. If d) holds, then two modes are identical just in case they are (nearly) the same linguistic expression. But Spinoza says that individuals, i.e. modes, have parts, for example here: "parts composing an individual" (E2L5). But Spinoza does not seem to have the parts of an linguistic expression in mind, but the parts of an individual. In general, both are different. The parts of the linguistic expression "human body" are for example "human" and "body". But the parts of a human body are it's head, arms, organs, etc. So d) cannot hold as well.
So apparently all the cases lead to contradictions. But why is this so? Is this considered to be a problem for the property-interpretation of Spinoza? Or could it even indicate a lacuna in the overview of S&O? Could it even be that Spinozas modes don't have any identity conditions? But then how can they be different?
[1] https://www.iep.utm.edu/spinoz-m/#SH3b : "Spinoza claims that one important consequence of this proof is that modes are properties of substance. The view that modes are properties of substance has been denied by at least one prominent interpreter of Spinoza (Curley 1988: 31-39). Curley’s view has, however, proven unpopular (See Carriero 1999; Malamed 2009.) The dominant interpretation today is that modes are properties of the one substance."
