Proposition 5 of Ethics
This proposition is ambiguous and its demonstration difficult to accept; however, it is the basis for demonstration for many subsequent propositions. I believe its demonstration is crucial to the whole of Ethics, but only in so far as one requires strict proof-solutions, or at least the chapter in which it rests. I will here attempt to study it, but we must keep in mind that the rest of the Ethics can still be understood intelligibly and used instructively even if some of its proofs are not valid. Even Bertrand Russell says that the Propositions can be read without drudging through the Demonstrations.
P5: In Nature there cannot be two or more substances of the same nature or attribute.
Demonstration: If there were two or more distinct substances, they would have to be distinguished from one another either by a difference in their attributes, or by a difference in their affections (by P4). If only by a difference in their attributes, then it will be conceded that there is only one of the same attribute. But if by a difference in their affections, then since a substance is prior in nature to its affections (by P1), if the affections are put to one side and [the substance] is considered in itself, that is (by D3 and A6), considered truly, one cannot be conceived to be distinguished from another, that is (by P4), there cannot be many, but only one [of the same nature or attribute], QED.
Antecedents:
D3: By substance I understand what is in itself and is conceived through itself, that is, that whose concept does not require the concept of another thing, from which it must be formed.
A6: A true idea must agree with its object.
P1: A substance is prior in nature to its affections.
P2: Two substances having different attributes have nothing in common with one another. (For my purposes.)
P4: Two or more distinct things are distinguished from one another, either by a difference in the attributes of the substances or by a difference in their affections.
Thoughts:
1 If two distinct substances exist, upon perceiving those substances (by D4), a distinction between them can only be made through appealing to their Attributes or through their Affections (which are the Modes, or ways, in which an Attribute is itself perceived of the substance; all by P4).
2 If one takes the former approach to distinguish the substances by an Attribute, presupposing that they do in reality bear different Attributes in order to distinguish them, then the substances must be understood as not having the same Attribute. Indeed, from this case, the distinction cannot be made because in order to distinguish the substances through an Attribute, supposing that an attempt at distinguishing presumes that the Attributes are exclusive identities and not the same in nature, they would have to have something in common. But they would not have anything in common (by P2). Therefore, if they have different Attributes, as stated, only one of them can be (”there is only one of the same attribute”). That is, one substance will be the only substance in so far as attempting to perceive its distinction from other substances because its distinction by distinguishing it presupposes that it is already distinct.
Simply: If their Attributes are different, then they cannot be distinguished from one other by those Attributes because their difference is already established in principle (by P2); they have no commonality thus no mutual Attribute.
This seems like it is unhelpful to distinguish them by their Attributes alone. We must look to their affections.