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I'm having some trouble comprehending Spinoza's Proposition 5, in Part I of the Ethics. Here's the excerpt (Curley translation):

P5: In Nature there cannot be two or more substances of the same nature or attribute.

Dem.: 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], q.e.d.

It seems to me that Spinoza is proceeding from contradiction: suppose there are two distinguishable substances, say S1 and S2, which share a common attribute. I honestly have no idea what this next sentence means:

If only by a difference in their attributes, then it will be conceded that there is only one of the same attribute.

What is "only one of the same attribute"? It also doesn't seem like his argument answers the case in which S1 and S2 are each consist of multiple attributes, but only share one attribute in common. This also ties into the second half of his argument, since if S1 and S2 each contain an attribute the other does not have, then it will conceive of modes that the other does not have. But then in this case considering the substance "in itself" and "truly" will lead back to the same issue.

I'd really appreciate it if someone could shed some light on this!

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  • This question is exactly answered on pages 48-49 in Michael Della Rocca's Spinoza.
    – Vasting
    Nov 29, 2022 at 4:56

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Def. IV. By attribute, I mean that which the intellect perceives as constituting the essence of substance.

Prop. II. Two substances, whose attributes are different, have nothing in common.

Prop. V. There cannot exist in the universe two or more substances having the same nature or attribute.

Thus, you are right: Spinoza is proceeding from contradiction. Assume that there are two substances having common attributes; by P.II, two substances having different attributes have nothing in common; contradiction.

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    I think your answer at this question (philosophy.stackexchange.com/questions/69985/…) helped clarify some things -- namely that two different substances cannot share a common attribute (as attributes relate to the essence of a substance).
    – Vasting
    Nov 21, 2022 at 22:57

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