Spinoza's Ethics: I.P2: Substance Commonality

If two substances have different attributes (from one another) (e.g., one has the attribute of thinking and one has the attribute of existing), then Spinoza is arguing that these two substances have nothing in common.

Clearly if two substances have different attributes (from one another), then they are essentially different. That follows from the definition of attribute (I.D4): "what the intellect perceives of substance as constituting its essence." If the perceived essences are different, then the substances are essentially different, assuming perception is reliable. So, for example, if one substance is perceived through the attribute of thinking and another substance is perceived through the attribute of existing, then the two substances are essentially different. But does that mean that the substances have "nothing in common"?

To have something in common is addressed in the definition of interactivity: (I.A5): "things that have nothing in common with one another also cannot be understood through one another." So, to have something in common is to be understood through one another. That means that either substance could be understood through the other. However, the definition of substance: (I.D3): "is in itself and is conceived through itself" denies this. Substance is only understood through itself not through another.

At this point, he does not seem to be stating there are actually two substances, but only is stating that if there were two substances did exist with different attributes (from one another), there would be an inability for such presumed substances to have anything in common with each other.

This seems to be simply a logical exercise, extending the implications of the definitions and axioms. However a key point seems to be made. If one substance were expressed with the attribute of thinking and one substance were expressed with the attribute of existing, then this two substance universe would have its thinking and existing having nothing to do with each other.   

Duae substantiae diversa attributa habentes nihil inter se commune habent.

Translated as,

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

Also translated as,

Two substances having different attributes have nothing in common with one another.

Demonstratio: Patet etiam ex definitione 3. Unaquaeque enim in se debet esse et per se debet concipi sive conceptus unius conceptum alterius non involvit.

Translated as,

This also evident from D3. For each ·substance· must be in itself and be conceived through itself, which is to say that the concept of the one doesn’t involve the concept of the other.

Also translated as,

Dem.: This is also evident from I.D3. For each must be in itself and be conceived through itself, or the concept of the one does not involve the concept of the other.