Arnauld and later Leibniz pressed that identity of nature requires intersubstitutability in all contexts, but Descartes treats corporeal substance as a substance-bearer while extension functions as an attribute, violating strict identity.

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Key Terms

Arnauld(as a historical figure in metaphysics): Antoine Arnauld was a 17th-century French philosopher and theologian who thought deeply about logic and how ideas relate to reality.
Descartes: # Descartes René Descartes was a French philosopher and mathematician from the 1600s who fundamentally changed how people think about knowledge and the mind. He's famous for the idea "I think, therefore I am" (cogito ergo sum), which means that the very fact that you can think proves you exist—a foundation for modern philosophy. He also invented the coordinate system used in mathematics (the x and y axes on a graph), which connects geometry and algebra in practical ways we still use today.
Identity of nature(in metaphysics (the study of what exists)): The idea that if two things are truly the same in their essential character, they should behave the same way in every situation.
Intersubstitutability in all contexts(in logic and metaphysics): The ability to swap one thing for another in any situation without changing the truth or meaning of what you're saying; if A and B are truly identical, you should be able to use them interchangeably everywhere.

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Leibniz

Leibniz is a German philosopher and mathematician from the 1600s-1700s who developed calculus (a powerful math tool for measuring change and areas) independently around the same time as Isaac Newton. He's famous for creating much of the notation we still use in mathematics today and for arguing that everything in the universe follows logical principles. His ideas profoundly influenced modern science, mathematics, and philosophy, making him one of history's most important thinkers.

Strict identity(in metaphysics and logic)

Perfect, absolute sameness in every way—if two things have strict identity, there is no meaningful difference between them whatsoever.

Substance-bearer(in metaphysics)

Something that 'holds' or 'carries' properties and qualities; like how a table is the substance that bears the properties of being brown, wooden, and four-legged.

attribute(Spinoza's metaphysics, contrasted with Descartes)

That which individuates substances, not merely types of substances; dropped the Cartesian qualifier 'principal' so that all attributes, not just the chief one, are individuating.

corporeal substance(Leibnizian metaphysics of living matter)

A body endowed with a basic entelechy or vital principle, of which it may be said in general that it is living.

extension(Semantics and philosophy of language)

Another term for reference, i.e., the object or set of objects a term picks out

substance(Spinoza's metaphysics; criteria include (i) necessity and (ii) self-subsistence)

The fundamental existent that is wholly necessary and self-subsistent, not depending on anything else for its existence

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Arnauld and later Leibniz pressed that identity of nature requires intersubstitutability in all contexts, but Descartes treats corporeal substance as a substance-bearer while extension functions as an attribute, violating strict identity.

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