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Bertrand Russell in 1903, in Principles of Mathematics, reflecting on the situation in the field of symbolic logic at the time, says that symbolic logic has three parts: The calculus of propositions, the calculus of classes and the calculus of relations. He also discussed an example demonstrating according to him that the calculus of propositions and the calculus of classes were two fundamentally different things.

This was presumably what most logicians believed in 1903, but is it still what most logicians believe today?

If so, is there any logician today who says that the logic of propositions, the logic of classes, and the logic of relations are just three applications of the same logic to distinct fields?

Thank you for any scholarly reference.

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Today, logic is usually divided up in a different way. What Russell called the calculus of relations we might call predicate logic or quantifier logic. This can be lumped together with propositional logic, and the result is sometimes called elementary logic. It is usually first order, though it is possible to extend it to higher orders.

Today it is common to distinguish the component parts of logic as proof theory, model theory, and computability theory (or recursion theory). Model theory and computability theory did not exist in 1903.

The calculus of classes is typically handled using set theory. There are many formal systems of set theory, including ZF/ZFC, NBG, MK, NF, etc. Depending on how you choose to think about it, set theory might be regarded as part of logic, or else as a theory that sits on top of logic.

This division of the subject is used, for example, in the Handbook of Mathematical Logic, Jon Barwise (ed), Elsevier, 1977.

Type theory and category theory provide other approaches to studying formal logic.

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  • I’m curious, in what way can set theory be understood as a part of logic?
    – PW_246
    Commented Nov 1 at 15:26
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    To my way of thinking set theory is not part of logic but a mathematical theory of collections, in the same way that Peano arithmetic is a theory of numbers and mereology is a theory of wholes and parts. But it seems many people do think of set theory as part of logic. The Barwise book I cited does, as do the Wikipedia entries on set theory and mathematical logic. I suppose the idea is that set theory can be used to formulate a foundation for mathematics so it shares the fundamental status of logic.
    – Bumble
    Commented Nov 1 at 15:42

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