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Is there some branch of mathematics that works with truth bearing statements instead of variables, and defines operations between them?

Basically I am looking for some well known system that defines true or false statements like variables and has some short notation used to denote the truth/false hood of these statements as well as the logical conjunction of several statements.

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Your description is rather vague, but propositional logic (sometimes called sentential logic or propositional calculus) seems to fit. If it's not what you have in mind, then please edit the question to make it more specific.

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