Abstract
We attempt to develop a proof theory for heterogeneous logic combining first-order formulas and diagrams. In proof theory, normal proofs and normalization play a central role, which makes it possible to analyze and characterize the structure of proofs in a given system. In light of the difference between linguistic reasoning and diagrammatic reasoning, we apply the traditional proof theory developed in symbolic logic to heterogeneous logic, and we give a characterization of the structure of heterogeneous proofs based on our normalization theorem.
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