 The proof-theoretic and model-theoretic perspectives have been considered as providing rival accounts of logical consequence. However, one can also view “logical consequence” and “validity” as expressing cluster concepts: “A number of different, closely related notions go by those names. They invoke matters of modality, meaning, effectiveness, justification, rationality, and form” (Shapiro 2014). One can also note that the division between the model-theoretic and the proof-theoretic perspectives is a modern one, and it was only made possible when tools for metamathematical investigations were developed. Frege’s Begriffsschrift, for instance, which predates the development of those tools, is formulated as an axiomatic proof system, but the meanings of the connectives are given via truth conditions.