Trivalence: Origins and Developments

This chapter gives some elements of the history of trivalent logics and presents the key technical notions. We stress that Boole and Frege were aware of reasons to go beyond bivalence, in ways that influenced Łukasiewicz in particular. Then we put particular emphasis on the 1930s as a pivotal moment in the application of trivalence to a range of interconnected phenomena, such as probability and hypothetical reasoning, quantum indeterminacy, computability theory, and the semantic paradoxes. The chapter goes on to present basic elements regarding the connection of trivalent logics to other logics, including intuitionistic logic, modal logics, fuzzy and many-valued logics, and finally to classical and non-classical logics more broadly. Along the way, we present basic systems of mixed consequence, and provide glimpses into the theory of ternary computation.