Gaggle Theory: An Abstraction of Galois Connections and Residuation with Applications to Negation, Implication, and Various Logical Operations

The results of this paper have been developed as a kind of hobby over many years. The ideas were first presented in a seminar in 1979 at the University of Victoria and in 1984 at a seminar at Carnegie-Mellon University. I benefited early on from discussions with Nuel Belnap and Gary Hardegree, and more recently from discussions with Gerard AUwein. Perhaps the person I learned the most from, Robert K. Meyer, claims not to know what a ':gaggle" is, but it was his paper with Richard Routley (1972) that was the most direct influence on my thinking. Despite the years of thinking about this topic, the present paper was prepared with some haste, and so proofs and definitions are sometimes more sketchy than I would like. I hope on some later occasion to improve this situation. Routine definitions from lattice theory ("prime filter," "prime ideal," etc.) are to be found in Birkhoff (1967).