Section 1 lays out the basics of free logic, explaining how it differs from classical predicate logic and how it is related to inclusive logic, which permits empty domains or “worlds.” Section 2 shows how free logic may be represented by each of three formal methods: axiom systems, natural deduction rules and tree rules. Varying conventions for calculating the truth values of atomic formulas containing empty singular terms yield three distinct species of free logic: negative, positive and neutral. These are surveyed in Section 3, along with supervaluations, which were developed to augment neutral logics. Section 4 is critical, examining three anomalies that infect most free logics. Section 5 samples applications to theories of description, logics of partial or non-strict functions, logics with Kripke semantics, logics of fiction and logics that are in a certain sense Meinongian. Section 6 takes a glance at free logic's history.
"Free Logics," in Philosophy of Logic, ed., Dale Jacquette, (Volume 5 of the Handbook of the Philosophy of Science under the general editorship of Dov Gabbay, Paul Thagard, and John Woods), Elsevier 2006, pp. 1023-1060.