A Simpler Proof Theory for Nominal Logic

Nominal logic is a variant of first-order logic equipped with a “fresh-name quantifier” И and other features useful for reasoning about languages with bound names. Its original presentation was as a Hilbert axiomatic theory, but several attempts have been made to provide more convenient Gentzen-style sequent or natural deduction calculi for nominal logic. Unfortunately, the rules for И in these calculi involve complicated side-conditions, so using and proving properties of these calculi is difficult. This paper presents an improved sequent calculus $NL^{\Rightarrow}$ for nominal logic. Basic results such as cut-elimination and conservativity with respect to nominal logic are proved. Also, $NL^{\Rightarrow}$ is used to solve an open problem, namely relating nominal logic's И-quantifier and the self-dual $\nabla$-quantifier of Miller and Tiu's $FO\lambda^{\nabla}$.