Variations on the Collapsing Lemma ∗ Thomas Macaulay Ferguson Cycorp, 7718 Wood Hollow Drive, Austin, TX 78731 Saul Kripke Center, 365 Fifth Avenue, New York, NY 10016 ABSTRACT Graham Priest has frequently employed a construction in which a classical first-order model A may be collapsed into a three-valued model A ∼ suitable for interpretations in Priest’s logic of paradox (LP). The source of this construction’s utility is Priest’s Collapsing Lemma, which guarantees that a formula true in the model A will continue to be true in A ∼ (although the formula may also be false in A ∼ ). In light of the utility and elegance of the Collapsing Lemma, extending variations of the lemma to other deductive calculi becomes very attractive. The aim of this paper is to map out some of the frontiers of the Collapsing Lemma by describing the types of expansions or revisions to LP for which the Collapsing Lemma continues to hold and a number of cases in which the lemma cannot be salvaged. Among what is shown is that the lemma holds for a strictly more expressive form of LP including nullary truth and falsity constants, that any conditional connective that can be added to LP without inhibiting the lemma must be theoremhood-preserving, and that the Collapsing Lemma extends to the paraconsistent weak Kleene logic PWK as well. 1. INTRODUCTION Since its introduction in [1], Graham Priest has frequently appealed to his Collapsing Lemma as a method of constructing inconsistent, first-order models while retaining some control over their theories. The lemma guarantees that the formulae true in a classical model will continue to be designated in a certain type of quotient of that model when the formulae are evaluated by the lights of the calculus Priest calls the logic of paradox LP, a system first introduced by Asenjo in papers such as [2] and [3]. 1 The lemma first appears in [1], in which Priest introduces the logic LP m —the minimal logic of paradox —and studies its properties. In that paper, Priest investigates whether or not various versions of LP m (propositional, first-order) enjoy the property of Reassurance, i.e., the property that if the closure of a set of sentences Γ under LP consequence is non-trivial then its closure under LP m is also non-trivial. In particular, the Collapsing Lemma is introduced as an intermediate step in proving that Reassurance holds for first-order LP m for languages with finite signatures. Independently of its use in proving Reassurance, there are a number of ways in which the Collapsing Lemma has proven interesting. For one, it has been a versatile tool for the construction of inconsistent yet nontrivial models extending classical theories. Beginning with [4], Priest applied the lemma in order to construct inconsistent and decidable models of true arithmetic including a greatest natural number; this project was more formally investigated in the papers [5] and [6]. In [7], the lemma was brought to bear on Zermelo-Frankel set theory to produce models of ZF satisfying the what Priest labels the “Axiom of Countability,” i.e., the formal statement that all sets are countable. The lemma is also notable for the generality of its underlying construction. The technique Priest uses to build inconsistent models is closely related to a similar method to construct K 3 (i.e., strong three-valued Kleene logic) models from classical models due to J. Michael Dunn in [8]. Dunn’s Theorem in Three-Valued Model Theory is dual to Priest’s result as nothing false in the classical model becomes true in the quotient, a feature studied closely in [9]. Further cousins to the Collapsing Lemma—many of which tie into Priest’s project of plurivalent logics of [10]—have been introduced in [11]. * This is the author’s version of a paper to appear in Graham Priest on Dialetheism and Paraconsistency. Some corrections, editing, and typesetting differences will exist between this and the final version. 1 I appreciate the input of a referee reminding me to also credit Asenjo here.