Variations on the Collapsing Lemma

Abstract

Graham Priest has frequently employed a construction in which a classical first-order model $\mathfrak {A}$ may be collapsed into a three-valued model $\mathfrak {A}^{\sim }$ suitable for interpretations in Priest’s logic of paradox ($\mathsf {LP}$). The source of this construction’s utility is Priest’s Collapsing Lemma, which guarantees that a formula true in the model $\mathfrak {A}$ will continue to be true in $\mathfrak {A}^{\sim }$ (although the formula may also be false in $\mathfrak {A}^{\sim }$). 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 $\mathsf {LP}$ 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 $\mathsf {LP}$ including nullary truth and falsity constants, that any conditional connective that can be added to $\mathsf {LP}$ without inhibiting the lemma must be theoremhood-preserving, and that the Collapsing Lemma extends to the paraconsistent weak Kleene logic $\mathsf {PWK}$ as well.