Contraction and closure

This is a paper about noncontractive logical systems: systems not closed under contraction. (Contraction is what allows us to conclude that Γ,A⊢B from the claim that Γ,A,A⊢B.)These systems form one family of substructural logical systems: systems that do not impose the full budget of what are called structural rules, or in which there are counterexamples to the structural metainferences associated with these rules. My main goal here is to consider the connection between consequence and closure, a connection recently emphasized in (Beall 2015), and point to a seeming difficulty faced by many familiar noncontractive consequence relations in upholding this connection. The trouble I will point out in what follows applies to approaches explored and defended in Barker (2010), Beall and Murzi (2013), Grišin (1982), Mares and Paoli (2014), Petersen (2000), Restall (1994), and Zardini (2011). It does not apply to every noncontractive approach—there are ways to avoid the trouble, which I will flag—but I think it applies to enough of the usual suspects to be of interest. I will not argue (and indeed I don’t believe) that the difficulty I point to here is fatal for these approaches; my point is simply to call attention to the difficulty that they must face.