The Pure Logic of Ground

I lay down a system of structural rules for various notions of ground and establish soundness and completeness. Ground is the relation of one truth holding in virtue of others. This relation is like that of consequence in that a necessary connection must hold between the relata if the relation is to obtain but it differs from consequence in so far as it required that there should also be an explanatory connection between the relata. The grounds must account for what is grounded. Thus even though P is a consequence of P & P, P & P is not a ground for P, since it does not account for the truth of P. It is the aim of this paper to develop a semantics and proof theory for the pure logic of ground. The pure logic of ground stands to ground as Gentzen’s structural rules stand to consequence. One prescinds from the internal structure of the propositions under consideration and simply asks what follows from what in virtue of the formal features of the underlying relation. Thus the claim that ground is transitive, that if P is a ground for Q and Q a ground for R then P should be a ground for R, is plausibly regarded as part of the pure logic of ground; but the claim that P is a ground for P & P will be part of the applied as opposed to the pure logic of ground, since it turns on the logical properties of &. The question of the logic of ground has been considered by a number of other authors (even if not exactly under this head) and so it may be helpful to bring out some of the more distinctive aspects of my own approach.1 I have, in the first place, attempted to be rigorous and systematic in the development of the logic. Rather than simply laying down various principles that I regard as true, I have specified various formal systems of which these principles are a part. These systems, in familiar fashion, thereby become the object of metalogical study; they can be assessed for consistency, assigned a semantics, compared for proof-theoretic strength, etc. There are two other, more particular, aspects of my approach. The first lies in its conceptual basis. Most other philosophers have worked almost exclusively with a concept of strict ground, under which a truth is not capable (or, at least, not normally capable) of being a ground for itself.2 But I believe that there is also an important concept of weak ground, under which a truth will automatically be a ground for itself. Very roughly, we may say that strict grounds must move us down in the explanatory order while weak grounds must not move us up. Thus P can weakly ground P, but P & P cannot weakly ground P given that P strictly grounds P & P. Received: September 22, 2009. 1 The authors include Audi (2011), Batchelor (2010), Correia (2005, 2011), Rosen (2010), Schnieder (2006, 2011), Fine (2011), and, much earlier, Mulligan et al. (1984). 2 An exception is Correia (2011). He explicitly works with a notion of ground-theoretic equivalence and implicit in his approach, I believe, is a notion of weak ground. He also provides a formal system and a semantics. c © Association for Symbolic Logic, 2011 1 doi:10.1017/S1755020311000086