ZU064-05-FPR rsl-submission2 April 19, 2018 21:12 The Review of Symbolic Logic Volume 0, Number 0, Month 2009 Pure Logic of Iterated Full Ground Jon Erling Litland University of Texas at Austin Abstract. This paper develops the Pure Logic of Iterated Full Ground (plifg), a logic of ground that can deal with claims of the form “φ grounds that (ψ grounds θ)”—what we call iterated grounding claims. The core idea is that some truths Γ ground a truth φ when there is an explanatory argument (of a certain sort) from premisses Γ to conclusion φ. By developing a deductive system that distinguishes between explanatory and non-explanatory arguments we can give introduction rules for operators for factive and non-factive full ground, as well as for a propositional “identity” connective. Elimination rules are then found by using a proof-theoretic inversion principle. §1. Introduction Fine (2012a,b), Correia (2010, 2014), Schnieder (2011), and Poggi- olesi (2016, 2018) have developed logics of ground where ground is treated as a sentential operator (to be read: “because”). All of these logics, however, have been logics of “simple” ground: they have nothing to say about claims of the form “φ grounds that (ψ grounds θ)”—what we may call iterated grounding claims. In the metaphysics literature, on the other hand, Bennett (2011), deRosset (2013a), and Dasgupta (2014b) have given accounts of iterated ground, but their accounts were not accompanied by logics of ground. This is perhaps not surprising—and not only because developing such a logic is a non-trivial matter. While one might accept that true grounding claims themselves need to be grounded, one might think that it is a substantive matter what the grounds of grounding claims are: if so, one has no right to expect a logic of iterated ground. In previous work (Litland, 2017b) I argued that by linking ground to a type of “ex- planatory argument” one naturally arrives at a logic of iterated ground. The key upshot of the proposed logic was that true non-factive grounding claims are zero-grounded in Fine’s sense. Unfortunately, the resulting logic was not satisfactory by its own lights. The problem is that what the connection between ground and explanatory argument gives us it not just that true non-factive grounding claims are zero-grounded, it gives us the stronger claim that the only ground they have is the empty ground. The system proposed in (Litland, 2017b) does not allow us to derive this stronger claim. Received ?? Material from this paper was presented at the Munich Center for Mathematical Philosophy and the University of Nottingham. Thanks to the audiences at both places. Thanks to J¨ onne Kriener, Michaela McSweeney, Sam Roberts and Øystein Linnebo for many conversations about ground in Oslo. Thanks to Kit Fine for some very helpful suggestions which allowed me greatly to simplify the presentation of the logic. Thanks to Louis deRosset for very helpful comments on earlier versions of this material. Thanks are especially due to an anonymous referee for extremely helpful and detailed comments on earlier versions of this paper and to the editors of this journal for their patience. c  2009 Association for Symbolic Logic 1 doi:10.1017/S1755020300000000