Axiomatic Theories of Partial Ground II

Abstract

This is part two of a two-part paper in which we develop an axiomatic theory of the relation of partial ground. The main novelty of the paper is the of use of a binary ground predicate rather than an operator to formalize ground. In this part of the paper, we extend the base theory of the first part of the paper with hierarchically typed truth-predicates and principles about the interaction of partial ground and truth. We show that our theory is a proof-theoretically conservative extension of the ramified theory of positive truth up to 0 and thus is consistent. We argue that this theory provides a natural solution to Fine's "puzzle of ground" about the interaction of truth and ground. Finally, we show that if we apply the truth-predicate to sentences involving our ground-predicate, we run into paradoxes similar to the semantic paradoxes: we get ground-theoretical paradoxes of self-reference.

Ground first: against the proof-theoretic definition of ground

“…A principled way of doing this is to reject that there is an untyped notion of propositional truth and rather adopt the predicativist line developed by Korbmacher (2018bKorbmacher ( , 2018a where propositions and truth-properties come in orders where a truth-property of a given order can only be applied to a propositions of lower orders. However, such a predicativist line is unlikely to sit well with (∨-Internality).…”

Formal explanations as logical derivations

“…1 In other terms, we can say that the consequence holds in virtue of the ground. Logical systems of various kinds have been employed for the characterisation and study of the general features of grounding, see for instance the works by Fine (2012aFine ( , 2012b and Korbmacher (2018aKorbmacher ( , 2018b. Moreover, much work is being devoted to characterise the relation that holds between a logically complex formula F and the formulae in virtue of which F holds.…”

“…In particular, we will consider the notion of complete logical grounding introduced by Poggiolesi (2016) and based on Bolzano's analysis of Abfolge, and we will study the resulting notion of formal explanation, that is, of derivation constructed by only employing the grounding rules that characterise this notion of logical grounding. Our 1 See Betti (2010); Correia (2014); Correia and Schnieder (2012); Fine (2012a); Korbmacher (2018b); McSweeney (2020); Poggiolesi (2016); Rosen (2010); Rumberg (2013); Schnieder (2011);Sider (2018).…”

In Defense of the (Moderate) Disunity of Grounding

“…This principle can be rejected on predicativist grounds—for a detailed development see Korbmacher 2018a, 2018b discussing this further will take us too far afield.…”