Axiomatic Theories of Partial Ground I

Abstract

This is part one 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. This allows us to connect theories of partial ground with axiomatic theories of truth. In this part of the paper, we develop an axiomatization of the relation of partial ground over the truths of arithmetic and show that the theory is a proof-theoretically conservative extension of the theory P T of positive truth. We construct models for the theory and draw some conclusions for the semantics of conceptualist ground.

Axiomatic Theories of Partial Ground II

“…represents the function that maps the code #t of a term t to the code #T r(t) of the application T r(t) of the truth predicate to the term t. Thus, we have arrived at a quantified axiomatization of the Aristotelian principles. 11 Unfortunately, as [6] shows, the Aristotelian principles are ground-theoretically inconsistent: 12 Lemma 1 (Puzzle of Ground) AP T and AP F are inconsistent over P GT .…”

“…The quantified principle AP F works analogously. 12 Fine works in an operational framework, but his argument can easily be adapted to the present framework. Fine's argument is discussed and refined by [2,14].…”

“…1 Partial ground, remember, is the relation of one truth holding (either 1 For part one, see [12] in this journal.…”

Grounding rules and (hyper-)isomorphic formulas

“…see Fine (2012b); Correia (2014)), as a predicate (e.g. see Korbmacher (2017)), but also as a meta-linguistic relation (e.g. see Poggiolesi (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.…”