Ground and Grain

Peter Fritz

doi:10.1111/phpr.12822

Abstract

AbstractCurrent views of metaphysical ground suggest that a true conjunction is immediately grounded in its conjuncts, and only its conjuncts. Similar principles are suggested for disjunction and universal quantification. Here, it is shown that these principles are jointly inconsistent: They require that there is a distinct truth for any plurality of truths. By a variant of Cantor’s Theorem, such a fine‐grained individuation of truths is inconsistent. This shows that the notion of grounding is either not in good standing, or that natural assumptions about it need to be revised.

Highlights

The facts we encounter in our lives are not completely chaotic
One option is to reject them wholesale, either by giving up the notion of immediate ground, or by giving up the idea of logical ground
If the idea of logical ground is given up, the claim that grounding leads to distinctions between necessarily equivalent propositions loses much of its appeal

Summary

| INTRODUCTION

The facts we encounter in our lives are not completely chaotic. From the fact that this is red, we can infer the fact that something is red. The argument for the inconsistency of the original seven principles of section 2.2 has the same general form as the one just given: An open formula with a plural propositional parameter is specified which according to the grounding principles expresses a distinct truth for any truths (used to interpret the parameter) To state this more formally, we adopt the common convention of writing, e.g., ψ(pp) for the result of replacing a contextually salient free variable in ψ by pp. Similar to the case of arbitrary conjunction, the seven paradigmatic principles of ground set out in section 2.2 require a degree of fineness of grain of truths which is logically inconsistent, demanding a distinct truth ψ(pp) for any truths pp. These are jointly inconsistent by plural Cantorian reasoning

LOGICAL GROUND

| CONCLUSION