Explicating Logical Independence

Abstract

Accounts of (complete) logical independence which coincide when applied in the case of classical logic diverge elsewhere, raising the question of what a satisfactory all-purpose account of logical independence might look like. ‘All-purpose’ here means: working satisfactorily as applied across different logics, taken as consequence relations. Principal candidate characterizations of independence relative to a consequence relation are (i) that there the consequence relation concerned is determined by (= sound and complete w.r.t.) only by classes of (bivalent) valuations providing for all possible truth-value combinations for the formulas whose independence is at issue, and (ii) that the consequence relation ‘says’ nothing special about how those formulas are related that it does not say about arbitrary formulas. (The latter approach, we associate with de Jongh, though it is closely related to Marczewski’s notion of general algebraic independence, as well as to the absence of non-trivial logical relations as conceived by Lemmon.) Each of these proposals returns counterintuitive verdicts in certain cases—the truth-value inspired approach classifying certain cases one would like to describe as involving failures of independence as being cases of independence, and the de Jongh approach counting some intuitively independent pairs of formulas as not being independent after all. In final section, a modification of the latter approach is tentatively sketched to correct for these misclassifications. The attention is on conceptual clarification throughout, rather than the provision of technical results. Proofs, as well as further elaborations, are lodged in the ‘longer notes’ in a final Appendix.