Truthmakers and Relevance for FDE, LP, K3, and CL
(2021) Kit Fine on Truthmakers, Relevance, and Non-Classical Logic — ISBN: [978-3-030-71430-7], accepted/in-press
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- Abstract
- In this paper, we first develop truthmaker semantics for four relevance logics defined as the X-relevant cores (as introduced in [34]) of the well-known propositional logics CL (classical logic), LP (the logic of paradox), K3 (strong Kleene logic) and FDE (first degree entailment). The semantics is similar to Kit Fine’s truthmaker semantics for classical logic, but we define the notion of exact verification similarly to Fine’s inexact notion of loose verification. Dropping Fine’s principle of the Downward Closure of the set of consistent states makes our verification notion nevertheless exact. In order to prove soundness and completeness of the logics w.r.t. the new semantics, we make a detour via a sequent calculus that is adequate both for the four relevance logics and for the corresponding semantics’ exact consequence notion, i.e. each exact verifier of the premises exactly verifies the conclusion. The sequent calculus is interesting in its own right. Finally, we argue that the four presented truthmaker semantics are also interesting semantics for the original (irrelevant) consequence relations FDE, LP, K3, and CL. The most interesting difference with Fine’s approach (seen as a semantics for CL) is the way in which tautologies are handled: next to their usual verifiers, they are also made true by the empty state. We provide philosophical arguments for the plausibility of such an account.
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Verdée, P. (2021). Truthmakers and Relevance for FDE, LP, K3, and CL. In Federico L.G. Faroldi and Frederik Van De Putte (ed.), Kit Fine on Truthmakers, Relevance, and Non-Classical Logic. Springer. https://hdl.handle.net/2078.5/224426