Abstract

In this paper, we first develop truthmaker semantics for four relevance logics defined as the non-transitive relevant cores [as introduced in Verdée et al. (Aust J Log 16:10–40, 2019)] of the well-known propositional logics CL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {CL}}$$\end{document} (classical logic), LP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {LP}}$$\end{document} (the logic of paradox), K3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {K3}}$$\end{document} (strong Kleene logic), and FDE\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {FDE}}$$\end{document} (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 notion of loose verification. Dropping Fine’s principle of Downward Closure of the set of consistent states nevertheless warrants the exactness of our verification notion. The semantics of the four non-transitive relevance logics shows that they are in fact straightforward cut-free substructural logics, despite their definition by means of a filtering criterion. We develop the associated sound and complete sequent calculi for these logics. Finally, we argue that the four presented truthmaker semantics are also interesting alternatives to the standard Kit Fine style truthmaker semantics for the original (irrelevant) consequence relations FDE\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {FDE}}$$\end{document}, LP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {LP}}$$\end{document}, K3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {K3}}$$\end{document}, and CL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {CL}}$$\end{document} themselves. The most interesting difference with Fine’s approach 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.