The Power of Generalized Clemens Semantics
Hitoshi Omori, Jonas R. B. Arenhart
TL;DR
The paper generalizes Clemens' ordered-pair semantics to $n$-tuples and extends it to first-order languages with quantifiers, situating the framework within Haack's intelligibility program. It proves precise correspondences between $n$-tuple semantics and classical, LP, and K3 readings, e.g. $\Gamma \models_{n,t} A$ iff $\Gamma \models_{\mathbf{LP}} A$, and shows how strict, bossy, and tolerant designations link to $\models_k$, $\models_2$, and $\models_l$. By interpreting nonclassical values as readings relative to classical truth with epistemic or contextual qualifiers (the agent reading, respects, etc.), the authors unify many-valued logics with classical intuition while clarifying the role of quantification and mixed consequences. Applications include analyses of informative contradictions per Égré and mixed-consequence frameworks per Cobreros et al., offering a coherent semantic basis and potential impact on the interpretation of many-valued logics.
Abstract
In this paper, we elaborate on the ordered-pair semantics originally presented by Matthew Clemens for LP (Priest's Logic of Paradox). For this purpose, we build on a generalization of Clemens semantics to the case of n-tuple semantics, for every n. More concretely, i) we deal with the case of a language with quantifiers, and ii) we consider philosophical implications of the semantics. The latter includes, first, a reading of the semantics in epistemic terms, involving multiple agents. Furthermore, we discuss the proper understanding of many-valued logics, namely LP and K3 (Kleene strong 3-valued logic), from the perspective of classical logic, along the lines suggested by Susan Haack. We will also discuss some applications of the semantics to issues related to informative contradictions, i.e. contradictions involving quantification over different respects a vague predicate may have, as advanced by Paul Égré, and also to the mixed consequence relations, promoted by Pablo Cobreros, Paul Égré, David Ripley and Robert van Rooij.