The quantified argument calculus (Quarc) is a novel logic which departs from mainstream first-order logic by having quantifiers bind unary predicates instead of variables. Moreover, it also contains devices for modelling anaphoras, active-passive-voice distinctions and sentence- versus predicate-negation. First presented in [1], it has since been the subject of multiple further research directions. In [2], a first foray into modal extensions of Quarc was presented. Quarc-analogues of the Barcan formulas were shown to be invalid and existence to be contingent across the board.

\indent However, while a proof system was sketched, its completeness was not demonstrated. My talk presents the first strongly sound and complete proof systems for modal extensions of Quarc, based on work done in [4]. A family of unlabelled, Gentzen-style natural deduction systems are presented, each being an analogue of the usual modal logics K, D, T, S4 and S5. Moreover, identity is incorporated and shown to be contingent by default. Lastly, while the base semantics are two-valued, the proof systems can also be proven to be strongly sound and complete with respect to strong-Kleene three-valued semantics, assuming ST-validity (cf. [3]). I argue that such semantics are crucial for capturing presupposition-failure with respect to quantification, and finish my talk with an outlook on these three-valued semantics.

Bibliography

1. Hanoch Ben-Yami,The Quantified Argument Calculus,The Review of Symbolic Logic,vol. 7 (2014), no. 1, pp. 120–146.
2. Hanoch Ben-Yami,The Barcan Formulas and NecessaryExistence: The View from Quarc,Synthese,vol. 198 (2021), pp. 11029–11064.
3. Pablo Cobreros, Paul Egré, David Ripley, Robert van Rooij,Tolerant, classical, strict,Journal of Philosophical Logic,vol. 41 (2012), no. 2, pp. 347–385.
4. Hongkai Yin and Hanoch Ben-Yami,The Quantified ArgumentCalculus with Two- and Three-valued Truth-valuationalSemantics,Studia Logica,vol. 111 (2023), pp. 281–320.