Abstract
We introduce a two-valued and a three-valued truth-valuational substitutional semantics for the Quantified Argument Calculus (Quarc). We then prove that the 2-valid arguments are identical to the 3-valid ones with strict-to-tolerant validity. Next, we introduce a Lemmon-style Natural Deduction system and prove the completeness of Quarc on both two- and three-valued versions, adapting Lindenbaum’s Lemma to truth-valuational semantics. We proceed to investigate the relations of three-valued Quarc and the Predicate Calculus (PC). Adding a logical predicate T to Quarc, true of all singular arguments, allows us to represent PC quantification in Quarc and translate PC into Quarc, preserving validity. Introducing a weak existential quantifier into PC allows us to translate Quarc into PC, also preserving validity. However, unlike the translated systems, neither extended system can have a sound and complete proof system with Cut, supporting the claim that these are basically different calculi.
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Yin, H., Ben-Yami, H. The Quantified Argument Calculus with Two- and Three-valued Truth-valuational Semantics. Stud Logica (2022). https://doi.org/10.1007/s11225-022-10022-5
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DOI: https://doi.org/10.1007/s11225-022-10022-5
Keywords
- Quantified argument calculus
- Truth-valuational semantics
- Substitutional quantification
- Lindenbaum’s lemma
- Three-valued semantics
- Strict-to-tolerant validity
- Completeness