The operators of first-order logic, including negation, operate on whole formulae. This makes it unsuitable as a tool for the formal analysis of reasoning with non-sentential forms of negation such as predicate term negation (e.g., negatively affixed gradable adjectives). We extend its language with negation operators whose scope is more narrow than an atomic formula. Exploiting the usefulness of subatomic proof-theoretic considerations for the study of subatomic inferential structure, we define intuitionistic subatomic natural deduction systems which have several subatomic operators and an additional operator for formula negation at their disposal. We establish normalization and subexpression (resp. subformula) property results for the systems. The normalization results allow us to formulate a proof-theoretic semantics for formulae composed of the subatomic operators. We illustrate the systems with applications to reasoning with combinations of sentential negation, predicate term negation (of adjectives, verbs, and common nouns), subject term negation, and antonymy.
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I would like to thank the three anonymous reviewers for their perceptive comments and suggestions. I also thank the discussants at the Wilhelm-Schickard-Institute’s logic colloquium (University of Tübingen) and those at the 10th Scandinavian Logic Symposium (University of Gothenburg) where earlier versions of this paper have been presented in 2018. Support by the DFG (Project: Proof-theoretic foundations of intensional semantics; Grant: WI 3456/4-1) is gratefully acknowledged.
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Więckowski, B. Subatomic Negation. J of Log Lang and Inf 30, 207–262 (2021). https://doi.org/10.1007/s10849-020-09325-4
- Affixal negation
- Predicate term negation
- Proof theory
- Proof-theoretic semantics