What Absurdity Is Not in Natural Deduction: Structural Approach and the Empty Set. - Pezlar, Ivo (Topoi: An International Review of Philosophy, 2026)
In this paper, I argue that on a structural view of absurdity within natural deduction frameworks for both classical and intuitionistic logic, absurdity cannot be identified with the empty set; that is, with something that contains nothing. Instead, absurdity should be seen as representing nothing; that is, as the empty space below the inference line, as Neil Tennant originally suggested. However, diverging from Tennant's view that absurdity should be purely a structural notion, I consider an approach that incorporates both structural and logical features. Specifically, this version of absurdity cannot be embedded into other propositions (just like absurdity in the structural approach) yet it is not merely a punctuation mark in the metalanguage but a proper part of the object language of the logic (just like absurdity in the logical approach). This will allow us to show that accepting the structural approach does not require adopting inferences with no conclusion, as was previously thought. Formally, I treat absurdity as a limit case of a signed proposition and, informally, it may be interpreted as a degenerate form of denial that lacks any propositional content.