Signed Depth Relevance of subDL

subDL is a logic developed for paraconsistent mathematics by Zach Weber (2021), combining elements of relevant logic and affine logic. Tore Øgaard (2026) shows that subDL satisfies two important relevance properties: the signed variable-sharing property, which requires premises and conclusions of a valid inference to share a propositional variable with the appropriate polarity, and the depth-relevance property, which requires such a shared variable to occur at matching implicational depths. He leaves open whether subDL satisfies the stronger signed depth-relevance property, which combines these two constraints by requiring a variable to occur with both the appropriate sign and the appropriate implicational depth. The result proved here answers Øgaard’s question affirmatively: subDL satisfies the signed depth-relevance property. The proof proceeds by constructing, from any counterexample to signed depth relevance, an interpretation for subDL under which the premises receive designated values while the conclusion does not, contradicting validity. The construction can also be viewed as a simplification of Brady’s ω-rule technique for establishing relevance properties.

subDL satisfies the signed depth relevance property, answering an open question posed by Øgaard (2026). More precisely, every valid inference in subDL contains a propositional variable that occurs in both the premises and conclusion with matching sign and at matching implicational depth.