This formalization verifies a decision procedure due to Cantone and Zarba for a quantifier-free fragment of set theory. The fragment is called multi-level syllogistic with singleton, or MLSS for short. Its syntax syntax includes the usual set operations union, intersection, difference, membership, equality as well as the construction of a set containing a single element. We specify the semantics of MLSS in terms of hereditarily finite sets and provide a sound and complete tableau calculus for it. We also provide an executable specification of a decision procedure that applies the rules of the calculus exhaustively and prove its termination. Furthermore, we extend the calculus with a light-weight type system that paves the way for an integration of the procedure into Isabelle/HOL.
- Logic/General logic/Classical propositional logic
- Logic/Set theory
- Logic/General logic/Decidability of theories
- Stevens, L. (2022). Towards a Verified Tableau Prover for a Quantifier-Free Fragment of Set Theory (Version 2). arXiv. https://doi.org/10.48550/ARXIV.2209.14133