Automated Theorem Generation in First-Order Logic by Construction from Triangular Standard Contradictions
Automated theorem generation (ATG) seeks to construct new theorems rather than merely prove given conjectures. Existing approaches typically generate candidate statements that require subsequent proof or formal verification. We propose a constructive framework for ATG in first-order logic based on triangular standard contradictions. The method constructs minimally unsatisfiable clause sets from selected first-order literals, then forms theorems by retaining some clauses as premises and taking the negation of one clause, or a conjunction of clauses, as the conclusion. Minimal unsatisfiability ensures that the remaining premises are satisfiable, avoiding trivial inference by explosion, while unsatisfiability of the full set guarantees the resulting entailment. Theorems are therefore correct by construction, without a separate proof procedure for each instance. Extending the previously developed $Δ_1$ generator, we introduce seven methods, $Δ_2$--$Δ_8$, forming a unified family of eight theorem-generation methods. Four generators, $Δ_1$--$Δ_4$, have been implemented and released. We also formulate a general $Δ$-class method encompassing the family. Combinatorial analysis demonstrates the capacity to generate large numbers of distinct theorems systematically. This framework offers a principled foundation for automated theorem discovery and the construction of logically guaranteed knowledge.