AI systems are usually characterized by non-standard logics. The respective calculi include nonlogical axioms expressing domain knowledge. Please see my relevant research papers:

• Ordered Inference in Sequent Calculi for Applied Logics
• Partially Ordered Derivations in Sequent Calculi for Nonstandard Logics
• Analytic Sequent Calculi With Nonlogical Axioms
• A Normal Form of Derivations for Quantifier-Free Sequent Calculi With Nonlogical Axioms
• Fuzzy Non-Horn Knowledge Bases: Calculi, Models, Inference
• Calculi and Models for Non-Horn Knowledge Bases Containing Neural and Evaluable Predicates
• Symbolic Inference for Non-Horn Knowledge Bases With Fuzzy Predicates
• A Logical Characterization of Evaluable Knowledge Bases
• Knowledge Base Logics
• Inference Methods for Evaluable Knowledge Bases
• Inference From Non-Horn Clauses With Partial Predicates (short paper)