11. A formal system is determined by a formal language and a deductive system ( axioms and rules of inference ). 12. The justification of rules of a deductive system depends on our judgements about whether to reject or accept specific deductive inferences. 13. These deductive systems are most often studied for first-order logic, but are of interest for other logics as well. 14. A key property of deductive systems is that they are purely syntactic, so that derivations can be verified without considering any interpretation. 15. A deductive system is "'sound "'if any formula that can be derived in the system is logically valid. 16. Curry's paradox, and other paradoxes arise in Lambda Calculus because of the inconsistency of Lambda calculus considered as a deductive system . 17. A deductive system for a logic is a set of inference rules and logical axioms that determine which sequences of formulas constitute valid proofs. 18. That is, that its content is based on some formal deductive system and that some of its elementary statements are taken as axioms. 19. Several deductive systems can be used for second-order logic, although none can be complete for the standard semantics ( see below ). 20. A converse to completeness is "'soundness, "'the fact that only logically valid formulas are provable in the deductive system .
