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