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2.5.1: Rules and Derivations
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Natural deduction proofs begin with assumptions. Inference rules are then applied. Assumptions are “discharged” by the \(\lnot\text{Intro}\), \({\rightarrow}\text{Intro}\), \(\lor\text{Elim}\) and \(\exists\text{Elim}\) inference rules.
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2.5.2: Propositional Rules
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Rules for ∧, ∨, →, ¬, and ⊥
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2.5.3: Quantifier Rules
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Rules for ∀ and ∃
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2.5.4: Derivations
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Each derivation either is an assumption on its own, or consists of one, two, or three derivations followed by a correct inference.
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2.5.5: Examples of Derivations
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Derivations of the sentences \((A \land B) \rightarrow A\) and \((\lnot A \lor B) \rightarrow (A \rightarrow B)\), and an example of the \(\bot_C\) rule
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2.5.6: Derivations with Quantifiers
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When dealing with quantifiers, we have to make sure not to violate the eigenvariable condition, and sometimes this requires us to play around with the order of carrying out certain inferences.
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2.5.7: Proof-Theoretic Notions
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Just as we’ve defined a number of important semantic notions (validity, entailment, satisfiability), we now define corresponding proof-theoretic notions.
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2.5.8: Derivability and Consistency
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We will now establish a number of properties of the derivability relation.
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2.5.9: Derivability and the Propositional Connectives
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2.5.10: Derivability and the Quantifiers
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2.5.11: Soundness
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A derivation system, such as natural deduction, is sound if it cannot derive things that do not actually follow.
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2.5.12: Derivations with Identity predicate
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Derivations with identity predicate require additional inference rules.
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2.5.13: Soundness with Identity predicate
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Natural deduction with rules for = is sound.
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2.5.14: Summary
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