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Natural Deduction in P

Learn how to apply quantifier rules in natural deduction by replacing variables with constants in substitution instances. Explore examples of Universal Elimination and Existential Introduction.

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Natural Deduction in P

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  1. Natural Deduction in P Gregory Chapter 7

  2. Quantifier in and out rules

  3. 7.1 Derivation Rules for the Quantifiers • How the Quantifier Rules Work: in applying a Q Rule, we knock off the leftmost quantifier from the sentence to which the rule is applied and replace each instance of the variable it binds with a constant—same for each substitution. Substitution Instance Let ℚ(𝕒/𝕩) indicate the WFF which is just like ℚ except for having the constant 𝕒 in every position where the variable 𝕩 appears in ℚ. Where ℙ is a closed WFF of the form (∀𝕩)ℚ or (∃𝕩)ℚ, then ℚ(𝕒/𝕩) is a substitution instance of ℙ, with 𝕒 as the instantiating constant.

  4. 7.1.1 Universal Elimination - ∀E All cats bite and scratch.Tse-Tse is a cat. Tse-Tse scratches. 1 (∀x)(Cx → (Bx ∧ Sx) P2 Ct P ⊢ St 3 Ct → (Bt ∧ St) 4 Bt ∧ St 2, 3 →E 5 St 4, ∧E Example

  5. 7.1.1 Existential Introduction - ∃I 1 P2 P ⊢ 3 4 Example

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