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PD : Natural Deduction In P

PD : Natural Deduction In P. Substitution Instance.

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PD : Natural Deduction In P

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  1. PD: Natural Deduction In P

  2. Substitution Instance Substitution Instance:Let P(a/x) indicate the wff which is just like P except for having the constant a in every position where the variable x appears in P. Where Q is a closed wff of the form (8x)P or (9x)P and a is a constant, then P(a/x) is a substitution instance of Q, with a as the instantiating constant.

  3. Given a universally quantified wff you may derive any substitution instance of it, i.e., remove the x-quantifier and replace all occurrences of x in P with the instantiating constant, a There is no restriction on the instantiating constant, you may use any constant In other words, you may derive any instance of a universal claim This rule is sometimes called Universal Instantiation Universal Elimination

  4. Given a wff containing one or more occurrences of a, you may replace one or more of those occurrences of a with x and append an existential x-quantifier In other words, you may existentially quantify into or existentially generalize over one or more of those a-positions This rule is sometimes called Existential Generalization Existential Introduction

  5. Given a wff containing one or more occurrences of a, and provided the restrictions are met, you may replace all of those occurrences of a with x and append a universal x-quantifier In other words, you may universally quantify into or universally generalize over all of those a-positions This rule is sometimes called Universal Generalization Universal Introduction

  6. Given an existentially quantified wff, and provided the restrictions are met, if you can produce a subderivation which leads from a substitution instance to some wff Q, then you may derive Q Note that the instantiating constant a must be “new” to the derivation, as per restrictions (i) and (ii) Further, the instantiating constant a must “die” with the subderivation, as per restriction (iii) Existential Elimination

  7. Rules of PD All the rules of SD plus

  8. Proof Theory in PD Derivable inPD, P:A wff P of S is derivable inPD from a set  of wffs of S iff there is a derivation in PD the primary assumptions of which are members of  and P depends on only those assumptions. Valid inPD:An argument of S is valid in PD iff the conclusion is derivable in PD from the set consisting of only the premises, otherwise it is invalid in PD. Theorem ofPD:A wff P is a theorem ofPD iff P is derivable in PD from the empty set; i.e., iff P. Equivalent in PD:Two wffs P and Q are equivalent in PD iff they are interderivablein PD; i.e., iff both PQ and QP. Inconsistent in PD:A set  of wffsis inconsistent inPD iff, for some wff P, both P and :P (this can be shown using a single derivation).

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