10.1 - Yuri Balashov, PHIL 2500 Lecture Notes Derivations...

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Yuri Balashov, PHIL 2500 Lecture Notes Derivations in Predicate Logic The Natural Deduction System PD Derivation rules of PD = Derivation rules of SD (applied to complete sentences of PL considered as wholes) + New rules for introduction and elimination of quantifiers (“intelim” rules for ( x) and ( x)) Derive: ( x) Ux 1 ( x) Hx ( y ) U y A s s u m . 2 ( x) ( y) (Sx x) Hx Assum. ————————————— 3 ( x ) H x 2 E 4 ( x ) U x 1 , 3 E 1
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Yuri Balashov, PHIL 2500 Lecture Notes 2 Elimination Rule for ‘( x)’ Allows deriving any substitution instance from a universally quantified sentence. For example: Given Derive ( x) Fx Fa (or Fb, Fc, …) ( y) Gby Gba,Gbb (notably!)… ( z) (Az Bzz) Aa Baa Ad Bdd … ( x) ( y) (Pxy & Qxx) ( y) (Pay & Qaa) ( y) (Pby & Qbb) … In general: Given Derive ( x ) P P ( a/x ) Universal Elimination ( E) ( x ) P P ( a/x )
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Yuri Balashov, PHIL 2500 Lecture Notes Derive: Kg 1 ( x) ( y) Hxy Assum. 2 Hcf K g A s s u m . —————— 3 4 5 Derive: Pi Ai 1 ( x) (Px T x ) A s s u m . 2 ( x) (Tx A x ) A s s u m . —————— 3 4 5 6 7 8 3
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Introduction Rule for ‘( x)’ Allows deriving an existentially quantified sentence from any of its substitution instances. For example: Given Derive Fa (or Fb, Fc, …) ( x) Fx G b a ( y) Gya ( z) Gbz G b b ( x) Gbx ( y) Gyb ( x) Gxx Aa B a a ( z) (Az Bzz) ( z) (Aa Bzz) ( z) (Aa Baz) ( z) (Aa Bza) ( z) (Az Bza) ( z) (Az Baz)
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This note was uploaded on 03/22/2009 for the course PHIL 2500 taught by Professor Staff during the Spring '08 term at University of Georgia Athens.

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10.1 - Yuri Balashov, PHIL 2500 Lecture Notes Derivations...

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