frege - Some History Surrounding our Deductive Apparatus...

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Unformatted text preview: Some History Surrounding our Deductive Apparatus for P Branden Fitelson 02/07/07 In the Begriffsschrift , Frege gives the following deductive apparatus (PS ) for P : Six (6) Axiom Schemata: (PS1 ) A (B A) (PS2 ) (A (B C)) ((A B) (A C)) (PS3 ) (A (B C)) (B (A C)) (PS4 ) (A B) ( B A) (PS5 ) A A (PS6 ) A A One (1) Rule of Inference (Schemata): Modus Ponens (MP). From A and A B , infer B . A couple of decades later, Lukasiewicz discovered simpler sets of axioms that would suffice for P (includ- ing the three-axiom basis of Hunters system PS see below.). Along the way, he also proved that Freges axiom (PS3 ) is redundant , since it can be deduced from (PS1 ) and (PS2 ) by (MP). The reason Frege didnt notice this is that it is not obvious ! Heres Lukasiewiczs proof of (PS3 ) from (PS1 ) and (PS2 ), in schematic form. This is (in a sense) the shortest possible proof of (PS3 ) from (PS1 ) and (PS2 )! Moral: these aint easy! [1] ((X (Y Z)) ((X Y) (X Z))) ((Y Z) ((X (Y Z)) ((X Y) (X Z)))) [PS1 ] [2] (Y Z) ((X (Y Z)) ((X Y) (X Z))) [MP, 1, PS2 ] [3] ((Y Z) ((X (Y Z)) ((X Y) (X Z)))) (((Y Z) (X (Y Z))) ((Y Z) ((X Y) (X Z)))) [??] [4] ((Y Z) (X (Y Z))) ((Y Z) ((X Y) (X Z))) [??] [5] (Y Z) (X (Y Z)) [??] [6] (Y Z) ((X Y)...
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This note was uploaded on 08/01/2008 for the course PHIL 140A taught by Professor Fitelson during the Spring '07 term at University of California, Berkeley.

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frege - Some History Surrounding our Deductive Apparatus...

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