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frege - Some History Surrounding our Deductive Apparatus...

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Some History Surrounding our Deductive Apparatus for P Branden Fitelson 02/07/07 In the Begriffsschrift , Frege gives the following deductive apparatus (PS 0 ) for P : Six (6) Axiom Schemata: (PS1 0 ) A (B A) (PS2 0 ) (A (B C)) ((A B) (A C)) (PS3 0 ) (A (B C)) (B (A C)) (PS4 0 ) (A B) ( B ⊃ ∼ A) (PS5 0 ) ∼∼ A A (PS6 0 ) 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 Hunter’s system PS – see below.). Along the way, he also proved that Frege’s axiom (PS3 0 ) is redundant , since it can be deduced from (PS1 0 ) and (PS2 0 ) by (MP). The reason Frege didn’t notice this is that it is not obvious ! Here’s Lukasiewicz’s proof of (PS3 0 ) from (PS1 0 ) and (PS2 0 ), in schematic form. This is (in a sense) the shortest possible proof of (PS3 0 ) from (PS1 0 ) and (PS2 0 )! Moral: these ain’t easy! [1] ((X (Y Z)) ((X Y) (X Z))) ((Y Z) ((X (Y Z)) ((X Y) (X Z)))) [PS1 0 ] [2] (Y Z) ((X (Y Z)) ((X Y) (X Z))) [MP, 1, PS2 0 ] [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) (X Z)) [??] [7] ((Y Z) ((X Y) (X Z))) (((Y Z) (X Y)) ((Y Z) (X Z))) [??] [8] ((Y Z) (X Y)) ((Y Z) (X Z)) [??] [9] (((X Y) Z)
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