63 negations Dave Ripley Universities of Connecticut and Melbourne - PowerPoint PPT Presentation

1/ 52 63 negations Dave Ripley Universities of Connecticut and Melbourne Australasian Association for Logic 2013 davewripley@gmail.com 63 negations

2/ 52 1024 1024 to 100 100 to 63 63 davewripley@gmail.com 63 negations

1024 DLL, SM 3/ 52 1024 DLL, SM davewripley@gmail.com 63 negations

1024 DLL, SM 4/ 52 (Bounded) DLL: Axioms: A ⊢ A A ⊢ ⊤ ⊥ ⊢ A A 0 ∧ A 1 ⊢ A i A i ⊢ A 0 ∨ A 1 A ∧ ( B ∨ C ) ⊢ ( A ∧ B ) ∨ ( A ∧ C ) Rules: A ⊢ B A ⊢ C A ⊢ C B ⊢ C A ⊢ B ∧ C A ∨ B ⊢ C A ⊢ B B ⊢ C A ⊢ C davewripley@gmail.com 63 negations

1024 DLL, SM 5/ 52 SM A ⊢ B − B ⊢ − A davewripley@gmail.com 63 negations

1024 DLL, SM 6/ 52 Dualizing a sequent: swap premise/conclusion, ∧ / ∨ , and ⊤ / ⊥ . Every axiom has a dual theorem. Every rule has a dual rule. So every proof has a dual proof: a proof of the dual theorem. davewripley@gmail.com 63 negations

1024 DLL, SM 7/ 52 Derived rule: If A ⊢ B and C ( ) is a positive context, then C ( A ) ⊢ C ( B ) . If C ( ) is negative, then C ( B ) ⊢ C ( A ) . (Proof: induction on C ( ) .) davewripley@gmail.com 63 negations

1024 10 principles 8/ 52 1024 10 principles davewripley@gmail.com 63 negations

1024 10 principles 9/ 52 Normality principles: N 1: ⊤ ⊢ −⊥ N 2: −⊤ ⊢ ⊥ N : N 1 + N 2 davewripley@gmail.com 63 negations

1024 10 principles 10/ 52 Antidistribution principles: A 1: − A ∧ − B ⊢ − ( A ∨ B ) A 2: − ( A ∧ B ) ⊢ − A ∨ − B A : A 1 + A 2 davewripley@gmail.com 63 negations

1024 10 principles 11/ 52 Double negation principles: D 1: A ⊢ − − A D 2: − − A ⊢ A D : D 1 + D 2 davewripley@gmail.com 63 negations

1024 10 principles 12/ 52 Minimal ex’ion principles: X 1: A ∧ − A ⊢ −⊤ X 2: −⊥ ⊢ A ∨ − A X : X 1 + X 2 Recall: −⊤ ⊢ − B , and − B ⊢ −⊥ . davewripley@gmail.com 63 negations

1024 10 principles 13/ 52 Full ex’ion principles: X + 1: A ∧ − A ⊢ ⊥ X + 2: ⊤ ⊢ A ∨ − A X + : X + 1 + X + 2 davewripley@gmail.com 63 negations

1024 10 principles 14/ 52 The 1 principles and 2 principles are dual. davewripley@gmail.com 63 negations

1024 10 principles 15/ 52 2 10 = 1024 specifications. How many distinct logics? davewripley@gmail.com 63 negations

1024 to 100 1024 to 256 16/ 52 1024 to 100 1024 to 256 davewripley@gmail.com 63 negations

1024 to 100 1024 to 256 17/ 52 Clearly: X 1 + N 2 entails X + 1. Clearly: X + 1 entails X 1. Less clearly: X + 1 entails N 2. davewripley@gmail.com 63 negations

1024 to 100 1024 to 256 18/ 52 −⊤ ⊢ ⊤ −⊤ ⊢ −⊤ X + 1: ∧ R: −⊤ ⊢ ⊤ ∧ −⊤ ⊤ ∧ −⊤ ⊢ ⊥ Cut: −⊤ ⊢ ⊥ davewripley@gmail.com 63 negations

1024 to 100 1024 to 256 19/ 52 Dually, X + 2 entails N 1. No need for X + principles. Down to 2 8 = 256. davewripley@gmail.com 63 negations

1024 to 100 256 to 100 20/ 52 1024 to 100 256 to 100 davewripley@gmail.com 63 negations

1024 to 100 256 to 100 21/ 52 D i entails N i : ⊥ ⊢ −⊤ − − ⊤ ⊢ −⊥ ⊤ ⊢ − − ⊤ ⊤ ⊢ −⊥ davewripley@gmail.com 63 negations

1024 to 100 256 to 100 22/ 52 D i entails A i : Part I − A ⊢ − A ∨ − B − ( − A ∨ − B ) ⊢ − − A D 2: − ( − A ∨ − B ) ⊢ A Part II − ( − A ∨ − B ) ⊢ A ∧ B − ( A ∧ B ) ⊢ − − ( − A ∨ − B ) D 2: − ( A ∧ B ) ⊢ − A ∨ − B davewripley@gmail.com 63 negations

1024 to 100 256 to 100 23/ 52 D i N i A i N i A i ∅ davewripley@gmail.com 63 negations

1024 to 100 256 to 100 24/ 52 This cuts 2 3 = 8 down to 5. 256 was ( 8 × 2 ) 2 . We reach ( 5 × 2 ) 2 = 100. davewripley@gmail.com 63 negations

100 to 63 Remaining entailments 25/ 52 100 to 63 Remaining entailments davewripley@gmail.com 63 negations

100 to 63 Remaining entailments 26/ 52 That exhausts one-principle entailments. Combinations remain. davewripley@gmail.com 63 negations

100 to 63 Remaining entailments 26/ 52 That exhausts one-principle entailments. Combinations remain. (Note: no use of distribution yet!) davewripley@gmail.com 63 negations

100 to 63 Remaining entailments 27/ 52 Classical: The pair of X + principles together entail all others. So X + N does too. Usual presentation of Boolean algebra. davewripley@gmail.com 63 negations

100 to 63 Remaining entailments 27/ 52 Classical: The pair of X + principles together entail all others. So X + N does too. Usual presentation of Boolean algebra. Of our 100, 15 are classical. davewripley@gmail.com 63 negations

100 to 63 Remaining entailments 28/ 52 Full X brings full A . Part I X 2: − ( A ∧ B ) ⊢ B ∨ − B Fiddling: − ( A ∧ B ) ⊢ − ( A ∧ B ) ∧ ( B ∨ − B ) Dist: − ( A ∧ B ) ⊢ ( − ( A ∧ B ) ∧ B ) ∨ ( − ( A ∧ B ) ∧ − B ) DRule ( ∧ elim): − ( A ∧ B ) ⊢ ( − ( A ∧ B ) ∧ B ) ∨ − B Part II X 2: − ( A ∧ B ) ⊢ A ∨ − A Fiddling: − ( A ∧ B ) ∧ B ⊢ ( − ( A ∧ B ) ∧ B ) ∧ ( A ∨ − A ) Dist: − ( A ∧ B ) ∧ B ⊢ ( − ( A ∧ B ) ∧ B ∧ A ) ∨ ( − ( A ∧ B ) ∧ B ∧ − A ) DR ( ∧ E): − ( A ∧ B ) ∧ B ⊢ ( − ( A ∧ B ) ∧ B ∧ A ) ∨ − A DR ( X 1, fiddling): − ( A ∧ B ) ∧ B ⊢ − A ∨ − A Fiddling: − ( A ∧ B ) ∧ B ⊢ − A davewripley@gmail.com 63 negations

100 to 63 Remaining entailments 29/ 52 Full X plus N i brings D i . Dist: − − A ∧ ( A ∨ − A ) ⊢ ( − − A ∧ A ) ∨ ( − − A ∧ − A ) D Rule ( X + 1 = X 1 N 2): − − A ∧ ( A ∨ − A ) ⊢ ( − − A ∧ A ) ∨ ⊥ ⊥ -drop: − − A ∧ ( A ∨ − A ) ⊢ − − A ∧ A ∧ -elim: − − A ∧ ( A ∨ − A ) ⊢ A davewripley@gmail.com 63 negations

100 to 63 Remaining entailments 30/ 52 Full N plus X i and A i brings D i . Dist: A ∧ ( − A ∨ − − A ) ⊢ ( A ∧ − A ) ∨ ( A ∧ − − A ) Fiddling (DRule, X 1, N 2): A ∧ ( − A ∨ − − A ) ⊢ − − A DRule ( A 2): A ∧ − ( A ∧ − A ) ⊢ − − A DRule ( X 1, N 2): A ∧ −⊥ ⊢ − − A DRule ( N 1): A ∧ ⊤ ⊢ − − A ⊤ -drop: A ⊢ − − A davewripley@gmail.com 63 negations

100 to 63 Remaining entailments 31/ 52 D i plus X i brings X i . X 2: −⊥ ⊢ A ∨ − A SM: − ( A ∨ − A ) ⊢ − − ⊥ DR( N 1): − ( A ∨ − A ) ⊢ −⊤ A ⊢ − − A Fiddling: A 1: − A ∧ A ⊢ − A ∧ − − A − A ∧ − − A ⊢ − ( A ∨ − A ) Cut: − A ∧ A ⊢ − ( A ∨ − A ) davewripley@gmail.com 63 negations

100 to 63 Remaining entailments 32/ 52 X i plus N i plus A i brings A i . Part I X 1: − A ∧ − − A ⊲ − ( A ∨ B ) Fiddling: ( − A ∧ − − A ) ∨ ( − A ∧ − ( A ∨ B )) ⊲ − ( A ∨ B ) Dist: − A ∧ ( − − A ∨ − ( A ∨ B )) ⊲ − ( A ∨ B ) DRule ( A 2): − A ∧ − ( − A ∧ ( A ∨ B )) ⊲ − ( A ∨ B ) Part II: ECQ ( X 1 + N 2): A ∧ − A ⊲ B Fiddling: ( − A ∧ A ) ∨ ( − A ∧ B ) ⊲ B Cut (Dist): − A ∧ ( A ∨ B ) ⊲ B davewripley@gmail.com 63 negations

100 to 63 Remaining entailments 33/ 52 That’s it! Down to 63. davewripley@gmail.com 63 negations

63 Some interesting critters 34/ 52 63 Some interesting critters davewripley@gmail.com 63 negations

63 Some interesting critters 35/ 52 CL is NX (= DX ). davewripley@gmail.com 63 negations

63 Some interesting critters 35/ 52 CL is NX (= DX ). FDE is D . davewripley@gmail.com 63 negations

63 Some interesting critters 36/ 52 Preminimal logic (the logic of compatibility frames) is N 1 A 1. Dual premin (the logic of exhaustiveness frames) is N 2 A 2. davewripley@gmail.com 63 negations

63 Some interesting critters 37/ 52 Ockham logic (the logic of the Routley star) is NA . davewripley@gmail.com 63 negations

63 Some interesting critters 38/ 52 D 1 X 1 is the strongest logic here sound for minimal logic, N 2 D 1 X 1 for intuitionist. D 2 X 2 is the strongest logic here sound for dual minimal logic, N 1 D 2 X 2 for dual intuitionist. davewripley@gmail.com 63 negations

63 Some interesting critters 38/ 52 D 1 X 1 is the strongest logic here sound for minimal logic, N 2 D 1 X 1 for intuitionist. D 2 X 2 is the strongest logic here sound for dual minimal logic, N 1 D 2 X 2 for dual intuitionist. Are they these logics? davewripley@gmail.com 63 negations

63 Some interesting critters 39/ 52 Other than CL, there are two “attractive” logics among the 100: L1: X 2 D 1 = XN 1 = XA 2 D 1 L2: X 1 D 2 = XN 2 = XA 1 D 2 Anybody recognize these? davewripley@gmail.com 63 negations

63 Organized by X 40/ 52 63 Organized by X davewripley@gmail.com 63 negations

63 Organized by X 41/ 52 No X : 25 logics 0 N 1 A 1 N 1 A 1 D 1 � X 0 ⋆ (Pre) ⋆ (Quas) ⋆ ⋆ ⋆ N 2 ⋆ ⋆ ⋆ ⋆ ⋆ A 2 ⋆ ⋆ ⋆ ⋆ ⋆ N 2 A 2 ⋆ (DPre) ⋆ (Ock) ⋆ ⋆ ⋆ D 2 ⋆ (DQuas) ⋆ (FDE) ⋆ ⋆ ⋆ davewripley@gmail.com 63 negations