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To prove that the EFHW basis and the Anderson and Belnap basis are the same system, one needs to prove both that the Anderson and Belnap [AB] basis can prove the EFHW basis, and that the EFHW basis can prove the Anderson and Belnap basis [AB]. Given both of those, they are both bases for the same system.

# Proof output notation =format native # ----------------- Proof: -------------------- # Produced Thu Jul 17 10:15:27 2008 # by the "testy" program, version 2008-07-09 0.01.15 # Input files: # "input.c4.anderson-belnap" # "wishlist.c4.EFHW-2base1" # Name of this logical system. =name C4 # Axiomatic Basis: =basis Anderson and Belnap # --- Axioms --- 1: ('> p p) # Axiom 1 2: ('> ('> p q) ('> r ('> p q))) # Axiom 2 3: ('> ('> p ('> q r)) ('> ('> p q) ('> p r))) # Axiom 3 # --- Wish List --- =wishlist "('> p ('> q q))" "EFHW 2base1 Axiom 1" =wishlist "('> ('> p ('> q r)) ('> ('> p q) ('> s ('> p r))))" "EFHW 2base1 Axiom 2" # --- Theorems --- # Wish list theorem: EFHW 2base1 Axiom 1 4: ('> p ('> q q)) # Condensed detachment, inference line 2 with line 1 5: ('> p ('> ('> q r) ('> s ('> q r)))) # Condensed detachment, inference line 2 with line 2 6: ('> ('> p ('> q r)) ('> p ('> s ('> q r)))) # Condensed detachment, inference line 3 with line 5 7: ('> p ('> ('> q ('> r s)) ('> ('> q r) ('> q s)))) # Condensed detachment, inference line 2 with line 3 8: ('> ('> p ('> q ('> r s))) ('> p ('> ('> q r) ('> q s)))) # Condensed detachment, inference line 3 with line 7 9: ('> ('> p q) ('> ('> r p) ('> r q))) # Condensed detachment, inference line 8 with line 2 10: ('> ('> ('> p q) ('> r p)) ('> ('> p q) ('> r q))) # Condensed detachment, inference line 3 with line 9 11: ('> ('> ('> ('> p q) ('> p r)) s) ('> ('> p ('> q r)) s)) # Condensed detachment, inference line 10 with line 7 # Wish list theorem: EFHW 2base1 Axiom 2 12: ('> ('> p ('> q r)) ('> ('> p q) ('> s ('> p r)))) # Condensed detachment, inference line 11 with line 6 # Final proof had 12 lines (9 steps) [Work 395] # ------------------ End proof -------------- # Summary: # 3 axioms given. # 410 generated theorems in the working set. # 16560 "one step away" theorems in the halo. # 9 of those we generated seemed to be interesting. # (Factor of 40.1 overhead for the halo. [16560 vs. 413])

# Proof output notation =format native # ----------------- Proof: -------------------- # Produced Thu Jul 17 10:59:52 2008 # by the "testy" program, version 2008-07-09 0.01.15 # Input files: # "wishlist.c4.anderson-belnap" # "input.c4.EFHW-2base1" # Name of this logical system. =name C4 # Axiomatic Basis: =basis EFHW 2 base 2 # --- Axioms --- 1: ('> p ('> q q)) # Axiom 1 2: ('> ('> p ('> q r)) ('> ('> p q) ('> s ('> p r)))) # Axiom 2 # --- Wish List --- =wishlist "('> p p)" "Anderson and Belnap, basis 1, axiom 1" =wishlist "('> ('> p q) ('> r ('> p q)))" "Anderson and Belnap, basis 1, axiom 2" =wishlist "('> ('> p ('> q r)) ('> ('> p q) ('> p r)))" "Anderson and Belnap, basis 1, axiom 3" # --- Theorems --- # Wish list theorem: Anderson and Belnap, basis 1, axiom 2 3: ('> ('> p q) ('> r ('> p q))) # Condensed detachment, inference line 2 with line 1 # Wish list theorem: Anderson and Belnap, basis 1, axiom 1 4: ('> p p) # Condensed detachment, inference line 1 with line 1 5: ('> p ('> q ('> r r))) # Condensed detachment, inference line 3 with line 1 6: ('> ('> ('> p ('> q r)) ('> p q)) ('> s ('> ('> p ('> q r)) ('> t ('> p r))))) # Condensed detachment, inference line 2 with line 2 7: ('> p ('> ('> q ('> ('> r r) s)) ('> t ('> q s)))) # Condensed detachment, inference line 6 with line 5 8: ('> ('> p ('> q ('> ('> r r) s))) ('> t ('> p ('> u ('> q s))))) # Condensed detachment, inference line 2 with line 7 9: ('> p ('> ('> q ('> r s)) ('> t ('> ('> q r) ('> q s))))) # Condensed detachment, inference line 8 with line 2 10: ('> ('> p ('> q r)) ('> s ('> ('> p q) ('> p r)))) # Condensed detachment, inference line 9 with line 1 11: ('> ('> ('> p q) p) ('> r ('> ('> p q) q))) # Condensed detachment, inference line 2 with line 4 12: ('> p ('> ('> ('> q q) r) r)) # Condensed detachment, inference line 11 with line 1 13: ('> ('> p ('> ('> q q) r)) ('> s ('> p r))) # Condensed detachment, inference line 2 with line 12 14: ('> p ('> ('> ('> ('> ('> q q) r) r) s) s)) # Condensed detachment, inference line 11 with line 12 15: ('> ('> p ('> ('> ('> ('> q q) r) r) s)) ('> t ('> p s))) # Condensed detachment, inference line 2 with line 14 16: ('> p ('> ('> q ('> ('> r r) s)) ('> q s))) # Condensed detachment, inference line 15 with line 13 17: ('> ('> p ('> ('> q q) r)) ('> p r)) # Condensed detachment, inference line 16 with line 1 # Wish list theorem: Anderson and Belnap, basis 1, axiom 3 18: ('> ('> p ('> q r)) ('> ('> p q) ('> p r))) # Condensed detachment, inference line 17 with line 10 # Final proof had 18 lines (16 steps) [work: 421] # ------------------ End proof --------------

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