Notation: Native | Polish | Psudo Otter | CS

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 otter # ----------------- Proof: -------------------- # Produced Thu Jul 17 10:16:35 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(i(p,p)). # Axiom 1 2: P(i(i(p,q),i(r,i(p,q)))). # Axiom 2 3: P(i(i(p,i(q,r)),i(i(p,q),i(p,r)))). # Axiom 3 # --- Wish List --- =wishlist "P(i(p,i(q,q)))." "EFHW 2base1 Axiom 1" =wishlist "P(i(i(p,i(q,r)),i(i(p,q),i(s,i(p,r)))))." "EFHW 2base1 Axiom 2" # --- Theorems --- # Wish list theorem: EFHW 2base1 Axiom 1 4: P(i(p,i(q,q))). # Condensed detachment, inference line 2 with line 1 5: P(i(p,i(i(q,r),i(s,i(q,r))))). # Condensed detachment, inference line 2 with line 2 6: P(i(i(p,i(q,r)),i(p,i(s,i(q,r))))). # Condensed detachment, inference line 3 with line 5 7: P(i(p,i(i(q,i(r,s)),i(i(q,r),i(q,s))))). # Condensed detachment, inference line 2 with line 3 8: P(i(i(p,i(q,i(r,s))),i(p,i(i(q,r),i(q,s))))). # Condensed detachment, inference line 3 with line 7 9: P(i(i(p,q),i(i(r,p),i(r,q)))). # Condensed detachment, inference line 8 with line 2 10: P(i(i(i(p,q),i(r,p)),i(i(p,q),i(r,q)))). # Condensed detachment, inference line 3 with line 9 11: P(i(i(i(i(p,q),i(p,r)),s),i(i(p,i(q,r)),s))). # Condensed detachment, inference line 10 with line 7 # Wish list theorem: EFHW 2base1 Axiom 2 12: P(i(i(p,i(q,r)),i(i(p,q),i(s,i(p,r))))). # Condensed detachment, inference line 11 with line 6 # Final proof had 12 lines (9 steps) # ------------------ 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 otter # ----------------- Proof: -------------------- # Produced Thu Jul 17 10:48:20 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(i(p,i(q,q))). # Axiom 1 2: P(i(i(p,i(q,r)),i(i(p,q),i(s,i(p,r))))). # Axiom 2 # --- Wish List --- =wishlist "P(i(p,p))." "Anderson and Belnap, basis 1, axiom 1" =wishlist "P(i(i(p,q),i(r,i(p,q))))." "Anderson and Belnap, basis 1, axiom 2" =wishlist "P(i(i(p,i(q,r)),i(i(p,q),i(p,r))))." "Anderson and Belnap, basis 1, axiom 3" # --- Theorems --- # Wish list theorem: Anderson and Belnap, basis 1, axiom 1 3: P(i(p,p)). # Condensed detachment, inference line 1 with line 1 # Wish list theorem: Anderson and Belnap, basis 1, axiom 2 4: P(i(i(p,q),i(r,i(p,q)))). # Condensed detachment, inference line 2 with line 1 5: P(i(p,i(q,i(r,r)))). # Condensed detachment, inference line 4 with line 1 6: P(i(i(i(p,i(q,r)),i(p,q)),i(s,i(i(p,i(q,r)),i(t,i(p,r)))))). # Condensed detachment, inference line 2 with line 2 7: P(i(p,i(i(q,i(i(r,r),s)),i(t,i(q,s))))). # Condensed detachment, inference line 6 with line 5 8: P(i(i(p,i(q,i(i(r,r),s))),i(t,i(p,i(u,i(q,s)))))). # Condensed detachment, inference line 2 with line 7 9: P(i(p,i(i(q,i(r,s)),i(t,i(i(q,r),i(q,s)))))). # Condensed detachment, inference line 8 with line 2 10: P(i(i(p,i(q,r)),i(s,i(i(p,q),i(p,r))))). # Condensed detachment, inference line 9 with line 1 11: P(i(i(i(p,q),p),i(r,i(i(p,q),q)))). # Condensed detachment, inference line 2 with line 3 12: P(i(p,i(i(i(q,q),r),r))). # Condensed detachment, inference line 11 with line 1 13: P(i(i(p,i(i(q,q),r)),i(s,i(p,r)))). # Condensed detachment, inference line 2 with line 12 14: P(i(p,i(i(i(i(i(q,q),r),r),s),s))). # Condensed detachment, inference line 11 with line 12 15: P(i(i(p,i(i(i(i(q,q),r),r),s)),i(t,i(p,s)))). # Condensed detachment, inference line 2 with line 14 16: P(i(p,i(i(q,i(i(r,r),s)),i(q,s)))). # Condensed detachment, inference line 15 with line 13 17: P(i(i(p,i(i(q,q),r)),i(p,r))). # Condensed detachment, inference line 16 with line 1 # Wish list theorem: Anderson and Belnap, basis 1, axiom 3 18: P(i(i(p,i(q,r)),i(i(p,q),i(p,r)))). # Condensed detachment, inference line 17 with line 10 # Final proof had 18 lines (16 steps) # ------------------ End proof -------------- # Summary: # 2 axioms given. # 500 generated theorems in the working set. # 13905 "one step away" theorems in the halo. # 4 of those we generated seemed to be interesting. # (Factor of 27.7 overhead for the halo. [13905 vs. 502])

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