unit6 - Hardegree Intermediate Logic Unit 6 Derivations in...

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Unformatted text preview: Hardegree, Intermediate Logic ; Unit 6: Derivations in Description Logic 1 of 9 Unit 6: Derivations in Description Logic 1. EXERCISES Directions: For each of the following, construct a formal derivation of the conclusion (indicated by "/") from the premises (if any). In cases in which two formulas are separated by '//', construct a derivation of each formula from the other. 1. EXERCISE SET A 1. 2200 xFx ; 5 x[x = t] / F[t] 2. 2200 xFx ; 5 x[x = x] / 5 xFx 3. F[t] ; 5 x[x = t] / 5 xFx 4. / 2200 x(Fx ² 5 yFy) 5. / 2200 x( 2200 yFy ² Fx) 6. / 2200 x 5 y[x = y] 7. / 2200 x 2200 y{x = y ² (Fx – Fy)} 8. ~ 5 x[x = x] // 2200 x(Fx & ~ Fx) 9. R[s,t] ; 5 x[x = s] / 5 xR[x,t] 10. ~ 5 x 5 y[x = y] // 2200 x[x & x] 2. EXERCISE SET B 11. / 2200 x 5 y[x = y] 12. / 2200 x(x = • yFy – 2200 y(Fy – y = x)) 13. 5 x[x = • xFx] / 2200 x(Fx – x =• xFx) 14. 5 x[x = • xFx] / F[ • xFx] 15. 5 x[x = • xFx] // 5 x 2200 y(Fy – y = x) 16. / 2200 x[x = • y[y = x]] 17. / 2200 x 2200 y { x=y ↔ •z [z =x ] = •z [ z=y ] } 18. / 2200 x{Fx – F[ • y[y = x]]} 19. / 2200 x 2200 y{x = y ² [ • zRxz = • zRyz]} 20. 5 x 2200 y(Fy – y = x) / 2200 x(Fx – x =• xFx) 21. G[ • xFx]; 5 x[x = • xFx] / 5 x(Fx & Gx) 22. G[ • xFx] ; 5 x[x = • xFx] / 5 x( 2200 y(Fy – y = x) & Gx) 23. G[ • xFx] ; ~ 5 xGx / ~ 5 x[x = • xFx] 24. 5 x 2200 y{Fy – y = x} / 5 x(Fx & x =• xFx) Hardegree, Intermediate Logic ; Unit 6: Derivations in Description Logic 2 of 9 3. EXERCISE SET C 25. 5 x{ 2200 y(Fy – y = x) & Gx} / G[ • xFx] 26. 2200 x{Fx ² x =• xFx} / 5 xFx ² F[ • xFx] 27. t = • xFx ; 5 x[x = t] / 5 x{Fx & 2200 y[Fy ² y = x]} 28. Fa ; 2200 x 2200 y{(Fx & Fy) ² x = y} / a = • xFx 29. t = • xFx ; 5 x 5 y{x & y & Fx & Fy} / ~ 5 x[x = t] 30. 5 x 2200 y([y & t] – y = x) / 2200 x([x = t] ∨ [x =• x[x & t]]) 31. 2200 x 5 y 2200 z(Rxz – z = y) ; 2200 x 2200 y(Rxy ² Ryx) / 2200 x[ • yRxy = • yRyx] 32. 5 x 2200 y(Fy ² y = x) / 2200 x(Fx ² x =• xFx) 33. ~ F[ • xFx] ; 5 xFx / 5 x 5 y{x & y & Fx & Fy} 34. 2200 x(Fx ² Gx) ; 5 x 2200 y(Gy – y = x) / 2200 x(Fx ² [x = • xGx]) 35. 5 x 2200 y{Fy ² y = x} / 2200 x(Fx ² F[ • xFx]) Hardegree, Intermediate Logic ; Unit 6: Derivations in Description Logic 3 of 9 2. ANSWERS TO UNIT 6 EXERCISES #1: (1) 2200 xFx Pr (2) 5 x[x = t] Pr (3) › : F[t] DD (4) | a = t 2, 5 O (5) | Fa 1, 2200 O (6) | F[t] 4,5,LL #2: (1) 2200 xFx Pr (2) 5 x[x = x] Pr (3) › : 5 xFx 5, 5 I (4) | a = a 2, 5 O (5) | Fa 1, 2200 O #3: (1) F[t] Pr (2) 5 x[x = t] Pr (3) › : 5 xFx 5, 5 I (4) | a = t 2, 5 O (5) | Fa 1,4,LL #4: (1) › : 2200 x(Fx ² 5 yFy) UCD (2) | Fa As (3) |› : 5 yFy 2, 5 I #5: (1) › : 2200 x( 2200 yFy...
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This note was uploaded on 04/09/2008 for the course PSYCH 201 taught by Professor Hardegree during the Spring '08 term at UMass (Amherst).

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unit6 - Hardegree Intermediate Logic Unit 6 Derivations in...

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