# unit4 - Hardegree, Intermediate Logic; Unit 4: Derivations...

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Hardegree, Intermediate Logic ; Unit 4: Derivations in Identity Logic 1 of 22 UNIT 4: DERIVATIONS IN IDENTITY 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. 2. EXERCISE SET A 1. / 2200 x 2200 y{[Fx & ~ Fy] ² x y} 2. / 2200 x 2200 y{[ ~ Fx & Fy] ² x y} 3. / 2200 x 2200 y{x = y ² [Fx Fy]} 4. / 2200 x 2200 y{x = y ² 2200 z{Rzx Rzy}} 5. / 2200 x 2200 y{x = y ² f(x) = f(y)} 6. / 2200 w 2200 x 2200 y 2200 z{[w = = z] ² s(x,z) = s(w,y)} 7. a = b // 2200 x{x = a ² x = b} 8. a = b // 5 x{x = = b} 9. / 2200 x{Fx 2200 y[y = x ² Fy]} 10. / 2200 x{Fx 5 y[y = 3. EXERCISE SET B 11. 2200 x(Fx ² Gx) / 2200 x(Fx ² 5 = h(y))]) 12. 5 x 2200 y{Fy ² y = x} ; 5 x{Fx & Gx} / 2200 x{Fx ² Gx} 13. 2200 xR[x,b(i)] ; 2200 x{R[b(i),x] x = i} / i = b(i) 14. 5 x 2200 y[x = y] // 2200 x 2200 y[x = y] 15. 5 x 5 y[x y] // 2200 x 5 y[x y] 16. ~ 5 x 2200 y[y = x] / 5 x 5 y[x y] 17. ~ 5 x 5 y[x y] / 5 xFx 2200 xFx 18. / 2200 x 5 y 2200 z{z = y z = x} 19. 2200 x(Fx ² Gx) ; 5 x 2200 y{Gy ² y = x} / 5 x 2200 y{Fy ² y = x} 20. 2200 x{Fx x a} ; 2200 x{Gx x = a} / ~ 5 21. 5 x 2200 y{Fy ² y = x} 5 x 2200 y{Gy ² y = x} / 5 x 2200 y{[Fy & Gy] ² y = x} 22: ~ 5 x 5 y[f(x) f(y)] ; 5 x 2200 y[f(y) = x] 23: 5 x 2200 y{Fy ² y = x} / 2200 x{Fx ² 2200 y(Fy y = x)}

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Hardegree, Intermediate Logic ; Unit 4: Derivations in Identity Logic 2 of 22 4. EXERCISE SET C 24. 5 x 2200 y{[Fy & Gy] y = x} / 5 25. 5 x 2200 y{Fy y = x} // 5 x{Fx & 2200 y[Fy ² y = x]} 26. 5 xFx ; 2200 x 2200 y{(Fx & Fy) ² x = y} / 5 x 2200 y{Fy y = x} 27. 5 xFx ; ~ 5 x 5 y(x y & [Fx & Fy]) / 5 x{Fx & ~ 5 y(y 28. 5 xFx ; ~ 5 x 5 y(x y & [Fx & Fy]) / 5 x 2200 y{Fy y = x} 29. ~ 5 x 2200 y{Fy y = x} // 5 xFx ² 5 x 5 y{x 30. 5 x 2200 y{Fy y = x} / 2200 x{Fx ² Gx} 5 31. 2200 x(Fx ² Gx) ; 5 x 2200 y{Gy y = x} / 5 xFx ² 5 x 2200 y{Fy y = x} 32. 2200 x{Fx ² x = a} // 5 xFx ² 2200 x{Fx x = a} 33. 2200 x 2200 y{[Fx & Fy] ² x = y} // 5 x 2200 y{Fy ² y = x} 34. 5 xFx ; ~ 5 x 5 y(x y & [Fx & Fy]) / 5 x 2200 y{Fy y = x} 35. 5 x 2200 y{Fy y = x} / 5 x{Fx & ~ 5 y[y 36. 5 x{ 2200 y(Gy y = x) & 2200 y(Fy ² Ryx)} / 2200 x 2200 y([Fx & Gy] ² Rxy) 37. 5 x 2200 y{Fy y = x} ; 2200 x{Fx ² Gx} / 5 x 2200 y{[Fy & Gy] y = x} 5. EXERCISE SET D 38. 5 xFx & 2200 x 2200 y{[Fx & Fy] ² x = y} // 5 x 2200 y{Fy y = x} 39. 2200 x 5 y{Fx y = x} / 5 x 5 y[x y] 2200 xFx 40 5 x{ 2200 y(Fy y = x) & Gx} / 5 x 2200 y{[Fy & Gy] y = x} 41 5 x 2200 y{Fy y = x} ; 2200 x(Fx ² Gx) / 5 x 2200 y{[Fy & Gy] y = x} 42. 5 x 2200 y{[Gy & Hy] y = x} ; 2200 x(Fx ² Gx) / 5 x 2200 y{[Fy & Hy]
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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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unit4 - Hardegree, Intermediate Logic; Unit 4: Derivations...

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