(3)t对n1、3在a中不自由 t对x2在a中自由 评分标准: (1)每个变元1分 (2)每个式子2分, (3)t对每个变元1分 七、(10分)写出N中下列公式的前束范式: Cr F(r1, 2)VVr2 F(a1, 2))-Vr1r2 F(a1, r2) 解: mF(1, r2)VVx2 F(1, r2))-Va1t2 F(ar1,2 (丑r3F(x3,x2)r4F(x1,x4)→Vrsr6F(x5,r6 E.r3V4(F(3, a2)V F(1, I4))-Vcs 6 F(as, I6) Va334((F(3, 2)V F(1, 24))-Vrs6F(r5,I6 HVr3-r4V.rs.6((F(3, r2)V F(1, T4))+F(s, 6)) 评分标准: (1)第二步中,不对约来变元换名的,每一处扣1分。 将自由变元换名的,每一处扣1分 (2)第三、四和五步中,如将量词换错名的,每处扣2分 3)第四和五步中,量词顺序有锴误的,每一处扣1分 八、(10分)设x不在公式a中自由出现。在Nc中证明 证:(1)(a→B)→xy,B,a+B (2)(a→3x)→xrn,B,ah彐xB (+) (3)(a→33)→rn,B}a→彐rB (4)(a→彐r)→vx,B}(a→3x)→Wx7(∈) (6)(a→3x)→rh3→V →+ (x不在前提中自由出现 5(3) t ✭ x1 ❁ x3 ✮ α ✯❱➑✄✦Ö➲➂❿ t ✭ x2 ✮ α ✯✄✦Ö➲➂❿ ♠✐❱♥❳♦❬ (1) ♣❣✪✧➂❵ 1 ✐❱❿ (2) ♣❣❳➉➂❞ 2 ✐❱❿ (3) t ✭♣❣✪✧➂❵ 1 ✐❱❿ ✰❁ (10 ❃) ❄❳❅ NL Ü➂↕àÚ❱③❂❍✛✱✓✚❽③ ❬ (∃x1F(x1, x2) ∨ ∀x2F(x1, x2)) → ∀x1x2F(x1, x2). ❩❱❬ (∃x1F(x1, x2) ∨ ∀x2F(x1, x2)) → ∀x1x2F(x1, x2) |−−| (∃x3F(x3, x2) ∨ ∀x4F(x1, x4)) → ∀x5x6F(x5, x6) |−−| ∃x3∀x4(F(x3, x2) ∨ F(x1, x4)) → ∀x5x6F(x5, x6) |−−| ∀x3∃x4  (F(x3, x2) ∨ F(x1, x4)) → ∀x5x6F(x5, x6)  |−−| ∀x3∃x4∀x5x6  (F(x3, x2) ∨ F(x1, x4)) → F(x5, x6)  ♠✐❱♥❳♦❬ (1) ✲➂❴✪✳✁✯➂❤✛➑✆✭✴★✆✩☛✧❥❵✓✵✛✶❆❭❈❤ ♣ ➓✓✷✙❐ 1 ✐❱❨ ✸ ✦Ö➲✁✧➂❵✓✵✛✶❆❭❥❤ ♣ ➓✓✷✙❐ 1 ✐❱❨ (2) ✲ ➡❁✺✹ ➊✆✻☛✳✛✯❈❤①➦✸✓✼✓✽ ✵❂➷✛✶❆❭❈❤ ♣ ✷❂❐ 2 ✐❱❨ (3) ✲ ✹ ➊✞✻✪✳✛✯❈❤ ✼✓✽✛✾✓✿❫✙➷✓❀➌❭❈❤ ♣ ➓✛✷❂❐ 1 ✐❱❿ ❁❁ (10 ❃) r x ❘❱Û❱Ú❱③ α Ü✪✘➂❲❱❅✞❂❂❨①Û NL ÜÝ❳Þ❬ (α → ∃xβ) → ∀xγ ` ∀x(β → ∀xγ). á❬ (1) (α → ∃xβ) → ∀xγ, β, α ` β (∈) (2) (α → ∃xβ) → ∀xγ, β, α ` ∃xβ (∃+) (3) (α → ∃xβ) → ∀xγ, β ` α → ∃xβ (→ +) (4) (α → ∃xβ) → ∀xγ, β ` (α → ∃xβ) → ∀xγ (∈) (5) (α → ∃xβ) → ∀xγ, β ` ∀xγ (→ −) (6) (α → ∃xβ) → ∀xγ ` β → ∀xγ (→ +) (7) (α → ∃xβ) → ∀xγ ` ∀x(β → ∀xγ) (∀+) (x ➑✮✁❃✁❄✯❅✦➹➲➌➪❇❆ 5