课程考试试题纸 课程名称：  数理逻辑 （ B 卷）  考试方式：  （闭卷） 印刷份数：  30  学 院：  哲 学 系 任课教师：  宋 伟  专业年级：  哲学2003级   题 号 一 二 三 四 五 六 七 八 总分 阅卷 教师  得 分            …………………………………………………………………………………………………… 得 分   一. Complete the following blanks.(each 2 points, total 20 points) 1. For a statement form involving n different statement variables, the truth table will have rows 2. if A and B are statement forms, A B if (A→B) is a tautology. 3. For any statement forms A and B, (～(A∧B)) is logically equivalent to . 4. The Soundness Theorem for the formal system L: . 5. Translate the statement ‘NO number is both odd and even.’ into symbols: . 6. Any formal system consists of a and a . 7. The formal system L is , i.e. there is an effective method for deciding, given any wf. of L, whether it is a theorem of L. 8. If an occurrence of a variable in a wf. is not bound it is said to be . 9. A set is if it can be put in one-one correspondence with the set of natural numbers. 10. (x1) ((x2)A(x1,x2) →(x3)A(x1,x3)) yields a Skolemised form: . 得 分   二. Solve to following problems. (each 10 points, total 60 points) 1. Find the disjunctive normal form and the conjunctive normal form which are logically equivalent to (((～p)∨q) →r). 2. Find a statement form involving only | which is logically equivalent to (p→q). 3.Determine whether the argument form ((~p)→ (~q)), q; ∴p is valid or invalid. 4. Using the Deduction Theorem for L, show that the following wf. is theorem of L, where A and B are any wfs. of L. ((A→(B→C)) →(B→(A→C))) 5. Find an interpretation in which the wf. (x1) (A(x1) →A(f(x1))) is interpreted by a statement which is false. 得 分   三. Proof. (each 10 points, total 30 points) 1. Prove that if the statement form ((A1∧…∧An) → A) is a tautology then the argument form A1, A2, …, An ; ∴A is valid. 2. Prove that is B is a contradiction then B cannot be a theorem of any consistent extension of L. 3. Prove that if, in a particular interpretation I, the wfs. A and (A→B) are true, then B is also true.