课程考试试题纸 课程名称：  数理逻辑 （ A 卷）  考试方式：  （闭卷） 印刷份数：  28  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, there are distinct truth functions with n places. 2. If A and B are statement forms, A is to B if (AB) is a tautology. 3. For the connectives ～,∨,∧,→ and , the pairs {～,∧}, , and are adequate sets of connectives. 4. The Adequacy Theorem for the formal system L: . 5. Translate the statement ‘Every number is either odd or even.’ into symbols: . 6. A formal language consists of a set of and a set of . 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. In the wf. (xi) A, we say that A is the of the quantifier. 9. Any infinite subset of a countable (or denumerable) is . 10. (x1) (x2) (x3)(x1,x2,x3) yields a Skolemised form: . 得 分   二. Solve the following problems. (each 10 points, total 50 points) 1. Find the disjunctive normal form and the conjunctive normal form which are logically equivalent to (pq). 2. Find a statement form involving only ↓ which is logically equivalent to (p→q). 3. Determine whether the argument form (p→ (q∨r))，(～r); ∴ ((～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)→((～(B→C)→ ～A)→ (A→C))) 5. Find an interpretation in which the wf. (x1)((x1,x2)→(x2,x1)) is interpreted by a statement which is false. 得 分   三. Proof. (each 10 points, total 30 points) 1. Prove that if the argument form A1,A2,…,An；∴A is valid then the statement form, (A1∧…∧An)→A) is a tautology. 2. Prove that the formal system L is consistent. 3. Prove that if, in a particular interpretation I, the wfs. A and (A→B) are true, then B is also true.