### 2019　Mathematical Logic

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Undergraduate major in Mathematical and Computing Science
Instructor(s)
Kashima Ryo
Course component(s)
Lecture
Day/Period(Room No.)
Tue3-4(W834)  Fri3-4(W834)
Group
-
Course number
MCS.T313
Credits
2
2019
Offered quarter
2Q
Syllabus updated
2019/3/18
Lecture notes updated
2019/8/15
Language used
Japanese
Access Index ### Course description and aims

In mathematics class, you learn logic as the language of mathematics (for example, the usage of ∀ and ∃). In this course, we study logic itself mathematically, and we investigate the ability and limitation of logic. We also give an overview of logics, especially modal logics, in various fields (for example, computer program verification). In mathematics and computer science, logic is the foundation and an important tool. This course gives correct understanding of logic.

### Student learning outcomes

At the end of this course, students will be able to:
(1) Write a logical formula that represents intended meaning correctly.
(2) Have a correct understanding of the basic results of mathematical logic, as follows: propositional logic, predicate logic, syntax, semantics, Gentzen's natural deduction, Goedel's completeness theorem, Goedel's incompleteness theorem, compactness, decidability, undecidability, normal form of formulas, etc.
(3) Have basic knowledge about various logics, especially modal logics, in computer science.

### Keywords

propositional logic, predicate logic, Gentzen's natural deduction, Goedel's completeness theorem, Goedel's incompleteness theorem, modal logic.

### Competencies that will be developed

 Intercultural skills Communication skills ✔ Specialist skills Critical thinking skills Practical and/or problem-solving skills

### Class flow

The course consists of lectures.
Homework assignments are given several times for checking your understanding.

### Course schedule/Required learning

Course schedule Required learning
Class 1 The language of mathematics. Logical formula. Problems in Chapter 1 of the textbook.
Class 2 Natural deduction (1): Introduction. Problems in Chapter 2 of the textbook.
Class 3 Natural deduction (2): Propositional logic. Problems in Chapter 2 of the textbook.
Class 4 Natural deduction (3): Predicate logic. Problems in Chapter 2 of the textbook.
Class 5 Truth, validity, and satisfiability of logical formulas. Problems in Chapter 3 of the textbook.
Class 6 Soundness of natural deduction. Problems in Chapter 4 of the textbook.
Class 7 Completeness of natural deduction (1): Consistent set. Problems in Chapter 5 of the textbook.
Class 8 Completeness of natural deduction (2): Maximal consistent set. Problems in Chapter 5 of the textbook.
Class 9 Completeness of natural deduction (3): Compactness. Problems in Chapter 5 of the textbook.
Class 10 Incompleteness Theorem (1): Arithmetic, Goedel numbering, and representability. Problems in Chapter 6 of the textbook.
Class 11 Incompleteness Theorem (2): First Incompleteness Theorem. Undecidability of arithmetic and logic. Problems in Chapter 6 of the textbook.
Class 12 Normal forms of propositional formulas. Problems in Chapter 7 of the textbook.
Class 13 Modal logic (1): Kripke models and K, S4, and S5. Instructed in the class.
Class 14 Modal logic (2): Temporal logics LTL and CTL. Instructed in the class.
Class 15 Summary of the course. Instructed in the class.

### Textbook(s)

Materials for Class 13 and Class 14 can be found on OCW-i.

### Reference books, course materials, etc.

Instructed in the class.

### Assessment criteria and methods

Final exams (80%) and exercise problems (20%).

### Related courses

• MCS.T201 ： Set and Topology I
• MCS.T204 ： Introduction to Computer Science
• MCS.T214 ： Theory of Automata and Languages
• MCS.T323 ： Theory of Computation

### Prerequisites (i.e., required knowledge, skills, courses, etc.)

No prerequisites. 