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## 8241005Philosophy and History of Science (Seminars)

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 Numbering Code G-LET32 78241 SJ34 Term 2020/First semester Number of Credits 2 credits Course Type Seminar Target Student Graduate Language Japanese Day/Period Tue.5 Instructor(s) YATABE SHUNSUKEPart-time Lecturer YATABE SHUNSUKE Outline and Purpose of the Course The ultimate objectives of this class are for students to become experienced in logical and clear thinking, to be able to clarify the reasoning on which any assertion they may make is based, and to be able to make arguments without errors or omissions. The subject of that practice will be philosophical logic, and particularly the question of "What is logic?". We make deductions in daily life, and often use the word "logical." Of course, being "logical" is required, but what exactly is "logic"? Questioning once again the meaning of concepts we think we know, and which we use habitually and thoughtlessly, is an important task in philosophy.In this seminar, we will introduce a notation-based system ("formal system") that can be called "logic," that allows us to simulate proofs of theorems in mathematics. Specifically, we will explain and do practice problems on the system of natural deduction in minimal predicate logic. Course Goals Be able to solve basic practice problems using natural deduction in minimal predicate logic. Through this, we ill understand how deductions proceed in formal systems, while at the same time understanding how everyday reasoning is simulated in formal systems. Schedule and Contents Minimal predicate logic only has rules for introducing and removing logical connectors and is one of the fundamental systems of logic. In the first half of the first semester, we will introduce the system of natural deduction in minimal predicate logic. Through practice problems, the objective is for each student to be able to do natural deduction proofs on their own. In the second half, using the example of the "minimal arithmetic Q" system of arithmetic in minimal logic, we will show that it is possible to execute many proofs in mathematics with minimal logic. At the same time, we will also introduce basic concepts in calculation, such as primitive recursion.The specific class plan is as follows.1. What Does Logic Do as a Discipline?2. Formal Language3. Minimal Propositional Logic's ⇒: Rules of Introduction and Removal4. Minimal Propositional Logic's ∧ and ∨: Rules of Introduction and Removal5. Minimal Propositional Logic Practice Problems6. Proofs Without Detours7. Quantifiers and Minimal Predicate Logic8. Minimal Predicate Logic's ∀: Rules of Introduction and Removal9. Minimal Predicate Logic's ∃: Rules of Introduction and Removal10. Minimal Predicate Logic Practice Problems11. Formal Natural Number Theory12. Primitive Recursive Functions and the Proof of "2+2=4"13. Numerical Notation Possibilities for Recursive Functions14. Comprehensive Practice15. Formal Logic and the Philosophy of Language Grading Policy Grades will follow the cumulative score of the homework that will be assigned nearly every class session. Prerequisites None Preparation and Review Handouts and other class materials will be uploaded to the website (the above class blog) in advance (1-2 days before class).Students should read over the materials before class. Textbook None; handouts will be distributed in each class session. Reference(s) 戸次大介 『数理論理学』 (東大出版会), 小野寛晰 『情報科学における論理』 (日本評論社), Dag Prawitz 『Natural Deduction: A Proof-Theoretical Study』