Title:

# Mathematical Logic

Code:MLD
Ac.Year:2019/2020
Sem:Summer
Curriculums:
ProgrammeField/
Specialization
YearDuty
CSE-PHD-4DVI4-Elective
Language of Instruction:Czech
Completion:examination (written&oral)
Type of
instruction:
Hour/semLecturesSeminar
Exercises
Laboratory
Exercises
Computer
Exercises
Other
Hours:260000
ExamsTestsExercisesLaboratoriesOther
Points:1000000
Guarantor:Šlapal Josef, prof. RNDr., CSc. (DADM)
Lecturer:Šlapal Josef, prof. RNDr., CSc. (DADM)
Faculty:Faculty of Mechanical Engineering BUT
Department:Department of Algebra and Discrete Mathematics FME BUT

Learning objectives:
The aim of the course is to acquaint students with the basic methods of reasoning in mathematics. The students should learn about general principles of  predicate logic and, consequently, acquire the ability of exact mathematical reasoning and expression. They should also get familiar with some other important formal theories utilizied in informatics too.
Description:
In the course, the basics of propositional and predicate logics will be taught. First, the students will get acquainted with the syntax and semantics of the logics, then the logics will be studied as formal theories with an emphasis on formula proving. The classical theorems on correctness, completeness and compactness will also be dealt with. After discussing the prenex forms of formulas, some properties and models of first-order theories will be studied. We will also deal with the undecidability of first-order theories resulting from the well-known Gödel incompleteness theorems. Finally, some further important logics will be discussed which have applications in computer science.
Knowledge and skills required for the course:
The knowledge acquired in the bachelor's study course "Discrete Mathematics" and the master's study course "Mathematical Structures in Informatics" is assumed.
Subject specific learning outcomes and competencies:
The students will acquire the ability of understanding the principles of axiomatic mathematical theories and the ability of exact (formal) mathematical expression. They will also learn how to deduct, in a formal way, new formulas and to prove given ones. They will realize the efficiency of formal reasonong and also its limits.
Generic learning outcomes and competencies:
The students will learn exact formal reasoning to be able to devise correct and efficient algorithms solving given problems. They will also acquire an ability to verify the correctness of given algorithms (program verification).
Syllabus of lectures:

1. Basics of set theory and cardinal arithmetics
2. Language, formulas and semantics of propositional calculus
3. Formal theory of the propositional logic
4. Provability in propositional logic, completeness theorem
5. Language of the (first-order) predicate logic, terms and formulas
6. Semantic of predicate logics
7. Axiomatic theory of the first-order predicate logic
8. Provability in predicate logic
9. Theorems on compactness and completeness, prenex normal forms
10. First-order theories and their models
11. Undecidabilitry of first-order theories, Gödel's incompleteness theorems
12. Second-order theories (monadic logic, SkS and WSkS)
13. Some further logics (intuitionistic logic, modal and temporal logics, Presburger arithmetic)
Fundamental literature:

• E. Mendelson, Introduction to Mathematical Logic, Chapman&Hall, 2001
• A. Nerode, R.A. Shore, Logic for Applications, Springer-Verlag 1993
• D.M. Gabbay, C.J. Hogger, J.A. Robinson, Handbook of Logic for Artificial Intelligence and Logic Programming, Oxford Univ. Press 1993
• G. Metakides, A. Nerode, Principles of logic and logic programming, Elsevier, 1996
• Melvin Fitting, First order logic and automated theorem proving, Springer, 1996
• Sally Popkorn, First steps in modal logic, Cambridge Univ. Press, 1994
Study literature:

• E. Mendelson, Introduction to Mathematical Logic, Chapman&Hall, 2001
• A. Nerode, R.A. Shore, Logic for Applications, Springer-Verlag 1993
• D.M. Gabbay, C.J. Hogger, J.A. Robinson, Handbook of Logic for Artificial Intellogence and Logic Programming, Oxford Univ. Press 1993
• G. Metakides, A. Nerode, Principles of logic and logic programming, Elsevier, 1996
• Melvin Fitting, First order logic and automated theorem proving, Springer, 1996
• Sally Popkorn, First steps in modal logic, Cambridge Univ. Press, 1994
• A. Sochor, Klasická matematická logika, Karolinum, 2001
• V. Švejnar, Logika, neúplnost a složitost, Academia, 2002