Title:

# Logic

Code:LOG
Ac.Year:2012/2013
Sem:Summer
Curriculums:
ProgrammeField/
Specialization
YearDuty
IT-MSC-2MBI-Elective
IT-MSC-2MBS-Elective
IT-MSC-2MGM-Elective
IT-MSC-2MIN-Elective
IT-MSC-2MIS-Elective
IT-MSC-2MMI-Elective
IT-MSC-2MMM-Compulsory
IT-MSC-2MPV-Elective
IT-MSC-2MSK1stCompulsory-Elective - group M
Language of Instruction:Czech
Credits:5
Completion:credit+exam
Type of
instruction:
Hour/semLecturesSeminar
Exercises
Laboratory
Exercises
Computer
Exercises
Other
Hours:2626000
ExamsTestsExercisesLaboratoriesOther
Points:60202000
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 refresh their knowledge of fundamentals of propositional logic and learn about the axiomatic approach about the logic. But they should mainly learn about general principles of  predicate logic and, consequently, acquire the ability of exact mathematical reasoning and expression. At the end of the course, the students will get acquainted with the problematic of incompeteness of predicate logic including the famous Gödel incompleteness theorems.
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. At the end of the course, we will deal with the undecidability of first-order theories resulting from the well-known Gödel incompleteness theorems.
Knowledge and skills required for the course:
The knowledge acquired in the bachelor's study course "Discrete Mathematics" 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 and formulas of propositional calculus
3. Semantics of propositional calculus
4. Formal theory of the propositional logic
5. Provability in propositional logic, completeness theorem
6. Language of the (first-order) predicate logic, terms and formulas
7. Semantic of predicate logics
8. Axiomatic theory of the first-order predicate logic
9. Provability in predicate logic
10. Prenex normal forms
11. First-order theories and their models
12. Theorems on compactness and completeness,
13. Undecidabilitry of first-order theories, Gödel's incompleteness theorems

Syllabus of numerical exercises:

1. Relational systems and universal algebras
2. Sets, cardinal numbers and cardinal arithmetic
3. Sentences, propositional connectives and their independence
5. Independence of propositional connectives, axioms of propositional logic
6. Deduction theorem and proving formulas of propositional logic
7. Terms and formulas of predicate logics
8. Interpretation, satisfiability and truth
9. Axioms and rules of inference of predicate logic
10. Deduction theorem and proving formulas of predicate logic
11. Transforming formulas into prenex normal forms
12. First-order theories and their models
13. Completeness and compactness of predicate logic
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:

1. E. Mendelson, Introduction to Mathematical Logic, Chapman&Hall, 2001
2. A. Nerode, R.A. Shore, Logic for Applications, Springer-Verlag 1993
3. D.M. Gabbay, C.J. Hogger, J.A. Robinson, Handbook of Logic for Artificial Intellogence and Logic Programming, Oxford Univ. Press 1993
4. G. Metakides, A. Nerode, Principles of logic and logic programming, Elsevier, 1996
5. Melvin Fitting, First order logic and automated theorem proving, Springer, 1996
6. Sally Popkorn, First steps in modal logic, Cambridge Univ. Press, 1994
7. A. Sochor, Klasická matematická logika, Karolinum, 2001
8. V. Švejnar, Logika, neúplnost a složitost, Academia, 2002
Progress assessment:
Check tests at the middle and at the end of semester.
Exam prerequisites:
Regular attendance at exercises and passing both check tests.