Course details

# Logic

LOG Acad. year 2009/2010 Winter semester 5 credits

In the course, the basics of propositional and predicate logics  will systematically be taught  First, the students will get acquainted with the syntax and semantics of the two logics, then the logics will be studied as formal theories with an emphasis on formula proving. The classical theorems on compactness and completeness will alsom 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.

Guarantor

Language of instruction

Czech

Completion

Examination

Time span

39 hrs lectures, 13 hrs exercises

Assessment points

60 exam, 20 half-term test, 20 exercises

Department

Lecturer

Subject specific learning outcomes and competences

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 of a theory and to prove given ones. They will realize the efficiency of formal reasoning and also its limits.

Generic learning outcomes and competences

The students will learn exact formal reasoning to be able to devise a correct and efficient algorithms solving given problems. They will also acquire an ability to verify the correctness of given algorithms (program verification).

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 formal (axiomatic) theories and, consequently, acquire the ability of exact mathematical reasoning and expression. They should also get familiar with some important formal theories utilizied in informatics.

Prerequisite kwnowledge and skills

The knowledge acquired in the bachelor's study course "Discrete Mathematics" is assumed.

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

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 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

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 incompleteness theorems
12. Second-order theories (monadic logic, SkS and WSkS)
13. Some further logics (intuitionistic logic, modal and temporal logics, Presburger arithmetic)

Syllabus of numerical exercises

1. Relational systems and universal algebras
2. Sets, cardinal numbers and cardinal arithmetic
3. Sentences, propositional connectives, truth tables,tautologies and contradictions
4. Independence of propositional connectives, axioms of propositional logic
5. Deduction theorem and proving formulas of propositional logic
6. Terms and formulas of predicate logics
7. Interpretation, satisfiability and truth
8. Axioms and rules of inference of predicate logic
9. Deduction theorem and proving formulas of predicate logic
10. Transforming formulas into prenex normal forms
11. First-order theories and some of their models
12. Monadic logics SkS and WSkS
13. Intuitionistic, modal and temporal logics, Presburger arithmetics

Progress assessment

A mid-term text.

Course inclusion in study plans