Course details

# Logic

LOG Acad. year 2022/2023 Summer semester 5 credits

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.

Guarantor

Course coordinator

Language of instruction

Czech

Completion

Credit+Examination (written)

Time span

• 26 hrs lectures
• 26 hrs exercises

Assessment points

• 60 pts final exam
• 20 pts mid-term test
• 20 pts numeric 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 and to prove given ones. They will realize the efficiency of formal reasonong and also its limits.
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).

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.

Prerequisite knowledge and skills

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

Study literature

• D.M. Gabbay, C.J. Hogger, J.A. Robinson, Handbook of Logic for Artificial Intellogence and Logic Programming, Oxford Univ. Press 1993
• V. Švejnar, Logika, neúplnost a složitost, Academia, 2002
• E. Mendelson, Introduction to Mathematical Logic, Chapman&Hall, 2001
• 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's 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 test.

Exam prerequisites

Regular attendance at exercises and passing both check tests.

Course inclusion in study plans