Course detail

Mathematical Logic

FSI-SML-AAcad. year: 2025/2026

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.

Language of instruction

English

Number of ECTS credits

5

Mode of study

Not applicable.

Guarantor

prof. RNDr. Josef Šlapal, CSc.

Department

Institute of Mathematics (ÚM)

Entry knowledge

Students are expected to have knowledge of the subjects General algebra and Methods of discrete mathematics taught in the bachelor's study programme.

Rules for evaluation and completion of the course

The course unit credit is awarded on condition of having attended the seminars actively and passed a written test. The exam has a written and an oral part. The written part tests student's ability to deal with various problems using the knowledge and skills acquired in the course. In the oral part, the student has ro prove that he or she has mastered the related theory.


The attendance at seminars is required and will be checked regularly by the teacher supervising a seminar. If a student misses a seminar due to excused absence, he or she will receive problems to work on at home and catch up with the lessons missed.

Aims

The aim of the course is to acquaint students with the basic methods of reasoning in mathematics. The students will learn about general principles of predicate logic and, consequently, acquire the ability of exact mathematical reasoning and expression. They will understand the general principles of construction of mathematical theories and proofs. The course will contribute students to better acquiring logical reasonong in mathematics and thus to better understanding mathematical knowledge. 


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.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

A. Nerode, R.A. Shore, Logic for Applications, Springer-Verlag 1993 (EN)
E.Mendelson, Introduction to Mathematical Logic, Chapman&Hall, 2001 (EN)

Recommended reading

G. Metakides, A. Nerode, Principles of logic and logic programming, Elsevier, 1996 (EN)
Vítězslav Švejnar, Logika - neúplnost,složitost a nutnost, Academia Praha, 2002 (CS)

Classification of course in study plans

  • Programme N-AIM-A Master's 2 year of study, summer semester, compulsory
  • Programme N-MAI-A Master's 1 year of study, summer semester, compulsory

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

prof. RNDr. Josef Šlapal, CSc.

Syllabus

1. Introduction to mathematical logic
2. Propositions and their truth, logical connectives
3. Language, formulas and semantics of propositional calculus
4. Principle of duality, applications of propositional logic
5. Formal theory of the propositional logic
6. Provability in propositional logic, completeness theorem
7. Language of the (first-order) predicate logic, terms and formulas
8. Semantic of predicate logic
9. Axiomatic theory of the first-order predicate logic
10.Provability in predicate logic
11.Prenex normal forms, first-order theories and their models
12. Theorems on compactness and completeness
13.Undecidability of the first-order theories, Gödel's incompleteness theorems

Exercise

26 hod., compulsory

Teacher / Lecturer

prof. RNDr. Josef Šlapal, CSc.

Syllabus

Relational systems and universal algebras
1. Sentences, propositional connectives, truth tables,tautologies and contradictions
2. Duality principle, applications of propositional logic
3. Complete systems and bases of propositional connectives
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.Theorems on completeness and compactness
13. Undecidability of first-order theories, Gödel's incompleteness theorems