Course detail

# Logic

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.

Language of instruction

Czech

Number of ECTS credits

5

Mode of study

Not applicable.

Learning outcomes of the course unit

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).

Prerequisites

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

Co-requisites

Not applicable.

Planned learning activities and teaching methods

Not applicable.

Assesment methods and criteria linked to learning outcomes

A mid-term test.

Course curriculum

Not applicable.

Work placements

Not applicable.

Aims

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.

Specification of controlled education, way of implementation and compensation for absences

Not applicable.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Not applicable.

Recommended literature

Melvin Fitting, First order logic and automated theorem proving, Springer, 1996
Sally Popkorn, First steps in modal logic, Cambridge Univ. Press, 1994
V. Švejnar, Logika, neúplnost a složitost, Academia, 2002
E. Mendelson, Introduction to Mathematical Logic, Chapman&Hall, 2001

D.M. Gabbay, C.J. Hogger, J.A. Robinson, Handbook of Logic for Artificial Intellogence and Logic Programming, Oxford Univ. Press 1993

Classification of course in study plans

• Programme IT-MSC-2 Master's

branch MBI , 0 year of study, summer semester, elective
branch MBS , 0 year of study, summer semester, elective
branch MGM , 0 year of study, summer semester, elective
branch MIN , 0 year of study, summer semester, elective
branch MIS , 0 year of study, summer semester, elective
branch MMM , 0 year of study, summer semester, compulsory
branch MPV , 0 year of study, summer semester, elective

• Programme MITAI Master's

specialization NADE , 0 year of study, summer semester, elective
specialization NBIO , 0 year of study, summer semester, elective
specialization NCPS , 0 year of study, summer semester, elective
specialization NEMB , 0 year of study, summer semester, elective
specialization NGRI , 0 year of study, summer semester, elective
specialization NHPC , 0 year of study, summer semester, elective
specialization NIDE , 0 year of study, summer semester, elective
specialization NISD , 0 year of study, summer semester, elective
specialization NISY up to 2020/21 , 0 year of study, summer semester, elective
specialization NMAL , 0 year of study, summer semester, elective
specialization NMAT , 0 year of study, summer semester, elective
specialization NNET , 0 year of study, summer semester, elective
specialization NSEC , 0 year of study, summer semester, elective
specialization NSEN , 0 year of study, summer semester, elective
specialization NSPE , 0 year of study, summer semester, elective
specialization NVER , 0 year of study, summer semester, elective
specialization NVIZ , 0 year of study, summer semester, elective
specialization NISY , 0 year of study, summer semester, elective

• Programme IT-MSC-2 Master's

branch MSK , 1 year of study, summer semester, compulsory-optional

• Programme MITAI Master's

specialization NEMB up to 2021/22 , 0 year of study, summer semester, elective

#### Type of course unit

Lecture

26 hod., optionally

Teacher / Lecturer

Syllabus

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)

Fundamentals seminar

26 hod., compulsory

Teacher / Lecturer

Syllabus

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