Course detail

# Logic

FIT-LOGAcad. year: 2018/2019

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.

Supervisor

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.

Recommended optional programme components

Not applicable.

Recommended or required reading

- 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

- 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

Planned learning activities and teaching methods

Not applicable.

Assesment methods and criteria linked to learning outcomes

A mid-term test.

Exam prerequisites:

Regular attendance at exercises and passing both check tests.

Language of instruction

Czech

Work placements

Not applicable.

Course curriculum

- Basics of set theory and cardinal arithmetics
- Language, formulas and semantics of propositional calculus
- Formal theory of the propositional logic
- Provability in propositional logic, completeness theorem
- Language of the (first-order) predicate logic, terms and formulas
- Semantic of predicate logics
- Axiomatic theory of the first-order predicate logic
- Provability in predicate logic
- Theorems on compactness and completeness, prenex normal forms
- First-order theories and their models
- Undecidabilitry of first-order theories, Gödel's incompleteness theorems
- Second-order theories (monadic logic, SkS and WSkS)
- Some further logics (intuitionistic logic, modal and temporal logics, Presburger arithmetic)
- Relational systems and universal algebras
- Sets, cardinal numbers and cardinal arithmetic
- Sentences, propositional connectives, truth tables,tautologies and contradictions
- Independence of propositional connectives, axioms of propositional logic
- Deduction theorem and proving formulas of propositional logic
- Terms and formulas of predicate logics
- Interpretation, satisfiability and truth
- Axioms and rules of inference of predicate logic
- Deduction theorem and proving formulas of predicate logic
- Transforming formulas into prenex normal forms
- First-order theories and some of their models
- Monadic logics SkS and WSkS
- Intuitionistic, modal and temporal logics, Presburger arithmetics

**Syllabus of lectures:**

**Syllabus of numerical exercises:**

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.

Classification of course in study plans

- Programme IT-MGR-2 Master's
branch MBI , any year of study, summer semester, 5 credits, elective

branch MPV , any year of study, summer semester, 5 credits, elective

branch MGM , any year of study, summer semester, 5 credits, elective

branch MIS , any year of study, summer semester, 5 credits, elective

branch MBS , any year of study, summer semester, 5 credits, elective

branch MIN , any year of study, summer semester, 5 credits, elective

branch MMI , any year of study, summer semester, 5 credits, elective

branch MMM , any year of study, summer semester, 5 credits, compulsory

branch MSK , 1. year of study, summer semester, 5 credits, compulsory-optional

#### Type of course unit

Lecture

26 hours, optionally

Teacher / Lecturer

Syllabus

- Basics of set theory and cardinal arithmetics

- Language, formulas and semantics of propositional calculus

- Formal theory of the propositional logic

- Provability in propositional logic, completeness theorem

- Language of the (first-order) predicate logic, terms and formulas

- Semantic of predicate logics

- Axiomatic theory of the first-order predicate logic

- Provability in predicate logic

- Theorems on compactness and completeness, prenex normal forms

- First-order theories and their models

- Undecidabilitry of first-order theories, Gödel's incompleteness theorems

- Second-order theories (monadic logic, SkS and WSkS)

- Some further logics (intuitionistic logic, modal and temporal logics, Presburger arithmetic)

Fundamentals seminar

26 hours, compulsory

Teacher / Lecturer

Syllabus

- Relational systems and universal algebras

- Sets, cardinal numbers and cardinal arithmetic

- Sentences, propositional connectives, truth tables,tautologies and contradictions

- Independence of propositional connectives, axioms of propositional logic

- Deduction theorem and proving formulas of propositional logic

- Terms and formulas of predicate logics

- Interpretation, satisfiability and truth

- Axioms and rules of inference of predicate logic

- Deduction theorem and proving formulas of predicate logic

- Transforming formulas into prenex normal forms

- First-order theories and some of their models

- Monadic logics SkS and WSkS

- Intuitionistic, modal and temporal logics, Presburger arithmetics