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Mathematical Logic

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Code Completion Credits Range Language
BI-LOG.21 Z,ZK 5 2P+2C Czech
The course cannot be taken simultaneously with:
Mathematical Logic (BI-MLO)
Garant předmětu:
Department of Applied Mathematics

The course focuses on the basics of propositional and predicate logic. It starts from the semantic point of view. Based on the notion of truth, satisfiability, logical equivalence, and the logical consequence of formulas are defined. Methods for determining the satisfiability of formulas, some of which are used for automated proving, are explained. This relates to the P vs. NP problem and Boolean functions in propositional logic. In predicate logic, the course further deals with formal theories, such as arithmetics, and their models. The syntactic approach to mathematical logic is demonstrated on the axiomatic system of propositional logic and its properties. Gödel's incompleteness theorems is explained.


Knowledge of basic mathematical structures from algebra and analysis.

Syllabus of lectures:

1.Historical introduction. Syntax and semantics of propositional logic. Proof by induction.

2.Logical equivalence. Full and minimal conjunctive and disjunctive normal forms.

3.Logical consequence. Tableau method for propositional logic.

4.Resolution method. SAT problem. P vs. NP problem.

5.Boole algebra. Boolean functions.

6.Predicate logic. Syntax. Interpretation.

7.Logical truth, satisfiability, contradictions. Logical equivalence.

8.Logical consequence. Tableau method for predicate logic.

9.Prenex normal forms. Resolution method for predicate logic.

10.First-order theories and its models. Ordering, equivalence, arithmetic.

11.Axiomatic system of propositional logic.

12.Consistency, correctness, completeness.

13.Gödel incompleteness theorems.

Syllabus of tutorials:

1.Propositional formulas. Truth tables. Formalization.

2.Basic logical laws. Universal system of connectives.

3.Disjunctive and conjunctive normal forms. Logical consequence.

4.Tableau method. Resolution method.

5.Boole algebra: properties, counting, ordering, atoms.

6.Predicate logic. Language, terms, formulas. Formalization.

7.Three levels of truth. Logical equivalence.

8.Interpretation. Satisfiable formulas.

9.Logical consequence. Tableau method.

10.Prenex form. Resolution method.

11.Theories and their models. Isomorphism and elementary equivalence.

12.Hilbert axiomatic system.


Study Objective:

The goal is to learn to work in formal mathematical logic, to understand its syntax and semantics. Work with theories as axiomatic systems and derive their consequences. Know what the correctness, completeness, consistency, and decidability of theories mean, and which problems are relied to. Understand Boolean algebra as a generalization of propositional logic.

Study materials:

1.Trlifajová K., Vašata D.,: Matematická logika. CVUT, 2017. ISBN 978-80-01-05342-3.

2.Mendelson E.: Introduction to Mathematical Logic (6th Edition). Chapman and Hall, 2015. ISBN 978-1482237726.

3.Bergmann M., Moor J., Nelson J.: The Logic Book (6th Edition). McGraw-Hill, 2013. ISBN 978-0078038419.

Further information:
No time-table has been prepared for this course
The course is a part of the following study plans:
Data valid to 2023-03-22
Aktualizace výše uvedených informací naleznete na adrese https://bilakniha.cvut.cz/en/predmet6615906.html