Mathematical Logic (FSI-SML-A)

Academic year 2018/2019
Supervisor: prof. RNDr. Josef Šlapal, CSc.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: English
Aims of the course unit:
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 mathematics and informatics, too.
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.
Prerequisites:
Students are expected to have knowledge of the subjects General algebra and Methods of discrete mathematics taught in the bachelor's study programme.
Course contents:
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.
Teaching methods and criteria:
The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures.
Assesment methods and criteria linked to learning outcomes:
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.
Controlled participation in lessons:
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.
Type of course unit:
    Lecture  13 × 2 hrs. optionally                  
    Exercise  13 × 2 hrs. compulsory                  
Course curriculum:
    Lecture 1. Introduction to mathematical logic
2. Propositions and their truth, logic operations
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 logics
9. Axiomatic theory of the first-order predicate logic
10.Provability in predicate logic
11. Theorems on compactness and completeness, prenex normal forms
12.First-order theories and their models
13.Undecidabilitry of first-order theories, Gödel's incompleteness theorems

    Exercise Relational systems and universal algebras
1. Sets, cardinal numbers and cardinal arithmetic
2. Sentences, propositional connectives, truth tables,tautologies and contradictions
3. Independence of propositional connectives, axioms of propositional logic
4. Deduction theorem and proving formulas of propositional logic
5. Terms and formulas of predicate logics
6. Interpretation, satisfiability and truth
7. Axioms and rules of inference of predicate logic
8. Deduction theorem and proving formulas of predicate logic
9. Transforming formulas into prenex normal forms
10.First-order theories and some of their models
11.Monadic logics SkS and WSkS
12.Intuitionistic, modal and temporal logics, Presburger arithmetics
Literature - fundamental:
1. E.Mendelson, Introduction to Mathematical Logic, Chapman&Hall, 2001
2. A. Nerode, R.A. Shore, Logic for Applications, Springer-Verlag 1993
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
M2A-A full-time study M-MAI Mathematical Engineering -- Cr,Ex 5 Compulsory 2 2 S