Title:  Logic 

Code:  LOG 

Ac.Year:  2009/2010 

Term:  Winter 

Study plans:  

Language:  Czech 

Credits:  5 

Completion:  examination 

Type of instruction:  Hour/sem  Lectures  Sem. Exercises  Lab. exercises  Comp. exercises  Other 

Hours:  39  13  0  0  0 

 Examination  Tests  Exercises  Laboratories  Other 

Points:  60  20  20  0  0 



Guarantee:  Šlapal Josef, prof. RNDr., CSc., DADM 

Lecturer:  Šlapal Josef, prof. RNDr., CSc., DADM 
Faculty:  Faculty of Mechanical Engineering BUT 

Department:  Department of Algebra and Discrete Mathematics FME BUT 

 Learning objectives: 

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 formal (axiomatic) theories and, consequently, acquire the ability of exact mathematical reasoning and expression. They should also get familiar with some important formal theories utilizied in informatics.  Description: 

In the course, the basics of propositional and predicate logics will systematically be taught First, the students will get acquainted with the syntax and semantics of the two logics, then the logics will be studied as formal theories with an emphasis on formula proving. The classical theorems on compactness and completeness will alsom be dealt with. After discussing the prenex forms of formulas, some properties and models of firstorder theories will be studied. We will also deal with the undecidability of firstorder theories resulting from the wellknown Gödel incompleteness theorems. Finally, some further important logics will be discussed which have applications in computer science.  Knowledge and skills required for the course: 

The knowledge acquired in the bachelor's study course "Discrete Mathematics" is assumed.  Subject specific 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 of a theory and to prove given ones. They will realize the efficiency of formal reasoning and also its limits.  Generic learning outcomes and competences: 

The students will learn exact formal reasoning to be able to devise a correct and efficient algorithms solving given problems. They will also acquire an ability to verify the correctness of given algorithms (program verification).  Syllabus of lectures: 

 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 (firstorder) predicate logic, terms and formulas
 Semantic of predicate logics
 Axiomatic theory of the firstorder predicate logic
 Provability in predicate logic
 Theorems on compactness and completeness, prenex normal forms
 Firstorder theories and their models
 Undecidabilitry of firstorder theories, Gödel incompleteness theorems
 Secondorder theories (monadic logic, SkS and WSkS)
 Some further logics (intuitionistic logic, modal and temporal logics, Presburger arithmetic)
 Syllabus of numerical exercises: 

 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
 Firstorder theories and some of their models
 Monadic logics SkS and WSkS
 Intuitionistic, modal and temporal logics, Presburger arithmetics
 Fundamental literature: 

 E. Mendelson, Introduction to Mathematical Logic, Chapman&Hall, 2001
 A. Nerode, R.A. Shore, Logic for Applications, SpringerVerlag 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
 Study literature: 

 E. Mendelson, Introduction to Mathematical Logic, Chapman&Hall, 2001
 A. Nerode, R.A. Shore, Logic for Applications, SpringerVerlag 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
 Progress assessment: 

A midterm text.  
