Title:  Logic 

Code:  QM4 

Ac.Year:  ukončen 2008/2009 

Term:  Summer 

Curriculums:  

Language of Instruction:  Czech 

Completion:  examination (verbal) 

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

Hours:  39  0  0  0  0 

 Examination  Tests  Exercises  Laboratories  Other 

Points:  0  0  0  0  0 



Guarantor:  Havel Václav, prof. RNDr., DrSc., DMAT 

Faculty:  Faculty of Electrical Engineering and Communication BUT 

Department:  Department of Mathematics FEEC BUT 

 Learning objectives: 

  The aim of the object is above all a methodological one: to make deeper the undergradual knowledges of the predicate logic by detailed analysis of specific reasonings in separate chapters of the subject.  Description: 

  Finiteness, countability, cardinalities, continuum hypothesis and axiom of choice. Semantics and syntax of proposition logic. Theorems: on compactness, on finiteness, on completeness. Semantics and syntax of the predicate logic of first order. Prenex formulas. Theorems on correctness and oncompleteness. Theorems: of Herbrand, of Hilbert and Ackermann, of Skolem. Interpretation of one langage in another one. Comments to temporal logic, to combinatorial logic and to logical programming.  Learning outcomes and competences: 

  Deeper understanding of specific reasonings in mathematical logic. Active dominating of its ideas and procedures for purposes of applications in informatics.  Syllabus of lectures: 


 Finite and countable sets, a mild axiomatic approach (FraenkelZermelo).
 Comparing of cardinalities. Continuum hypothesis, axiom of choice.
 Semantics and syntax of proposition logic.
 Compactness theorem (with a turning into general topology), finiteness theorem, completeness theorem.
 Semantics and syntax of first order predicate logic.
 Classic questions on prenex formulas.
 Correctness theorem and completeness theorem. Several words about Kurt Gödel and Alfred Tarski.
 Theorem of Henkin, theorem of Lindenbaum, theorem on compactness.
 Theorem of Herbrand, theorem of Hilbert and Ackermann, theorem of Skolem.
 Interpretation of one language in another one.
 Comments to temporal and modal logic.
 Comments to combinatorial logic.
 Comments to logical programming.
 Syllabus  others, projects and individual work of students: 


 One prescribed seminar work.
 Fundamental literature: 


 Stepanek, Mathematical logic (in Czech), Prague 1982.
 Brabec, Mathematical logic (in Czech), Prague 1975.
 Delahay, Outils logiques pour l'Intelligence artificielle, Paris, 1988.
 SalatSmital, Set theory (in Slovak), Bratislava, 1985.
 Bukovsky, Sets and various things about them (in Slovak), Bratislava 1986.
 van Leeuwen, Handbook of theoretical computer science, Amsterdam 1990.
 Engeler, Metamathematik der Elementatmathematik, Berlin, 1983.
 R.M.Smullyan:Gödel´s Incompleteness Theorems,Oxford University Press,New YorkOxford,1992
 J.L.Bell: Notes on Formal Logic; viz http://publish.uwo.ca/~jbell/LNOTES.pdf
 S. Biliniuk,A Problem Course in Mathematical Logic,Trent University Ontario, 2006;viz http://euclid.trentu.ca/math/sb/pcml/
 Greg Restall:Relevant and Substructural Logics, pp.289398 in Handbook of the History of Logic,vol.7 (ed. D.Gabbay and J.Woods).Elsevier, 2006
 Study literature: 


 An original introduction to predicate logic by Peter Vopenka, appearing in 1977 under title "Sets and natural numbers" (in Czech) in State pedagogic publishing house, Prague, with purposeful omitting of the name of the author.
 ErshovPaljutin, Mathematical logic, Nauka, Moscow (in Russian).
 LavrowMaksimova, Problems in set theory,mathematical logic and algorithm theory, Nauka, Moscow (in Russian)
 PottmannWallner, Computational Line Geometry, BerlinHeidelbergNew York, 2001.
 Leitsch, The Resolution Calculus, BerlinHeidelbergNew York 1997,i nv.č. 5330.
 
