Title:

# Logic

Code:QM4
Ac.Year:ukončen 2008/2009
Sem:Summer
Curriculums:
ProgrammeFieldYearDuty
CSE-PHD-4DVI4-Elective
Language of Instruction:Czech
Completion:examination (verbal)
Type of
instruction:
Hour/semLecturesSeminar
Exercises
Laboratory
Exercises
Computer
Exercises
Other
Hours:390000
ExamsTestsExercisesLaboratoriesOther
Points:00000
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 competencies:
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 (Fraenkel-Zermelo).
• 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.
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.
• Salat-Smital, 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 York-Oxford,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.289-398 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.
• Ershov-Paljutin, Mathematical logic, Nauka, Moscow (in Russian).
• Lavrow-Maksimova, Problems in set theory,mathematical logic and algorithm theory, Nauka, Moscow (in Russian)
• Pottmann-Wallner, Computational Line Geometry, Berlin-Heidelberg-New York, 2001.
• Leitsch, The Resolution Calculus, Berlin-Heidelberg-New York 1997,i nv.č. 5330.