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192.094 Refutation Systems
This course is in all assigned curricula part of the STEOP.
This course is in at least 1 assigned curriculum part of the STEOP.

2023W, VO, 2.0h, 3.0EC, to be held in blocked form


  • Semester hours: 2.0
  • Credits: 3.0
  • Type: VO Lecture
  • Format: Presence

Learning outcomes

After successful completion of the course, students are able to

  • to name and explain different refutation systems,
  • to select suitable methods and techniques for a given problem, and
  • to critically assess relevant solutions and formalisms.

Subject of course

The course deals with systems for the axiomatic rejection of propositions, i.e., with proof systems which characterise the invalid formulas of a logic. Rejection systems, also called refutation systems,  therefore axiomatise the complement of the valid formulas of a given logic and in this sense they are also often referred to as complementary calculi. Thus, in a formal derivation in a complementary calculus, invalid statements are deduced from other invalid statements.

The first mention of the term rejection in modern logic was done by Jan Lukasiewicz in 1920 and he subsequently developed the first axiomatic rejection system in the context of his study on Aristotelian syllogistic. 

In this course, we will discuss rejection calculi for different logics, like classical logic, many-valued logics, modal logics, etc. Moreover, we also discuss the concept of an anti-consequence operator,  as introduced by Lukasiewicz's student Jerzy Slupecki and subsequently studied, e.g., also together with Grzegorz Bryll and Urszula Wybraniec-Skardowska, all well-known researchers from the famous Lvov-Warsaw school of logic. Finally, we also discuss applications of rejection systems for nonmonotonic logics.

Teaching methods

Frontal lecture.

Mode of examination


Additional information


  • No lecture October 11!

ECTS breakdown: 3 ECTS = 75 hours

  • Lecture 24h
  • Lecture introduction 0.5h
  • Preparation for exam 50h
  • Oral exam 0.5h



Course dates

Wed16:00 - 18:0004.10.2023 - 24.01.2024EI 3A Hörsaal Lecture
Wed18:00 - 19:0004.10.2023 - 20.12.2023EI 3A Hörsaal Lecture
Refutation Systems - Single appointments
Wed04.10.202316:00 - 18:00EI 3A Hörsaal Lecture
Wed04.10.202318:00 - 19:00EI 3A Hörsaal Lecture
Wed11.10.202316:00 - 18:00EI 3A Hörsaal Lecture
Wed11.10.202318:00 - 19:00EI 3A Hörsaal Lecture
Wed18.10.202316:00 - 18:00EI 3A Hörsaal Lecture
Wed18.10.202318:00 - 19:00EI 3A Hörsaal Lecture
Wed25.10.202316:00 - 18:00EI 3A Hörsaal Lecture
Wed25.10.202318:00 - 19:00EI 3A Hörsaal Lecture
Wed08.11.202316:00 - 18:00EI 3A Hörsaal Lecture
Wed08.11.202318:00 - 19:00EI 3A Hörsaal Lecture
Wed22.11.202316:00 - 18:00EI 3A Hörsaal Lecture
Wed22.11.202318:00 - 19:00EI 3A Hörsaal Lecture
Wed29.11.202316:00 - 18:00EI 3A Hörsaal Lecture
Wed29.11.202318:00 - 19:00EI 3A Hörsaal Lecture
Wed06.12.202316:00 - 18:00EI 3A Hörsaal Lecture
Wed06.12.202318:00 - 19:00EI 3A Hörsaal Lecture
Wed13.12.202316:00 - 18:00EI 3A Hörsaal Lecture
Wed13.12.202318:00 - 19:00EI 3A Hörsaal Lecture
Wed20.12.202316:00 - 18:00EI 3A Hörsaal Lecture
Wed20.12.202318:00 - 19:00EI 3A Hörsaal Lecture
Course is held blocked

Examination modalities

Oral exam.

Course registration

Begin End Deregistration end
12.08.2023 08:00 04.01.2024 23:55 04.01.2024 23:55


Study CodeObligationSemesterPrecon.Info
066 931 Logic and Computation Mandatory elective


No lecture notes are available.