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.

2022W, 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. Rejection systems are also referred to as complementary calculi as they axiomatise the complement of the valid formulas of a given logic. 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


    • This semester, the lecture will be given as a presence course with a possible fallback to an online format depending on the COVID situation.
    • The start of the lecture is planned for November 3.

ECTS breakdown: 3 ECTS = 75 hours

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



Course dates

Thu15:00 - 18:0003.11.2022 - 26.01.2023FH Hörsaal 4 Lecture
Fri13:00 - 16:0016.12.2022Seminarraum FAV EG C (Seminarraum Gödel) Lecture
Refutation Systems - Single appointments
Thu03.11.202215:00 - 18:00FH Hörsaal 4 Lecture
Thu10.11.202215:00 - 18:00FH Hörsaal 4 Lecture
Thu17.11.202215:00 - 18:00FH Hörsaal 4 Lecture
Thu24.11.202215:00 - 18:00FH Hörsaal 4 Lecture
Thu01.12.202215:00 - 18:00FH Hörsaal 4 Lecture
Thu15.12.202215:00 - 18:00FH Hörsaal 4 Lecture
Fri16.12.202213:00 - 16:00Seminarraum FAV EG C (Seminarraum Gödel) Lecture
Thu22.12.202215:00 - 18:00FH Hörsaal 4 Lecture
Thu12.01.202315:00 - 18:00FH Hörsaal 4 Lecture
Thu19.01.202315:00 - 18:00FH Hörsaal 4 Lecture
Thu26.01.202315:00 - 18:00FH Hörsaal 4 Lecture
Course is held blocked

Examination modalities

Oral exam.

Course registration

Begin End Deregistration end
15.08.2022 08:00 05.01.2023 23:55 05.01.2023 23:55


Study CodeObligationSemesterPrecon.Info
066 931 Logic and Computation Mandatory elective


No lecture notes are available.