Lectures on Paraconsistent Logic


•These lectures give an overview on Paraconsistent Logics.• Paraconsistent Logics are logics that allow for inconsistency. •Allowing for that makes them non-classical, even more so – one is inclined to say – than are, for example, many valued approaches. •Nonetheless they are worth considering:
- Logical systems are worth considering in their own right, since so we learn about very abstract structural properties of logics and the concepts employed within them (e.g., negation, necessity, consistency ...)
- Some non-classical logics are especially of interest from a philosophical perspective, since only by them some philosophical problems can be solved or even be stated.
•This introduction argues from a philosophical perspective why (some) paraconsistent logics are interesting or even the best candidates for treating some specific philosophical problems. •Although logic is seen from the point of view of its philosophical use, various formal systems are described, compared and employed.

•This introduction is written for:
- Philosophers who have heard of these new systems, having some idea what they might be good for, but are still looking for a readable more general introduction.
- Information scientists interested in the treatment of inconsistent data and their consequence relation.
- Mathematicians who care about the foundational problems of set theory, or have even heard of  “inconsistent arithmetic” or such things.
- Linguists interested in formal methods and the philosophical foundations of semantics or the theory of formal languages.
- Anyone interested in the paradoxes of semantics, set theory ..

•It is assumed that the reader has mastered a course in classical (or as I rather say “standard”) First Order Logic, has heard at least in general about meta-logical properties of logical systems (like soundness) and about the idea of modal logics and their semantics (possible worlds...). •It is surely helpful if the reader has some background knowledge in the philosophy of logic and language (especially semantics), or such philosophical fields as epistemology, ontology or meta-ethics.


The lectures are divided into the following parts:

              1.    Introduction    (last update: 17.06.2005)


This part presents the basic philosophical motivation for paraconsistent logics (abbreviated “PL”), introduces some methodological guidelines for developing or choosing PLs and describes the different ways (i.e. systems) how one can develop PLs.

•            2. Semantic Closure    (last update: 30.03.2007)

            •3. Methodological Considerations (& First Applications)        (last update: 21.12.2005)

•            4. LP and Relatives         (last update: 22.04.2005) 

PROLOG implemented LP decision procedure (13.06.2005)

            •5. Relevant Logics        (last update: 28.02.2008

            •6. A Sequential Calculus        (last update: 22.04.2005)

            •7. Adaptive Logics        (last update: 31.10.2004) 

            •8. LFI and DaCosta Systems        (last update: 22.04.2005 ) 

            •9. A Bunch of Other Systems        (last update: 20.07.2006 )


This part employs some of the PLs or extensions thereof to deal with topics in set theory, meta-logic, semantics, epistemology, and meta-ethics.

•         10. Semantics        (last update: 08.06.2004)

          •11. Set Theory        (last update: 19.08.2008)

         •12. Inconsistent Mathematics        (last update: 27.05.2004)

         •13. Meta-Logic         (last update: 30.03.2006)

         •14. Deontic Logic and Meta-Ethics        (last update: 08.07.2004) 

         •15. Epistemology and Philosophy of Science        (last update: 10.08.2007) 

Discussions, Alternatives and Perspectives

This part confronts the whole approach with some problems (ontology, hypercontradictions) and discusses some alternatives to PLs.

            •16. Hypercontradictions        (last update: 22.08.2008)  

                   see also the paper Hypercontradictions (last update: 14.12.2005)
                   and the related paper Assertion and Rejection (last update 21.11.2007)

            •17. Inconsistent Ontology        (last update: 11.05.2006) 

            •18. Alternative Solutions of the Paradoxes?        (last update: 30.10.2009) 

            •19. Paraconsistency and Programming        (last update: 06.02.2007) 

            •20. Universal Logic        (last update: 22.08.2008) 

see also the updated overview on the universal logic UL4 (last update: 01.05.2009)


            21.     Epilogue   (last update: 31.05.2004)


•Each chapter
- contains at the end one slide with some questions on the topic of that chapter inviting you to reflect on some of the issues or to grasp the crucial points more clearly
- contains one slide with more formal exercises asking you to employ the methods just dealt with
- contains a list of further readings; here the primary sources of the chapter are given and texts with which you might continue on the topic in question (general references are listed in these Further References)

•The chapters in Part I are more or the less self-contained, the chapters in Part II obviously presuppose the PL-systems introduced in Part I.



You may find most of the materials in these slides in my corresponding introductory text book (
published with Lang [New York et al.] 2005). The book contains some additional material.
Changes made after September 2004 are only reflected on the slides here. 


(C) Manuel Bremer, 2003-2009


Chapters are revised and sometimes material is added (this applies especially to some more or the less empty slides.)
Each chapter/lecture can be viewed as pdf-based slides in Acrobat Reader (you can get for free at Adobe). 
[Within the Reader choose "Full Screen"[Ctrl & "L"] as "View" option.] Each file is between ½ and 1 MB large.
It is not allowed to distribute the material presented here; the copyright remains with the author. 

P.S. some applied paraconsistency: