Codes: BMA-LOTD-312, BMI-LOTD-312E
Title: Paraconsistent logic
Lecturer: William Brown
Location and time: i/224, Tuesday 12:00 - 13:30
Consultation and email: williamjosephbrown (at) gmail . com
Prerequisites: Familiarity with basic concepts of classical logic
Requirements: Oral exam or presentation at the end of the semester
In many logical systems, such as Propositional Logic, whenever contradictory premises are assumed, every formula of the system can be inferred. The behavior of such an inference relation is called explosive. The explosive principle has been called Ex Contradictione Quodlibet (ECQ), meaning that from contradictions anything follows. In many logics, ECQ is a valid inference. In a logic where ECQ holds, inconsistent premises can't be handled, since they render the system trivial by entailing every formula of the logic. Paraconsistent logics are those logics where ECQ doesn't hold, ie. from inconsistent premises not everything can be inferred. As such they can handle inconsistent premises.
Paraconsistent logics arose mostly in the last hundred year. Aristotle formulated the Principle of Non Contradiction (PNC) which states that contradictory statements can't be true at the same time. Until the early 20th century, the PNC was rarely questioned, and even fiercly defended ("Anyone who denies the law of non-contradiction should be beaten and burned until he admits that to be beaten is not the same as not to be beaten, and to be burned is not the same as not to be burned" Avicenna, Metaphysics, I). As such, the PNC held true in most logical systems, including those developed in the 19th and early 20th century. Once we allow ourselves to build logical systems where the PNC doesn't hold, it becomes easier to develop paraconsistent logics.
We will start by studying Aristotle's Principle of Non Contradiction, and its analysis and critique made by Łukasiewicz in his 1910 monograph (O zasadzie sprzeczności u Arystotelesa).
We will then look more closely at the concept of paraconsistency, what it is, its relation to ECQ and PNC, as well as some motivations why we might want to reason with inconsistent premises.
Finally, we are going to study formally various paraconsistent logics (Jaskowski's discussive logic, LP, FDE, and possibly a few others), as well as some of their distinctive properties, use, and how they might help solve or shed some light on various logical problems and paradoxes.
Beall J. C., van Fraassen B. C., Possibilities and Paradox, Oxford University Press, 2003.
Priest G., "Paraconsistent Logic", Handbook of Philosophical Logic (Second Edition), Vol. 6, D. Gabbay and F. Guenthner (eds.), Dordrecht: Kluwer Academic Publishers, 2002, pp. 287–393.
Priest G., An Introduction to Non-classical Logic (Second Edition), Cambridge University Press, 2008.
Priest G., Tanaka K., Weber Z., Paraconsistent logic, Stanford Encyclopedia of Philosophy, 2013, http://plato.stanford.edu/entries/logic-paraconsistent/
Aristotle, Metaphysics, book Γ.
Łukasiewicz J., Du principe de contradiction chez Aristote, Éditions de l'éclat, Paris, 2000. [Original version: O zasadzie sprzeczności u Arystotelesa, Panstwowe Wydawnictwo Naukowe, Warsaw, 1987 ; unfortunately there is no English translation of this text, however, apart from the original Polish, and the French translation, there is a German translation (Hildesheim: Olms, 1993) and an Italian one (Macerata: Quodlibet, 2003)]