Universal logic explained

Universal Logic is an emerging interdisciplinary field involving logic, non-classical logic, categorical logic, set theory, foundation of logic, and the philosophy and history of logic. The goal of the field is to develop an understanding of the nature of different types of logic. The expression Universal logic was coined by analogy with the expression Universal algebra by Jean-Yves Béziau. The aim was to develop Universal logic as a field of logic that studies the features common to all logical systems, aiming to be to logic what Universal algebra is to algebra, and guided by the features of "unity, generality, abstraction, and undetermination".^[1] A number of approaches to universal logic in this sense have been proposed since the twentieth century, using model theoretic and categorical approaches.

History and Development

The roots of universal logic as general theory of logical systems may go as far back as some work of Alfred Tarski in the early twentieth century and Paul Herz in 1922, but the modern notion was first presented in the 1990s by Swiss logician Jean-Yves Béziau.^{[2] [3]} The term 'universal logic' has also been separately used by logicians such as Richard Sylvan and Ross Brady to refer to a new type of (weak) relevant logic.^[4]

In the context defined by Béziau, three main approaches to universal logic have been explored in depth:^[5]

• An abstract model theory system axiomatized by Jon Barwise,^[6]
• a topological/categorical approach based on sketches (sometimes called categorical model theory),^[7]
• a categorical approach originating in Computer Science based on Goguen and Burstall's notion of institution.^[8]

While logic has been studied for centuries, Mossakowski et al commented in 2007 that "it is embarrassing that there is no widely acceptable formal definition of "a logic".^[9] These approaches to universal logic thus aim to address and formalize the nature of what may be called 'logic' as a form of "sound reasoning".^[9]

Community

Since 2005, Béziau has been organizing world congresses and schools on universal logic.

• First World Congress and School on Universal Logic, 26 March–3 April 2005, Montreux, Switzerland. Participants included Béziau, Dov Gabbay, and David Makinson. (Secret Speaker: Saul Kripke.)
• Second World Congress and School on Universal Logic, 16–22 August 2007, Xi'an, China.
• Third World Congress and School on Universal Logic, 18–25 April 2010, Lisbon, Portugal. (Secret Speaker: Jaakko Hintikka.)
• Fourth World Congress and School on Universal Logic, 29 March–7 April 2013, Rio de Janeiro, Brazil.
• Fifth World Congress and School on Universal Logic, 20–30 June 2015, Istanbul, Turkey.
• Sixth World Congress and School on Universal Logic, 16–26 June 2018, Vichy, France.
• Seventh World Congress and School on Universal Logic, 1–11 April 2022, Crete.

Publications in the field

A journal dedicated to the field, Logica Universalis, with Béziau as editor-in-chief started to be published by Birkhäuser Basel (an imprint of Springer) in 2007.^[10] Springer also started to publish a book series on the topic, Studies in Universal Logic, with Béziau as series editor.^[11]

An anthology titled Universal Logic was published in 2012, giving a new light on the subject.^[12]

See also

• Universal algebra
• Abstract algebraic logic
• Conceptions of logic
• Category theory

External links

• Logica Universalis

Notes and References

1. Web site: Beziau . Jean-Yves . "Universal Logic" .
2. The Road to Universal Logic: Festschrift for 50th Birthday of Jean-Yves Béziau Volume I,edited by Arnold Koslow and Arthur Buchsbaum 2014 Birkhäuser pp 2–10
3. Book: Jean-Yves Béziau. Logica universalis: towards a general theory of logic. 2007. Springer. 978-3-7643-8353-4. 2nd.
4. Brady, R. 2006. Universal Logic. Stanford: CSLI Publications. .
5. Book: Răzvan Diaconescu. Institution-independent model theory. 2008. Birkhäuser. 978-3-7643-8707-5. 2–3.
6. Jon Barwise. Axioms for abstract model theory. Annals of Mathematical Logic,7:221–265, 1974
7. Steffen Lewitzka "A Topological Approach to Universal Logic" Logica Universalis 2007 Birkhauser pp 35–61
8. Razvan Diaconescu, "Three decades of institution theory"in Universal Logic: An Anthology edited by Jean-Yves Béziau 2012 Springer pp 309–322
9. T. Mossakowski, J. A. Goguen, R. Diaconescu, A. Tarlecki, "What is a Logic?", Logica Universalis 2007 Birkhauser, pp. 113–133.
10. Web site: Logica Universalis.
11. Web site: Studies in Universal Logic.
12. Book: Jean-Yves Béziau. Universal Logic: an Anthology - From Paul Hertz to Dov Gabbay. 2012. Springer. 978-3-0346-0144-3.