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In the same way that universal algebra is a general theory of algebraic structures, universal logic is a general theory of logical structures.
In the same way that universal algebra is a general theory of algebraic structures, universal logic is a general theory of logical structures.
During the 20th century, numerous logical systems have been created: intuitionistic logic, deontic logic, many-valued logic, relevant logic, linear logic, non monotonic logic, etc. Universal logic is not a new logical system, it is a way of unifying this multiplicity of logical systems by developing general tools and concepts that can be applied to all logics.
During the 20th century, numerous logical systems have been created: intuitionistic logic, deontic logic, many-valued logic, relevant logic, linear logic, non monotonic logic, etc. Universal logic is not a new logical system, it is a way of unifying this multiplicity of logical systems by developing general tools and concepts that can be applied to all logics.
Universal logic helps to clarify basic concepts explaining what is an extension and what is a deviation of a given logic, what does it mean for a logic to be equivalent or translatable into another one. It allows to give precise definitions of notions often discussed by philosophers: truth-functionality, extensionality, logical form, identity, existence, negation, etc.
Universal logic helps to clarify basic concepts explaining what is an extension and what is a deviation of a given logic, what does it mean for a logic to be equivalent or translatable into another one. It allows to give precise definitions of notions often discussed by philosophers: truth-functionality, extensionality, logical form, identity, existence, negation, etc.
One aim of universal logic is to determine the domain of validity of such and such metatheorem (e.g. the completeness theorem) and to give general formulations of metatheorems. This is very useful for applications and helps to make the distinction between what is really essential to a particular logic and what is not, and thus gives a better understanding of this particular logic. Universal logic can also be seen as a toolkit for producing a specific logic required for a given situation, e.g. a paraconsistent deontic temporal logic.
One aim of universal logic is to determine the domain of validity of such and such metatheorem (e.g. the completeness theorem) and to give general formulations of metatheorems. This is very useful for applications and helps to make the distinction between what is really essential to a particular logic and what is not, and thus gives a better understanding of this particular logic. Universal logic can also be seen as a toolkit for producing a specific logic required for a given situation, e.g. a paraconsistent deontic temporal logic.
The idea of universal logic is not to build a monolithic system of logic but to develop comparative study of ways of reasoning and their systematizations, promoting better understanding and knowledge of the logical realm and its connections with other fields.
The idea of universal logic is not to build a monolithic system of logic but to develop comparative study of ways of reasoning and their systematizations, promoting better understanding and knowledge of the logical realm and its connections with other fields.
1. History and Philosophy of Universal Logic
1. History and Philosophy of Universal Logic
In this first lesson, we will present the historical development and philosophical motivation of universal logic as a general theory of logical structures.
In this first lesson, we will present the historical development and philosophical motivation of universal logic as a general theory of logical structures.
We will discuss the analogy with universal algebra, presenting a short history of this field, the coining of this expression by J.J.Sylvester, the first book on the topic by A.N.Whitehead, the promotion by Garrett Birkhoff of abstract algebras as structures obeying, at the most general level, no axioms.
We will discuss the analogy with universal algebra, presenting a short history of this field, the coining of this expression by J.J.Sylvester, the first book on the topic by A.N.Whitehead, the promotion by Garrett Birkhoff of abstract algebras as structures obeying, at the most general level, no axioms.
We will then present the different ways logic have been and can be considered as structures, in particular Tarski's theory of the consequence operator, Suszko's notion of abstract logic, multiple-conclusion logics.
We will then present the different ways logic have been and can be considered as structures, in particular Tarski's theory of the consequence operator, Suszko's notion of abstract logic, multiple-conclusion logics.
We will emphasize the distinctions and relations between a logical system conceived as an abstract structure and proof-theoretical or model-theoretical presentations.
We will emphasize the distinctions and relations between a logical system conceived as an abstract structure and proof-theoretical or model-theoretical presentations.
We wil explain in which sense universal logic can be considered as metalogic and the question of universality and relaivity of logic.
We wil explain in which sense universal logic can be considered as metalogic and the question of universality and relaivity of logic.
2. Logic as Mathematical Structures
2. Logic as Mathematical Structures
In this second lesson, we will present in details logics considered as mathemarical structures, the main concepts and tools related to this framwork, and we will examine some core problems of universal logic:
In this second lesson, we will present in details logics considered as mathemarical structures, the main concepts and tools related to this framwork, and we will examine some core problems of universal logic:
- Combination of logics
- Translation between logic
- Identity between logics
3. Univeral Logic: Results and Applications
3. Univeral Logic: Results and Applications
In this third lesson, we will present general concepts and results that can be developed within the framework of universal logic and show how they can be usefully applied to many different logical systems.
In this third lesson, we will present general concepts and results that can be developed within the framework of universal logic and show how they can be usefully applied to many different logical systems.
We will focus in particular on a general formulation of the completeness theorem from which we can derive many particular completeness theorems for specific systems.
We will focus in particular on a general formulation of the completeness theorem from which we can derive many particular completeness theorems for specific systems.
References
References
> Hilan Bensusan, Alexandre Costa-Leite and, Edelcio G. de Souza, “Logics and their galaxies”, in A.Koslow and A.Buchbaum (eds), The Road to Universal Logic - Vol 2, Birkhäuser, Basel, 2015, pp.243-252.
> Hilan Bensusan, Alexandre Costa-Leite and, Edelcio G. de Souza, “Logics and their galaxies”, in A.Koslow and A.Buchbaum (eds), The Road to Universal Logic - Vol 2, Birkhäuser, Basel, 2015, pp.243-252.
> Jean-Yves Beziau, “Universal logic”, in Logica’94 - Proceedings of the 8th International Symposium, T.Childers & O.Majer (eds), Prague, 1994, pp.73-93.
> Jean-Yves Beziau, “Universal logic”, in Logica’94 - Proceedings of the 8th International Symposium, T.Childers & O.Majer (eds), Prague, 1994, pp.73-93.
> Jean-Yves Beziau, “A paradox in the combination of logics’, in Workshop on Combination of Logics: Theory and Applications, W.A.Carnielli, F.M.Dionisio and P.Mateus (ed), IST, Lisbon, 2004, pp.75-78.
> Jean-Yves Beziau, “A paradox in the combination of logics’, in Workshop on Combination of Logics: Theory and Applications, W.A.Carnielli, F.M.Dionisio and P.Mateus (ed), IST, Lisbon, 2004, pp.75-78.
> Jean-Yves Beziau, “From consequence operator to universal logic: a survey of general abstract logic”, in Logica Universalis: Towards a general theory of logic, Birkhäuser, Basel, 2005, pp.3-17.
> Jean-Yves Beziau, “From consequence operator to universal logic: a survey of general abstract logic”, in Logica Universalis: Towards a general theory of logic, Birkhäuser, Basel, 2005, pp.3-17.
> Jean-Yves Beziau, “13 Questions about universal logic”, Bulletin of the Section of Logic, 35 (2006), pp.133-150.
> Jean-Yves Beziau, “13 Questions about universal logic”, Bulletin of the Section of Logic, 35 (2006), pp.133-150.
> Jean-Yves Beziau, “What is a logic ? - Towards axiomatic emptiness", Logical Investigations, 16 (2010), pp.272-279.
> Jean-Yves Beziau, “What is a logic ? - Towards axiomatic emptiness", Logical Investigations, 16 (2010), pp.272-279.
> Jean-Yves Beziau (ed) , Universal Logic: An Anthology - From Paul Hertz to Dov Gabbay, Birkhäuser, Basel, 2012. Preface available here.
> Jean-Yves Beziau (ed) , Universal Logic: An Anthology - From Paul Hertz to Dov Gabbay, Birkhäuser, Basel, 2012. Preface available here.
> Jean-Yves Beziau, “The relativity and universality of logic”, Synthese - Special Issue Istvan Németi 70th Birthday, 192 (2015), pp. 1939-1954.
> Jean-Yves Beziau, “The relativity and universality of logic”, Synthese - Special Issue Istvan Németi 70th Birthday, 192 (2015), pp. 1939-1954.
> Jean-Yves Beziau, “Metalogic, Schopenhauer and Universal Logic” in J.Lemanski (ed), Language, Logic, and Mathematics in Schopenhauer, Birkhäuser, Basel, 2020, pp.207-257.
> Jean-Yves Beziau, “Metalogic, Schopenhauer and Universal Logic” in J.Lemanski (ed), Language, Logic, and Mathematics in Schopenhauer, Birkhäuser, Basel, 2020, pp.207-257.
> Jean-Yves Beziau, “Logical structures from a model-theoretical viewpoint” in A.Costa-Leite (ed), Abstract Consequence and Logics - Essays in Honor of Edelcio G. de Souza , College Publications, London, 2020. pp.21-34.
> Jean-Yves Beziau, “Logical structures from a model-theoretical viewpoint” in A.Costa-Leite (ed), Abstract Consequence and Logics - Essays in Honor of Edelcio G. de Souza , College Publications, London, 2020. pp.21-34.
> Jean-Yves Beziau, and Arthur Buchsbaum, “Let us be Antilogical: Anti-Classical Logic as a Logic”, in A.Moktefi, A.Moretti and F.Schang (eds), Soyons logiques / Let us be Logical, College Publication, London, 2016, pp.1-10.
> Jean-Yves Beziau, and Arthur Buchsbaum, “Let us be Antilogical: Anti-Classical Logic as a Logic”, in A.Moktefi, A.Moretti and F.Schang (eds), Soyons logiques / Let us be Logical, College Publication, London, 2016, pp.1-10.
> Garrett Birkhoff, “Universal algebra”, in Comptes Rendus du Premier Congres Canadien de Mathématiques, Presses de l’Université de Toronto, Toronto, 1946, pp. 310–326.
> Garrett Birkhoff, “Universal algebra”, in Comptes Rendus du Premier Congres Canadien de Mathématiques, Presses de l’Université de Toronto, Toronto, 1946, pp. 310–326.
> Garret Birkhoff, “Universal algebra”, in G.-C. Rota, J.S. Oliveira (eds), Selected Papers on Algebra and Topology by Garrett Birkhoff, Birkhäuser, Basel, 1987. P.111- 116.
> Garret Birkhoff, “Universal algebra”, in G.-C. Rota, J.S. Oliveira (eds), Selected Papers on Algebra and Topology by Garrett Birkhoff, Birkhäuser, Basel, 1987. P.111- 116.
> Dov M. Gabbay (ed), What Is a Logical System? , Oxford University Press, Oxford, 1994.
> Dov M. Gabbay (ed), What Is a Logical System? , Oxford University Press, Oxford, 1994.
> Jean Porte, Recherches sur la théorie générale des systèmes formels et sur les systèmes connectifs, Gauthier-Villars, Paris, Nauwelaerts, Louvain.
> Jean Porte, Recherches sur la théorie générale des systèmes formels et sur les systèmes connectifs, Gauthier-Villars, Paris, Nauwelaerts, Louvain.
> Jacques Riche, “From Universal Algebra to Universal Logic”, in J.-Y. Beziau, A. Costal-Leite (eds), Perspectives on Universal Logic, Polimetra, Monza, 2009, pp.3-39.
> Jacques Riche, “From Universal Algebra to Universal Logic”, in J.-Y. Beziau, A. Costal-Leite (eds), Perspectives on Universal Logic, Polimetra, Monza, 2009, pp.3-39.
> Louis Rougier, “The relativity of logic”, Philosophy and Phenomenological Research, Vol. 2. 1941. pp.137-158, Reprinted in Anthology of Universal Logic, with a presentation by Mathieu Marion.
> Louis Rougier, “The relativity of logic”, Philosophy and Phenomenological Research, Vol. 2. 1941. pp.137-158, Reprinted in Anthology of Universal Logic, with a presentation by Mathieu Marion.
> Alfred Tarski, “Remarques sur les notions fondamentales de la méthodologie des mathématiques". In Annales de la Société Polonaise de Mathématique) (Vol. 7, 1929, pp. 270–272), translated as “Remarks on Fundamental Concepts of the Methodology of Mathematics” with a presentation of Jan Zygmunt in .
> Alfred Tarski, “Remarques sur les notions fondamentales de la méthodologie des mathématiques". In Annales de la Société Polonaise de Mathématique) (Vol. 7, 1929, pp. 270–272), translated as “Remarks on Fundamental Concepts of the Methodology of Mathematics” with a presentation of Jan Zygmunt in .
> Alfred North Whitehead, A treatise of universal algebra., Cambridge University Press, Cambridge 1898.
> Alfred North Whitehead, A treatise of universal algebra., Cambridge University Press, Cambridge 1898.
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