Abstract
I argue that the celebrated Square of Opposition is just a shadow of a much deeper relationship on duality, complementarity, opposition and quaternality expressed by algebraic means, and that any serious attempt to make sense of squares and cubes of opposition must take into account the theory of finite groups. By defining a group as triadic if all its elements, other than the identity, have order 3, I show that a natural notion of triality group acting on three-valued structures emerges, generalizing the intuitions of duality and quaternality.
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Notes
- 1.
A Boolean polynomial is the analogous of an ordinary polynomial, employing a finite number of Boolean operations ∧ and ∨ on a finite number of elements in a Boolean algebra.
- 2.
I am not here interested in Piaget’s theory, nor in the abundant criticism around it: the only thing interesting here is the coincidence involving Klein’s group V.
- 3.
It should be clear that the operation of multiplication is interpreted as conjunction or meet ∧, and addition is interpreted as exclusive disjunction or symmetric difference \(\veebar\). Addition does not represent disjunction ∨, which would perhaps explain how exclusive disjunction, rather than standard inclusive disjunction, would be a more natural basis for logic.
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Acknowledgements
I acknowledge support from FAPESP Thematic Project LogCons 2010/51038-0, Brazil and from the National Council for Scientific and Technological Development (CNPq), Brazil. I am thankful to the five anonymous referees who have read, commented and criticized this paper.
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Carnielli, W. (2017). Groups, Not Squares: Exorcizing a Fetish. In: Béziau, JY., Basti, G. (eds) The Square of Opposition: A Cornerstone of Thought. Studies in Universal Logic. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-45062-9_14
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