Boolean Subtypes of the U4 Hexagon of Opposition
This paper investigates the so-called ‘unconnectedness-4 (U4) hexagons of opposition’, which have various applications across the broad field of philosophical logic. We first study the oldest known U4 hexagon, the conversion closure of the square of opposition for categorical statements. In particul...
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| Format: | Article |
| Language: | English |
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MDPI AG
2024-01-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/13/2/76 |
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| author | Lorenz Demey Atahan Erbas |
| author_facet | Lorenz Demey Atahan Erbas |
| author_sort | Lorenz Demey |
| collection | DOAJ |
| description | This paper investigates the so-called ‘unconnectedness-4 (U4) hexagons of opposition’, which have various applications across the broad field of philosophical logic. We first study the oldest known U4 hexagon, the conversion closure of the square of opposition for categorical statements. In particular, we show that this U4 hexagon has a Boolean complexity of 5, and discuss its connection with the so-called ‘Gergonne relations’. Next, we study a simple U4 hexagon of Boolean complexity 4, in the context of propositional logic. We then return to the categorical square and show that another (quite subtle) closure operation yields another U4 hexagon of Boolean complexity 4. Finally, we prove that the Aristotelian family of U4 hexagons has no other Boolean subtypes, i.e., every U4 hexagon has a Boolean complexity of either 4 or 5. These results contribute to the overarching goal of developing a comprehensive typology of Aristotelian diagrams, which will allow us to systematically classify these diagrams into various Aristotelian families and Boolean subfamilies. |
| first_indexed | 2024-03-07T22:42:20Z |
| format | Article |
| id | doaj.art-25e24e0adbb54330843092b4cf30d46d |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-07T22:42:20Z |
| publishDate | 2024-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-25e24e0adbb54330843092b4cf30d46d2024-02-23T15:07:20ZengMDPI AGAxioms2075-16802024-01-011327610.3390/axioms13020076Boolean Subtypes of the U4 Hexagon of OppositionLorenz Demey0Atahan Erbas1Center for Logic and Philosophy of Science, KU Leuven, 3000 Leuven, BelgiumCenter for Logic and Philosophy of Science, KU Leuven, 3000 Leuven, BelgiumThis paper investigates the so-called ‘unconnectedness-4 (U4) hexagons of opposition’, which have various applications across the broad field of philosophical logic. We first study the oldest known U4 hexagon, the conversion closure of the square of opposition for categorical statements. In particular, we show that this U4 hexagon has a Boolean complexity of 5, and discuss its connection with the so-called ‘Gergonne relations’. Next, we study a simple U4 hexagon of Boolean complexity 4, in the context of propositional logic. We then return to the categorical square and show that another (quite subtle) closure operation yields another U4 hexagon of Boolean complexity 4. Finally, we prove that the Aristotelian family of U4 hexagons has no other Boolean subtypes, i.e., every U4 hexagon has a Boolean complexity of either 4 or 5. These results contribute to the overarching goal of developing a comprehensive typology of Aristotelian diagrams, which will allow us to systematically classify these diagrams into various Aristotelian families and Boolean subfamilies.https://www.mdpi.com/2075-1680/13/2/76Aristotelian diagramsquare of oppositionhexagon of oppositionlogical geometrysyllogisticsconversion |
| spellingShingle | Lorenz Demey Atahan Erbas Boolean Subtypes of the U4 Hexagon of Opposition Axioms Aristotelian diagram square of opposition hexagon of opposition logical geometry syllogistics conversion |
| title | Boolean Subtypes of the U4 Hexagon of Opposition |
| title_full | Boolean Subtypes of the U4 Hexagon of Opposition |
| title_fullStr | Boolean Subtypes of the U4 Hexagon of Opposition |
| title_full_unstemmed | Boolean Subtypes of the U4 Hexagon of Opposition |
| title_short | Boolean Subtypes of the U4 Hexagon of Opposition |
| title_sort | boolean subtypes of the u4 hexagon of opposition |
| topic | Aristotelian diagram square of opposition hexagon of opposition logical geometry syllogistics conversion |
| url | https://www.mdpi.com/2075-1680/13/2/76 |
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