Aristotelian Diagrams for the Proportional Quantifier ‘Most’

In this paper, we study the interaction between the square of opposition for the Aristotelian quantifiers (‘all’, ‘some’, ‘no’, and ‘not all’) and the square of opposition generated by the proportional quantifier ‘most’ (in its standard generalized quantifier theory reading of ‘more than half’). In...

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Main Authors: Hans Smessaert, Lorenz Demey
Format: Article
Language:English
Published: MDPI AG 2023-02-01
Series:Axioms
Subjects:
Online Access:https://www.mdpi.com/2075-1680/12/3/236
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author Hans Smessaert
Lorenz Demey
author_facet Hans Smessaert
Lorenz Demey
author_sort Hans Smessaert
collection DOAJ
description In this paper, we study the interaction between the square of opposition for the Aristotelian quantifiers (‘all’, ‘some’, ‘no’, and ‘not all’) and the square of opposition generated by the proportional quantifier ‘most’ (in its standard generalized quantifier theory reading of ‘more than half’). In a first step, we provide an analysis in terms of bitstring semantics for the two squares independently. The classical square for ‘most’ involves a tripartition of logical space, whereas the degenerate square for ‘all’ in first-order logic (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">FOL</mi></semantics></math></inline-formula>) involves a quadripartition, due to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">FOL</mi></semantics></math></inline-formula>’s lack of existential import. In a second move, we combine these two squares into an octagon of opposition, which was hitherto unattested in logical geometry, while the meet of the original tri- and quadripartitions yields a hexapartition for this octagon. In a final step, we switch from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">FOL</mi></semantics></math></inline-formula> to a logical system, which does assume existential import. This yields an octagon of the well known Lenzen type, and its bitstring semantics is reduced to a pentapartition.
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spelling doaj.art-4f4ecb32694e4e7da76723246859dbb02023-11-17T09:34:40ZengMDPI AGAxioms2075-16802023-02-0112323610.3390/axioms12030236Aristotelian Diagrams for the Proportional Quantifier ‘Most’Hans Smessaert0Lorenz Demey1Department of Linguistics, KU Leuven, 3000 Leuven, BelgiumCenter for Logic and Philosophy of Science, KU Leuven, 3000 Leuven, BelgiumIn this paper, we study the interaction between the square of opposition for the Aristotelian quantifiers (‘all’, ‘some’, ‘no’, and ‘not all’) and the square of opposition generated by the proportional quantifier ‘most’ (in its standard generalized quantifier theory reading of ‘more than half’). In a first step, we provide an analysis in terms of bitstring semantics for the two squares independently. The classical square for ‘most’ involves a tripartition of logical space, whereas the degenerate square for ‘all’ in first-order logic (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">FOL</mi></semantics></math></inline-formula>) involves a quadripartition, due to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">FOL</mi></semantics></math></inline-formula>’s lack of existential import. In a second move, we combine these two squares into an octagon of opposition, which was hitherto unattested in logical geometry, while the meet of the original tri- and quadripartitions yields a hexapartition for this octagon. In a final step, we switch from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">FOL</mi></semantics></math></inline-formula> to a logical system, which does assume existential import. This yields an octagon of the well known Lenzen type, and its bitstring semantics is reduced to a pentapartition.https://www.mdpi.com/2075-1680/12/3/236square of oppositionoctagon of oppositionlogical geometrybitstring semanticsproportional quantificationAristotelian quantifiers
spellingShingle Hans Smessaert
Lorenz Demey
Aristotelian Diagrams for the Proportional Quantifier ‘Most’
Axioms
square of opposition
octagon of opposition
logical geometry
bitstring semantics
proportional quantification
Aristotelian quantifiers
title Aristotelian Diagrams for the Proportional Quantifier ‘Most’
title_full Aristotelian Diagrams for the Proportional Quantifier ‘Most’
title_fullStr Aristotelian Diagrams for the Proportional Quantifier ‘Most’
title_full_unstemmed Aristotelian Diagrams for the Proportional Quantifier ‘Most’
title_short Aristotelian Diagrams for the Proportional Quantifier ‘Most’
title_sort aristotelian diagrams for the proportional quantifier most
topic square of opposition
octagon of opposition
logical geometry
bitstring semantics
proportional quantification
Aristotelian quantifiers
url https://www.mdpi.com/2075-1680/12/3/236
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