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Full Idea
All the main set theories deny that there is a set of which everything is a member. No interpretation has a domain with everything in it. So the universal quantifier never gets to mean everything all at once; 'all' does not mean all.
Gist of Idea
The universal quantifier can't really mean 'all', because there is no universal set
Source
William D. Hart (The Evolution of Logic [2010], 4)
Book Ref
Hart,W.D.: 'The Evolution of Logic' [CUP 2010], p.111
A Reaction
Could you have an 'uncompleted' universal set, in the spirit of uncompleted infinities? In ordinary English we can talk about 'absolutely everything' - we just can't define a set of everything. Must we 'define' our domain?
| 8079 | Aristotelian logic has two quantifiers of the subject ('all' and 'some') [Aristotle, by Devlin] |
| 7742 | Frege reduced most quantifiers to 'everything' combined with 'not' [Frege, by McCullogh] |
| 7730 | Frege introduced quantifiers for generality [Frege, by Weiner] |
| 6061 | Existence is entirely expressed by the existential quantifier [Russell, by McGinn] |
| 6069 | 'Partial quantifier' would be a better name than 'existential quantifier', as no existence would be implied [McGinn] |
| 11115 | 'All horses' either picks out the horses, or the things which are horses [Jubien] |
| 13392 | Philosophers reduce complex English kind-quantifiers to the simplistic first-order quantifier [Jubien] |
| 13506 | The universal quantifier can't really mean 'all', because there is no universal set [Hart,WD] |
| 8312 | It is better if the existential quantifier refers to 'something', rather than a 'thing' which needs individuation [Lowe] |