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Definite: Difference between revisions
add túq, (tó)ꝡaı; flesh out denotations for the ℩… functionals
m (hóbo → hobo) |
(add túq, (tó)ꝡaı; flesh out denotations for the ℩… functionals) |
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! {{t|báq}} | ! {{t|báq}} | ||
| Always resolves to the associated [[kind]]. {{t|báq kanı}} is always a singular ‘rabbit-kind’ | | Always resolves to the associated [[kind]]. {{t|báq kanı}} is always a singular ‘rabbit-kind’ | ||
|- | |||
! {{t|túq}} | |||
| ‘all’; ‘the totality of’ | |||
|- | |- | ||
! {{t|cúaq||#9NfM8JXeD}} | ! {{t|cúaq||#9NfM8JXeD}} | ||
| (Unofficial:) ‘the concept of satisfying property’. Always refers to that one concept | | (Unofficial:) ‘the concept of satisfying property’. Always refers to that one concept | ||
|- | |||
! {{t|ꝡáı||#xvC8-FhEy}} | |||
| (Unofficial:) ‘the (aforementioned or not)’. To be understood as a hypernym of {{t|hú}} and {{t|ké}} | |||
|- | |||
! {{t|tóꝡaı||#SaSWb3ZIb}} | |||
| (Unofficial:) ‘the unique’ | |||
|} | |} | ||
{{done|2}} is definite when it refers to a phrase that appears in the same sentence and that phrase is definite. Otherwise, it does the same thing as {{t|ké}} does, which is definite, too. | {{done|2}} is definite when it refers to a phrase that appears in the same sentence and that phrase is definite. Otherwise, it does the same thing as {{t|ké}} does, which is definite, too. | ||
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as | as | ||
<blockquote> | <blockquote> | ||
𝑃( | 𝑃(℩<sub>◻</sub>(𝑄)) | ||
</blockquote> | </blockquote> | ||
where | where ℩<sub>◻</sub> exists and is some ⟨⟨𝚎, 𝚝⟩, 𝚎⟩ – then we say that the quantifier ◻ (and its associated determiner {{class|sá}}) is definite. In other words, ◻ can be mapped to some ℩<sub>◻</sub> such that for any 𝑄 of our choosing, ℩<sub>◻</sub>(𝑄) resolves to a single plural constant, which can then directly be plugged into 𝑃 to judge the truth value of the entire clause. And indeed for the determiners listed above we have: | ||
* ℩<sub>⟦{{t|ꝡáı}}⟧</sub>(𝑄) = the contextually appropriate 𝑄 | |||
* ℩<sub>⟦{{t|hú}}⟧</sub>(𝑄) = the aforementioned 𝑄 | |||
* ℩<sub>⟦{{t|ké}}⟧</sub>(𝑄) = the contextually appropriate 𝑄, not aforementioned | |||
* ℩<sub>⟦{{t|báq}}⟧</sub>(𝑄) = 𝑄-kind; the kind of 𝑄s | |||
* ℩<sub>⟦{{t|túq}}⟧</sub>(𝑄) = such an 𝑥 : 𝑄𝑥 for which [∀𝑥′ : 𝑄𝑥′] 𝑥′ ⊑ 𝑥. In other words, an 𝑥 : 𝑄𝑥 which “subsumes” all other 𝑥′ : 𝑄𝑥′ | |||
* ℩<sub>⟦{{t|cúaq}}⟧</sub>(𝑄) = 𝑄 itself (as an ⟨𝚎, 𝚝⟩ property) ≈ ⟦{{t|lä}} ⟧𝑄⟦ {{t|já}}⦄ | |||
* ℩<sub>⟦{{t|tóꝡaı}}⟧</sub>(𝑄) = the unique 𝑥 for which 𝑄𝑥; more precisely, entails [∀𝑥′ : 𝑄𝑥′] 𝑥′ = 𝑥. Equivalent to [[wikipedia:Definite_description#Mathematical_logic|Russell’s iota notation]] – ℩𝑥(𝑄𝑥) – by which the ℩<sub>…</sub> notation in this article is inspired | |||
Note that these are not required to ''successfully'' resolve to a plural constant. They may also produce a presuppositional failure, e.g., {{t|ꝡáı/hú/ké sıapuı}} cannot resolve to anything because {{t|sıapuı}} is trivially false, and {{t|tóꝡaı req}} is going to fail in a world where {{t|puq túq req}} holds. What’s important is that in none of these cases the [◻𝑥 : 𝑄𝑥] 𝑃𝑥 form can succeed if the 𝑃(℩<sub>◻</sub>(𝑄)) form fails. | |||
Then, any free determiner phrase is definite if all indices inside it are | Then, any free determiner phrase is definite if all indices inside it are | ||