Definite: Difference between revisions

Revision as of 09:58, 7 February 2024

Determiners

The following determiners are definite, which means that they refer to one concrete things and not multiple possible thingses, like with sá. In other words, they act like constants.

hú	‘the aforementioned’. Always resolves to one concrete thing, even if that thing might not be clear to the hearer
ké	‘the not aforementioned’. Same as above
báq	Always resolves to the associated kind. báq kanı is always a singular ‘rabbit-kind’
túq	‘all’; ‘the totality of’
cúaq	(Unofficial:) ‘the concept of satisfying property’. Always refers to that one concept
ꝡáı	(Unofficial:) ‘the (aforementioned or not)’. To be understood as a hypernym of hú and ké
tóꝡaı	(Unofficial:) ‘the unique’

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 ké does, which is definite, too.

Semantics jank

We say a noun phrase is definite if it’s a function of just one plural constant.

For a clause like P sá Q, where the quantifier is represented as ◻, if we’re able to rephrase the usual denotation

[◻𝑥 : 𝑄𝑥] 𝑃𝑥

as

𝑃(℩_◻(𝑄))

where ℩_◻ exists and is some ⟨⟨𝚎, 𝚝⟩, 𝚎⟩ – then we say that the quantifier ◻ (and its associated determiner sá) is definite. In other words, ◻ can be mapped to some ℩_◻ such that for any 𝑄 of our choosing, ℩_◻(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:

℩_{⟦ꝡáı⟧}(𝑄) = the contextually appropriate 𝑄
℩_⟦hú⟧(𝑄) = the aforementioned 𝑄
℩_⟦ké⟧(𝑄) = the contextually appropriate 𝑄, not aforementioned
℩_⟦báq⟧(𝑄) = 𝑄-kind; the kind of 𝑄s
℩_⟦túq⟧(𝑄) = such an 𝑥 : 𝑄𝑥 for which [∀𝑥′ : 𝑄𝑥′] 𝑥′ ⊑ 𝑥. In other words, an 𝑥 : 𝑄𝑥 which “subsumes” all other 𝑥′ : 𝑄𝑥′
℩_⟦cúaq⟧(𝑄) = 𝑄 itself (as an ⟨𝚎, 𝚝⟩ property) ≈ ⟦lä ⟧𝑄⟦ já⦄
℩_{⟦tóꝡaı⟧}(𝑄) = the unique 𝑥 for which 𝑄𝑥; more precisely, entails [∀𝑥′ : 𝑄𝑥′] 𝑥′ = 𝑥. Equivalent to Russell’s iota notation – ℩𝑥(𝑄𝑥) – by which the ℩_… 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., ꝡáı/hú/ké sıapuı cannot resolve to anything because sıapuı is trivially false, and tóꝡaı req is going to fail in a world where puq túq req holds. What’s important is that in none of these cases the [◻𝑥 : 𝑄𝑥] 𝑃𝑥 form can succeed if the 𝑃(℩_◻(𝑄)) form fails.

Then, any free determiner phrase is definite if all indices inside it are

bound inside the 𝑛P
bound outside the 𝑛P, to a phrase that is itself transitively definite

So for example, the following examples are all definite:

pó sá (sá is contained within po’s internal CPᵣₑₗ, hence does not escape)
ké kune bï, cho jí réo hobo (hóbo points to ké kune, which is itself definite)

@@ Line 10: / Line 10: @@
 ! {{t|báq}}
 | Always resolves to the associated [[kind]]. {{t|báq kanı}} is always a singular ‘rabbit-kind’
+|-
+! {{t|túq}}
+| ‘all’; ‘the totality of’
 |-
 ! {{t|cúaq||#9NfM8JXeD}}
 | (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.
@@ Line 25: / Line 34: @@
 as
 <blockquote>
-𝑃(℩◻(𝑄))
+𝑃(℩<sub>◻</sub>(𝑄))
 </blockquote>
-where ℩◻ exists and is some ⟨⟨𝚎, 𝚝⟩, 𝚎⟩ – then we say that the quantifier ◻ (and its associated determiner {{class|sá}}) is definite. In other words, ℩◻(𝑄) resolves to a single plural constant, which can then directly be plugged into 𝑃 to judge the truth value of the entire clause.
+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