4.
Sá poaqse po Tóaqzu
/
A fragment of Toaq
Starting small
To recap: Kuna is built on the idea that meaning is compositional, and we derive the meaning of a sentence by successively combining the meanings of all the words in it.
To get started studying this phenomenon, we’ll imagine a simple fragment of Toaq (let’s call it Póaqzu) where all the sentences are of the form “tense, verb, noun, speech act particle.”
Jıa suaq líq nha.
Naı dom pícarıaq da.
Pu puagı cháq ꝡeı.
They all have this syntax tree:
SAP
TP
T
VP
V
N
SA
Here, SA means “speech act”, T means “tense”, V means “verb”, and N means “noun”. The corresponding labels ending in P mean “… phrase.”
Types in Póaqzu
What are the types of the denotations of all the constituents (subtrees) in this tree?
From our discussion of intension in the last chapter, We know that líq denotes an intensional entity ○, and jıa suaq líq an intensional truth value ◐. We also know that an SAP, a “speech act phrase”, is supposed to denote a speech act ( ! ), and we want the verb-and-noun to denote an event predicate.
We can summarize this data in the following table:
| Label | Text | Denotation | Type |
|---|---|---|---|
| SAP | Jıa suaq líq nha. | speech act | ! |
| TP | jıa suaq líq | proposition | ◐ |
| VP | suaq líq | event predicate | ✲ › ◐ |
| N | líq | noun | ○ |
Deriving the other types
As described in chapter 1, we’d like to use function application to combine our meanings as we go up the tree. This means we can use the following strategy to help us get an idea of what types the remaining words should have.
If 𝑋 : α combines with 𝑌 to form 𝑍 : β, then 𝑌 might have a type like α › β.
(In other words: if we want type-driven compositionality, then while designing or discovering our semantics, we adopt compositionality-driven types.) Here’s an example:
!
◐
◐ › !
Indeed, it makes intuitive sense for ⟦nha⟧ to be a function that turns a proposition into a promise.
The derivation of the other remaining types is similar:
| Label | Text | Denotation | Type |
| V | suaq | verb | ○ › ✲ › ◐ |
| T | jıa | tense | ✲ › ◐ › ◐ |
| SA | nha | speech act maker | ◐ › ! |
⟦suaq⟧ turns a noun 𝑁 into an event predicate describing 𝑁-singing-events. And ⟦jıa⟧ turns an event predicate 𝑄 into a proposition, which claims some future event satisfies 𝑄.
(Both of these descriptions paper over intensionality: there’s actually some “λ𝑤” and “in 𝑤” involved.)
A denoted tree
Now we can denote the entire tree, writing a formula that has the right type at each node.
nha(λ𝑤 líq will sing in 𝑤)
!
λ𝑤 líq will sing in 𝑤
◐
λ𝑃 λ𝑤 ∃𝑒: 𝑃 𝑤 𝑒 ∧ future(𝑒)
✲ › ◐ › ◐
λ𝑤 λ𝑒 𝑒 is a
líq𝑤-singing-event in 𝑤
✲ › ◐
λ𝑎 (λ𝑤 λ𝑒 𝑒 is an
𝑎(𝑤)-singing-event in 𝑤)
○ › ✲ › ◐
λ𝑤 líq𝑤
○
nha
◐ › !
You can verify that each formula is the result of applying one of the two child formulas to the other and simplifying the resulting expression through substitution. For example:
⟦suaq líq⟧
= ⟦suaq⟧ ⟦líq⟧
= [λ𝑎 (λ𝑤 λ𝑒 𝑒 is an 𝑎(𝑤)-singing-event in 𝑤)] [λ𝑤 líq𝑤]
= (λ𝑤 λ𝑒 𝑒 is a [λ𝑤 líq𝑤](𝑤)-singing-event in 𝑤)
= (λ𝑤 λ𝑒 𝑒 is a líq𝑤-singing-event in 𝑤)
Looking forward
Hooray! We’ve denoted a sentence of Póaqzu. Importantly, if we change just one word of the sentence, we can see how changing the denotation of the leaf has a ripple effect on the denotation of the whole sentence. (Try drawing your own tree for Jıa marao líq nha, for example.)
Of course, we’ve made loads of simplifications. For the rest of the guide, we will undo them, until our little Póaqzu catches up with Toaq itself.