3.
Báq sıo bẹmuıdıu
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Concepts from semantics
Intension
In philosophy, the word intension (spelled with an s: not “intention”) refers to definitions in terms of properties, or “necessary and sufficient conditions”, rather than extension, a definition by enumerating the referents.
Furthermore, in semantics, an intensional interpretation of a phrase is one that retains the sense of the words used, whereas an extensional interpretation is one that directly gets at the thing referred to.
See: Sense and reference on Wikipedia.
Why do we need intension?
It turns out that extension, i.e. identifying expressions with their referents, fails to capture some of what we say. Suppose that ⟦Líma⟧ and ⟦ké joqdoaq po Pérugua⟧ are equal, and refer to Lima (the capital of Peru). Then consider the following sentence:
Chı Náo, ꝡá eq ké joqdoaq po Chílegua Líma.
Nao thinks the capital of Chile is Lima.
This describes a reasonable belief. But if Líma and ké joqdoaq… mean the same thing, we should be able to swap one out for the other without changing the meaning of the sentence. Instead, we get
Chı Náo, ꝡá eq ké joqdoaq po Chílegua ké joqdoaq po Pérugua.
Nao thinks the capital of Chile is the capital of Peru.
which is pretty absurd. There is something about these descriptions that we are not capturing, when we skip straight to referents or extensions in our denotation of definite references.
Intension in Kuna
The solution to this problem used by Kuna is to make the denotations be functions from possible worlds to entities: ○ instead of ○.
⟦Líma⟧ = λ𝑤 Lima in world 𝑤
⟦ké joqdoaq po Pérugua⟧ = λw the capital of Peru in world 𝑤
We can imagine possible worlds in which Lima had been founded elsewhere, or Peru had ended up with other borders and picked another capital. When given such worlds as input, these functions will differ in meaning. In this way the functions capture the sense of what was said.
Similarly, ⟦ꝡá eq ké joqdoaq po Chílegua Líma⟧ does not just equal false, because the principle of compositionality would say that the above sentence would reduce to “Nao thinks false”, which isn’t what we want.
Instead, propositions get the type ◐ rather than ◐: their truth value depends on the world. By denoting the content clause as
λ𝑤 whether, in world 𝑤, the capital of Chile is Lima
we retain the sense of what is said.
Here is a glimpse at the reason for the “squiggle” notation: world-dependence, or intensionality, is expressive and omnipresent in our semantics, while at the same time being a “layer” we don’t want to draw too much attention to. We may think of ○ › ◐ as a special flavor of function from ○ to ◐, that incidentally carries its “sense” with it by depending on the world variable.
Discourse and deixis
What is a conversation, or discourse, made up of? The common view says speech acts: questions, requests, statements, and so on. We denote these with the type ! , and for now that is about all we do with them.
Speech acts update the discourse state, which describes which participants are committed to the truth of which facts, which questions have been raised, which variables are bound, and so on. Kuna does not yet model the discourse state beyond some support for cross-sentence anaphora. For our purposes, words like da and ba “wrap” a proposition up into a speech act, but we have no way of running or combining speech acts: we can only look inside them.
Sometimes, though, the meaning of a word or sentence hinges not just on the world it’s said in and the words that came before it, but even on its speaker and their surroundings. A word like súq or naı doesn’t mean much without the context of who it’s said to, or when it’s said. Such words are called deictic, and using deictic words is called deixis.
Kuna models deixis as constituents “depending on the deictic state”, the same way it models intensionality as constituents “depending on the world variable.” Again we have special notation: placing a curved arrow before a type marks it as a function from deictic state to that type, and so something like ↪ ◐ describes the type of deictic propositions.
Event predicates
A final concept we should get comfortable with before stepping into the next chapter is the idea of an event predicate. Conceptually, this is a predicate that describes what kind of event something is, without yet situating it in time.
A sentence like Jıa suaq Mía nha promises the existence of a Mía-singing-event in the future. In this case, the event predicate is a function that has type ✲ › ◐ and describes the event involved:
⟦suaq Mía⟧ = λ𝑤 λ𝑒 𝑒 is a Mía-singing-event in 𝑤
It, too is intensional (depends on the world variable): this is what lets us tie events to a possible world that they might or might not happen in.
The essential structure of any Toaq clause is that the verb-plus-nouns describe an event predicate, which is then modified by the aspect, tense, and complementizer nodes to describe a proposition.
✲ › ◐
◐