Semantics

Basic types and variables

	Type	Variables
individuals	e	a, b, c, ..., ráı, ...
truth-values	t
events	v	e, e', e'', ...
time intervals	i	t, t', t'', ...
worlds	s	w, w', w'', ...

Complex types (selection)

	Type
Vintrans, Vtrans	<e,<v,t>>
Vditrans	<e,<e,<v,t>>>
Q	<<e,t>,<<e,t>,t>>
&(róı)	<e,<e,e>>
Asp	<<v,t>,<i,t>>
T	<i,t>
Propositions	<s,t>
Properties	<e,t> or <s,<e,t>>

Tree notation

gold = node labels and types

:: separates label from type, e.g. DP :: e

Semantic denotations are colored red, e.g. λx. hao(x).

Toaq words and phrases are light blue like everywhere else.

Functional Application (FA)

If β is a head and γ its complement, then
⟦α⟧ = ⟦β⟧(⟦γ⟧)

Functional Application is the most common composition rule. It applies to the following head classes:

V, D, Asp, T, Modal(P), Σ, &, SA, Q, C

Predicate Modification (PM)

If β or γ is an adjunct, then
⟦α⟧ = λx. ⟦β⟧(x) ∧ ⟦γ⟧(x)

Predicate Modification applies to:

CP_rel, AdjunctP, aP

Predicate Abstraction (PA)

If β is a QP or FocusAdvP with the index i, then
⟦β_i γ⟧^g = ⟦β⟧(λx. ⟦γ⟧^g[x/i])

g[x/i] is that variable assignment which is identical to g except that every constituent with the index i is replaced by x.

Predicate Abstraction applies to:

QP, FocusAdvP

Event Identification (EI)

If β is a non-vacuous v head and γ is a VP, then
⟦α⟧ = λx.λe. ⟦β⟧(x)(e) ∧ ⟦γ⟧(e)

Event Identification applies to:

v

Personal pronoun denotations

The personal pronouns follow this pattern:

⟦jí⟧^g,c = the speaker in c
⟦súq⟧^g,c = the listener in c
⟦úmo⟧^g,c = the speaker in c + the listener in c
...

Anaphoric pronouns and bound variables

For any other DP, the denotation is as follows:

If a DP α is an anaphoric pronoun or a bound variable, g is a variable assignment, and i is the DP's index, then
⟦α_i⟧^g = g(i)

For example:
⟦kúne₅⟧^g = g(5)

See also: determiner phrase denotation

VP denotation

The verb composes with its internal argument to form a VP.

The internal argument satisfies the object place, leaving the VP with one open event place.

⟦V⟧ = λx.λe. V(x)(e)
⟦VP⟧ = ⟦V⟧(<object>)
= [λx.λe. V(x)(e)](<object>)
= λe. V(<object>)(e)

For example:

⟦gaı súq⟧
= ⟦gaı⟧(súq)
= [λx.λe. gaı(x)(e)](súq)
= λe. gaı(súq)(e)
= λe. e is an event of perceiving the listener

Intransitive verbs

The subject of an unaccusative verb merges as the complement of the verb, where it receives theme or experiencer.

⟦V⟧(⟦DP⟧)
= [λx.λe. shua(x)(e)](g(9))
= λe. shua(g(9))(e)
= λe. e is an event of g(9) falling, defined only if g(9) is rain

The subject of an unergative verbs originates in Spec,vP, where it receives its agent role (via Event Identification):

⟦v⟧(⟦VP⟧)
= [λx.λe. agent(e) = x](λx.λe. shua(x)(e))
= λx.λe. saqsu(e) ∧ agent(e) = x

[λx.λe. saqsu(e) ∧ agent(e) = x](jí)
= λe. saqsu(e) ∧ agent(e) = jí
= λe. e is an event of the speaker whispering

Transitive and ditransitive verbs

Transitives and ditransitives have an external agent argument which originates in Spec,vP. It receives its role from the v head. v combines with its sister via Event Identification:

λe. buja(g(1))(e) ∧ agent(e) = súq
= λe. e is an event of kissing g(1) ∧ the listener is the agent of e,
defined only if g(1) is an equine

With ditransitive verbs, the indirect object precedes the direct object:

λe. do(súq, g(6))(e) ∧ agent(e) = jí
= λe. e is an event of giving g(6) to the listener ∧ the speaker is the agent of e,
defined only if g(6) is a book

Cleft verb denotation

The cleft verb nä allows a single argument to be fronted. This argument must appear in the complement of nä.

nä is used purely for manipulating word order. It does not add any semantic content.

Event accessor denotation

The event accessor ë turns the event argument of a vP into a surface argument in order to make it accessible for surface arguments or to serve as the complement of a determiner.

márao "the one who dances"
é marao "the event of dancing"

Predicatizer denotation

Predicatizers are quantificationally opaque: if the complement is a quantified DP, the DP is QP-bound locally rather than by a QP in the clause that contains the predicatizer. To properly model this behavior, we posit the presence of a CP, which acts as a scope boundary.

The above example has a referring expression (jí "the speaker") as the complement. The following example features a quantified DP.

This example shows how sá paı takes local scope, yielding:

λx_i. ∃x : paı(x). hao(x_i, x) , which means
"X is such that there exists a friend with which X is related via the salient relation",
for the phrase po sá paı.

Determiner phrase denotation

Determiner phrases are complex structures of type e. They are composed of a determiner (D) followed by an nP. The nP consists of an unprounced n head, which contains a phi feature (animacy) and an index. Both are passed up the tree to give the projected DP an index and a phi feature.

The phi feature is presuppositional:

⟦n_i⟧ = λP: φ(g(i)). P, i.e., the function returns P if g(i) has the feature φ, and is undefined otherwise.

There is only one denotation for every determiner, or put another way, there is only one determiner in Toaq: bound-the.

⟦D nP_i⟧ = λP: P(g(i)). g(i)

This again contains a presupposition: if g(i) satisfies the predicate P, then the function returns g(i), otherwise it is undefined. This ensures that the main semantic content (in CP_rel) makes a contribution, while the denotation of the DP is just a variable.

For example, the DP tú poq

has the denotation g(3), is presupposed to satisfy λx.poq(x), and is animate.

The nP also moves and merges with a higher Q head, which is where the DP variable actually gets bound (see below).

Quantifier scope pattern

Every quantifier phrase (QP) binds a DP in its scope.

The scope of a QP is its sister.

QP pattern

A quantifier phrase (QP) is a quantifier followed by an nP.

The nP originates in a DP, from where it moves to the complement of Q to restricts the quantifier. The nP carries an index, which is passed to the QP.

Every QP binds a DP with the same index.

Here, the nP ruq moves out of the DP and into the complement of Q to restrict the quantifier ∃, and to provide the QP with an index.

Quantifiers are of type <<e,t>,<<e,t>,t>>, meaning they take two one-place predicates as arguments. The first predicate (the nP) acts as a restriction on the domain of the quantifier. The second predicate is the scope of the quantifier.

A QP combines with its sister via Predicate Abstraction.

⟦QP₇ TP⟧^g,c
= ⟦QP₇⟧(λx. ⟦TP⟧^g[x/7])
= [λS. ∃x : ruq(x). S(x)](λx. ⟦TP⟧^g[x/7])
= [λS. ∃x : ruq(x). S(x)](λx. ∃e. shua(x)(e))
= ∃x : ruq(x). ∃e. shua(x)(e)
= There exist(s) some x such that x is rain, such that there exists an event e of x falling.

⟦nuo⟧^g,c,w = λx.λe.λw. nuo_w(x)(e)
which has the type <e,<v,<s,t>>>

Propositional verbs

The denotation of the CP ꝡä nuo kúne "that the dog is asleep" is
λw. ∃e'. nuo_w(kúne)(e'), i.e.,
a function which returns True if there is an event of the dog being asleep in w (at time t, ...). This CP combines with the verb via Intensional Functional Application:

If β is an intension-taking function, and γ is an intensional complement, then, for any possible world w:
⟦α⟧^w = ⟦β⟧^w(λw'. ⟦γ⟧^w')

What the verb does with the proposition depends on the definition of the verb. For chı, we can say that

chı(jí, {λw. ∃e'. nuo_w(kúne)(e')}) expresses that

"Every possible world w which is compatible with the beliefs of the speaker is one in which the dog is asleep at time t", or in short, "The speaker believes that the dog is asleep."

Adverbial adjunct denotation

Adverbials are adjuncts to vPs. They combine with their sister via Predicate Modification. They act as restrictions on the event variable of the vP.

⟦[_vP [vP] [_AdjunctP Adjunct VP]]⟧^g,c
= λx. ⟦vP⟧(x) ∧ ⟦AdjunctP⟧(x)
= λx. ⟦nuokuaı súq⟧(x) ∧ ⟦bîe jóafao⟧(x)
= λx. [λe. nuokuaı(súq)(e)](x) ∧ [λx. bıe(x, jóafao)](x)
= λe. nuokuaı(súq)(e) ∧ bıe(e, jóafao)
= λe. e is an event of the listener being tired ∧ e is after the weekend

Relative clause denotation

Relative clauses are adjuncts to nPs. They combine with their sister via Predicate Modification:

⟦[_nP [nP CP_rel]]⟧^g,c
= λx. ⟦nP⟧(x) ∧ ⟦CP_rel⟧(x)
= λx. ⟦poq⟧(x) ∧ ⟦ꝡë kaqgaı jí hóa⟧(x)
= λx. [λx. poq(x)](x) ∧ [λx. kaqgaı(jí, x)](x)
= λx. poq(x) ∧ kaqgaı(jí, x)
= λx. x is a person ∧ the speaker sees x

Negation

Polarity (Σ) is a flexible category which can attach to many kinds of phrases. The type-agnostic definitions are:

bu = λP. ¬P

jeo = λP. †P

Aspect and Tense

Tense and aspect work together. Aspect acts on the vP first, and deals with the vP's event variable by relating it to a time interval. This interval is then supplied by a tense.

Aspect denotations

Aspect	Denotation
tam	λP. λt. ∃e. τ(e) ⊆ t ∧ P(e)
chum	λP. λt. ∀w' ∈ IW(w,t'). ∃e. t ⊆ τ(e) ∧ Pw'(e)
luı	λP. λt. ∃e. τ(e) < t ∧ P(e)
za	λP. λt. ∃e. τ(e) > t ∧ P(e)
hoaı	λP. λt. ∃e. t ⊆ τ(e) ∧ t > ExpEnd(e) ∧ P(e)
haı	λP. λt. ∃e. t ⊆ τ(e) ∧ t < ExpStart(e) ∧ P(e)
hıq	λP. λt. ∃e. τ(e) <near t ∧ P(e)
fı	λP. λt. ∃e. τ(e) >near t ∧ P(e)

Symbols:
< : the "before" relation: a time inverval t is before a time interval t' if every moment in t temporally precedes every moment in t'.
> : the "after" relation
⊆ : the "subinterval" relation: interval t is included in interval t'
τ() : the temporal trace function, which maps an event to its temporal trace / its runtime
IW(w,t) : the set of inertia worlds for world w at time t, where an inertia world w' is a world which, up to t, is exactly like w, but then it continues at t in a way that is normal or natural for the given verb. This modal treatment of chum makes it possible to use this aspect even if the event never ends up getting completed. Chum baı jí báq anıjıo "I was building a sand castle... when I remembered I had somewhere else to be". The event never fully takes place in w, but it does in w'.

Tense denotations

The tenses pu, naı, jıa are pronominal: they refer to a (salient) time interval, and the temporal position of this interval is presupposed to be before, including, or after, the utterance time.

Tense	Denotation
pu	⟦pu⟧g,c is only defined if c provides a time interval t such that t < t0. If defined, then ⟦pu⟧g,c = t.
naı	⟦naı⟧g,c is only defined if c provides a time interval t such that t ⊆ t0. If defined, then ⟦naı⟧g,c = t.
jıa	⟦jıa⟧g,c is only defined if c provides a time interval t such that t > t0. If defined, then ⟦jıa⟧g,c = t.

Symbols:
< : the "before" relation
> : the "after" relation
⊆ : the "subinterval" relation
t₀ : the utterance time / speech time

∃e. τ(e) < t ∧ tısha(tíqra, jáqgala)(e) reads:
"There is an event e whose runtime is before t and e is an event of the tiger arriving in the jungle,
where t is presupposed to be included in the utterance time t₀."

Naı luı tısha tíqra jáqgala.
"The tiger has arrived in the jungle."

The tenses sula, mala, jela are existential and non-presuppositional:

Tense	Denotation
sula	λP. ∃t P(t)
mala	λP. ∃t : t < t0. P(t)
jela	λP. ∃t : t > t0. P(t)

Focus adverb denotation

Focusing adverbs (see also the syntax section on them) have a structure that is similar to that of QPs, but with rather different denotations.

An interesting property of Focus particles is that they contain presuppositions. For example, mao “also” does not add meaning to the claim itself, but contains the presupposition that other contextually relevant things satisfy the predicate, too.

The following example shows the general structure:

f stands for the focus-marked constituent, and A^f is the set of contextually available alternatives to the focus-marked constituent.

In the above example, [only] has the denotation
λf. λP. ∀x : (x ∈ A^f ∧ x ≠ f). ¬P(x),
but it also contains a presupposition, namely that the f-marked constituent satisfies the predicate. Below is a table containing each focusing adverbs along with presupposition and claim (the presupposition being the part between colon and period):

	Presupposition	Claim
[only] (tó) λf. λP:	P(x).	∀x : (x ∈ Af ∧ x ≠ f). ¬P(x)
[also] (máo) λf. λP:	∃x : (x ∈ Af ∧ x ≠ f). P(x).	P(x)
[even] (júaq) λf. λP:	∀x : (x ∈ Af ∧ x ≠ f). P(x) >likely P(f).	P(x)

ModalP denotation

Toaq's modals are, in essence, quantifiers over possible worlds. Because of this, we can use the same structure that QPs use:

Q : [restriction]. [scope]

The restriction restricts the possible worlds to those in which the modal complement is true.

The scope specifies what the worlds in that restricted set are like.

The ModalP combines with its sister via Functional Application.

ao is a function which creates the "ao-worlds" of a world w. It returns the set of minimally different worlds supplied by context c in which the complement is true, and it is only defined if the complement is false in w₀ (or whatever reference world we're currently in). This last part is what makes ao a "subjunctive" conditional.

The above example arrives at
∀w ∈ ao(w) : ∃e. tı_w(súq, ní)(e).
∃e'. buja_w(súq)(e') ∧ agent(e') = jí

"Every ao-world in which you are here
is a world in which I kiss you." (Presupposition: "You are not here")

"If you were here, I would kiss you."

The following table lists the four basic modal denotations:

Modal	Denotation
she	λR.λS. ∀w ∈ shew'(w) : R(w). S(w)
daı	λR.λS. ∃w ∈ shew'(w) : R(w). S(w)
ao	λR.λS. ∀w ∈ aow'(w) : R(w). S(w)
ea	λR.λS. ∃w ∈ aow'(w) : R(w). S(w)

she-worlds are "indicative", i.e. factual, while ao-worlds are "subjunctive", i.e. counterfactual.

DP conjuncts

The basic, uninflected, plural coordinate structure looks as follows:

róı can be inflected to indicate the presence of a higher QP which quantifies over the referents of the conjuncts:

In other words, the change from róı to rú has resulted in a change from

∃e. nuo([súq & jí])(e)
= ∃e. e is an event of [the listener & the speaker] being asleep

to

∀x : mea(x, [súq & jí]). ∃e. nuo(x)(e)
= ∀x : x is among [the listener & the speaker]). ∃e. e is an event of x being asleep

I.e., every individual among the referents of the coordinate DP súq róı jí is claimed to satisfy the predicate.

Likewise for rá:

Adverbial conjuncts

Preposition conjuncts

These are identical to full adverbial conjuncts, except that the conjuncts share the same complement:

⟦nîe rú gûq tíaı⟧^g,c
= λx. x is in the box ∧ x is under the box

Relative clause conjuncts

⟦[_&P CPrel [_&' & CPrel]]⟧^g,c
= λx. x likes flowers ∧ x is wearing no clothes

Tense phrase conjuncts

⟦[_&P TP [_&' & TP]]⟧^g,c
= ⟦[_&P [_TP joaı súq báq rua] [_&' [_& rú] [_TP sea jí]]]⟧^g,c
= (gen r : rua(r). ∃e. joaı(súq, r)(e)) ∧ (∃e'. sea(jí)(e'))
= (∃e. e an event of looking for flowers and the agent of e is the listener in c) ∧ (∃e'. e' is an event of the speaker in c relaxing)

(tense and aspect omitted)