Interés general según la CE

First, we will introduce the type system and the frame that semantic evaluations ofInqD

will be based on. Explicit formal definitions of the notions discussed in the previous section will then be given here in the type-theoretic framework.

3.2.1.1 Types and Frames

First let’s lay out the type system ofInqD that we will operate on. Besides basic types

from Ty2, namely the types of individuals e, worlds sand truth values t, a basic type

of discourse referentsr is also included here. In constructing complex types, besides functional types, i.e. types of a function that maps an object of one type to another, we also include relational types in the construction, i.e. for any two types σ and τ, their cartesian product (σ× τ) is also a type, categorizing a relation between them. The relational type will typically be used in constructing context.

Definition3.1. (Inq_Dtypes)

(i) InqDhas four basic types: tfor truth-values,sfor possible worlds,efor individuals,

r for discourse referents;

(ii) The set of allInqD typesTypesis the smallest set containing the basic types, and

such that for any two type σ, τ ∈ Types, there is also ⟨σ, τ⟩ ∈ Types (sometimes abbreviated as (στ)) and (σ×τ)∈Types.

The semantics ofInqDexpressions will always be evaluated on anInqDmodel, which

in turn resides in anInqD frame.

Definition3.2. (Inq_DFrames)

AnInqDframe is a set (of domains){Dτ|τ∈Types}such that: (i) De, Ds, Dt, Drare pairwise disjoint;

(ii) De is the set of all non-empty subsets of a given set of entitiesE, i.e.De =℘+(E) :=

℘(E)\∅;

(iv) Dsis a non-empty set of possible worlds;

(v) Dt ={0,1};

(vi) For any (στ)∈Types,D(στ) is the set of all functions fromDσtoDτ; (vii) For any (σ×τ)∈Types,Dσ×τ is the set of all pairs inDσ×Dτ

An Inq_D model, then, can be defined as anInq_D frame Fpaired with an Interpreta- tion functionI over constants of each type, and a variable assignment functionθ over variables of each type. The semantic sentences then can be given in the same manner as first-order logic, which we will omit here for now.

Some remarks on Definition 3.2 (ii): note that an individual inInqDis now defined as

a subset of the set of entitiesE. This upgrade enables us to define atomic individuals as singleton sets in De, and plural individuals as non-singleton ones. We can also define

the sum (⊕) operation and parthood (≤) relation in set-theoretic terms. Namely, the sum of two individuals (atomic or plural)d,d′ ∈Deis their union, denoted asd⊕d′; the sum

of a set of individualsI ⊂ Deis defined similarly, ⊕I := ∪

I. The parthood relation≤is defined in terms of subset relation, i.e.d ≤ d′ if d ⊆ d′; and its proper counterpart< as well, i.e.d<d′ ifd⊂d′.

Note that discourse referents are stored in the domainDr, thereforedrefassignment

functions, which map discourse referents to individuals, are elements ofD(re). Since a

drefassignment function only hasactivedrefs in its domain, we allow it to be a partial function, and its domain will be denoted byδ⊆Dr. Further, we define adrefassignment

Definition3.3. (drefAssignment Functions and Matrices)

LetFbe anInqD frame andδ⊆Dra set of discourse referents inF.

(i) Adrefassignment function is apartialfunction g∈D(re) withdom(g) :=δ⊆Dr;

(ii) Adrefassignment matrix is a non-empty set ofdrefassignment functionsG∈D(re)t

with a same domainδ; we abbreviate the type (re)tasm.

Note that adrefassignment matrix is by definition non-empty, and if nodrefs have been introduced yet, we assume that the drefassignment matrix is {∅}, i.e. a singleton set containing an empty function. This stipulation will benefit subsequent definitions regarding context extension.

With everything at hand, we can now regenerate the step-by-step introduction of

InqDcontext in §3.1 with formal definitions.

Definition3.4. (Possibilities)

For any set of discourse referentsδ, apossibilitywith domainδis a pair⟨w,G⟩ ∈Ds×m

wherewis a possible world andGadrefassignment matrix with domainδ. Definition3.5. (Information States)

Aninformation state is a set of possibilities, thus of type ((s×m)t), abbreviated as i. Definition3.6. (Contexts)

(i) Downward closure: a setSof information states is downward closedifffor every

s∈ S, every subset ofsis also inS.

(ii) Acontextis a non-empty, downward closed set of information states, thus of type (it), abbr.k.

Let’s end this series of definitions of thedomainof information states and context, as it corresponds directly to the set of activedrefs, thus is important for discussions below regarding context update.

Definition3.7. (The Domain of an Information State and a Context)

(i) Thedomainof an information statesis the union of the domains of the possibilities ins.

(ii) Thedomainof a contextcis the union of the domains of the information states in

c.

To summarize, we list the types and their abbreviations corresponding to the notions defined above in the following Table 3.1.