Formalism

Types

All types

s situations
e individuals
t truth values
χ propositional completions
ρ discourse change records
σ conversational scoreboards
(αβ): α,β ∈ T complex

Discourse referent types

s situations
π = (st)t common ground partitions
χ propositional completions
ξ = χ(st) propositional radicals
ρ discourse change records

Pragmatics

Drs

φ

=

φ^d

φ^c

is

a discourse representation structure (DRS)

Drefs

φ^d

is

a sequence of drefs introduced in the discourse

Conditions

φ^c

is

a sequence of conditions, imposing constraints on the
model against which sentences are evaluated

Contributions

For all a sentence S, uttered in discourse D, represented by DRS φ,

Drefs

S^d

is

the sequence of drefs contributed by S,

Conditions

S^c

is

the sequence of conditions contributed by S.

Sentences

D ⇒ S

=

φ^d ⊕ S^d

φ^c ⊕ S^c

Information state

⟨

iR

⟨

dref

⟩

⟨

sand

⟩

⟨

rec

⟩

is

an information state for the discourse

Common ground

i : s

is

the common ground

Relation set

R : (πt)

is

a set of partitions of the common ground

Drefs

⟨

dref

⟩

is

a sequence of discourse referents

Sandbox

⟨

sand

⟩

is

a sequence of propositional radical/completion pairs
representing the collaborative space of the discourse

Record

⟨

rec

⟩

is

a sequence of discourse referents of type σ that
provides a record of changes made to the information
state throughout the discourse

Cooperative update

Informative update

C_st

⇒

i

⋂

⟦

C_st

⟧

ⁱ

λσ( i_σ ⋂ C_s = i_σ )

Structuring update

C_(st)t

⇒

R

⋊

⟦

C_(st)t

⟧

^R

λσ( R^σ ⋂ C_st = R_σ )

Dref introduction

u₁..u_n

⇒

u₁..u_n

⧀

⟨

dref

⟩

λσ( ∀(k ∈ [0..n])( δ_σ[k] = g(u_k ) )

Collaborative update

Initialization

U^init

↥_σ

⟨

i_σ

⟨

dref

⟩

_σ

⟩

_sand

Propagation

U^prop

↦_σ

dref^prop

U^prop_(st)t

Present

dref^prop

⧀

dref^sand

Partition

R^prop

⋊

R^sand

Merge

U^merge

↧_σ

⟨

i_s += C_ξ(k_χ)i += C_ξ(k_χ)dref_σ += dref_sand

⟩

_σ

Speech acts

For all I : σ, U, an utterance in discourse,

Assert

I

⇒

!U

=

I

Question

I

⇒

?U

=

I

Conjecture

I

⇒

▵̣U

=

I

Correct

I

⇒

▿̣U

=

I

Bare plurals

For all Ps : et, Q : et,

Semantic structure

Primitives

Constants

Cons

is

a set of sets of constants of each type in T

Variables

Vars

is

a set of sets of variables of each type in T

Semantic value

⟦

·

⟧

igM

is

the semantic value function

Index of evaluation

i

:

s

is

an index of evaluation

Assignment function

g

:

Vars

→

D

is

an assignment function

Model

M

=

⟨

DV

⟩

is

a model for the language

Universe

D

=

{

e

∈

D_α

}

:

α

∈

T

}

is

a type structured universe of the discourse

Valuation

V

:

Cons

→

D

is

a valuation function that assigns type-relevant objects
to each constant

Semantic rules

Primitives

For all α∈T, cα∈Cons, uα∈Vars,

Constants

⟦

c_α

⟧

M

=

V(c_α)

Variables

⟦

u_α

⟧

g

=

g(u_α)

Top

⟦

⊤_α

⟧

=

D_α

Application

For all α∈T, u_α∈Vars, d_α∈D, termsA,

⟦

λu_α(A)

⟧

g

=

⟦

A

⟧

g[u_α→d]

Identity

For all α∈T, termsAandB,

⟦

A_α

=

B_α

⟧

=

1

if

⟦

A_α

⟧

=

⟦

B_α

⟧

0

otherwise

Booleans

For all A : (et), B : (et)t,

⟦

¬φ

⟧

=

⟦

⊤

⟧

/

φ

⟦

φ∧ψ

⟧

=

⟦

φ

⟧

⋂

⟦

ψ

⟧

⟦

φ∨ψ

⟧

=

⟦

φ

⟧

⋃

⟦

ψ

⟧

Quantifiers

For all u_e∈Vars, φ : (st),

⟦

∀u_eφ

⟧

g

=

1

if

⟦

φ

⟧

g[x→u_e]

=

1 for all u_e

0

otherwise

⟦

∃u_eφ

⟧

g

=

1

if

⟦

φ

⟧

g[x→u_e]

=

1 for some u_e

0

otherwise

Bare plurals

For all A : (et), B : (et)t, R : π,

⟦

∀_R(A)(B)

⟧

=

1

if

⟦

A

⟧

∈

R

⟦

A

⟧

⊆

⟦

B

⟧

0

otherwise

A model

Domains

Situations

D_s

=

{

s₁s₂s₃…

}

Individuals

D_e

=

{

abcde…

}

Predicates

Bears

V(bears)

=

s₁

⇒

{

abe

}

s₂

⇒

{

abcd

}

s₃

⇒

{

a

}

Dangerous

V(dangerous)

=

s₁

⇒

{

abd

}

s₂

⇒

{

a

}

s₃

⇒

{

bd

}

Grizzly

V(grizzly)

=

s₁

⇒

{

a

}

s₂

⇒

{

ac

}

s₃

⇒

{

a

}

Koala

V(koala)

=

s₁

⇒

{

b

}

s₂

⇒

{

b

}

s₃

⇒

{

b

}

Polar

V(polar)

=

s₁

⇒

{

}

s₂

⇒

{

d

}

s₃

⇒

{

e

}

Formalism

Site navigation

Discourse as collaborative inquiry by Erik Hoversten Table of contents Search sources Chapters Project overview Knowledge in the image of inquiry Collaborative discourse dynamics Dispute in discourse Appendices Bibliography Formalism Glossary Predication in CU Typologies

Types

Pragmatics

Semantic structure

Semantic rules

A model

Discourse as collaborative inquiry
by Erik Hoversten

Table of contents

Search sources

Chapters
Project overview

Knowledge in the image of inquiry

Collaborative discourse dynamics

Dispute in discourse

Appendices
Bibliography

Formalism

Glossary

Predication in CU

Typologies