WiSe22/23: Constraint-based Semantics 2

HPSG-neutral notation for LRS

Constraints

Metavariables: α, β, ɣ, ..., φ, ψ, ...

Contribution constraints:

• call(x)
  The semantic representation of a sign with a contribution constraint of the form call(x) must be an expression containing call(x) as a subexpression
• ¬α
  The semantic representation of a sign with a contribution constraint of the form ¬α must be an expression containing ¬α as a subexpression where α can be any expression.

Embedding constraints:

• α[call(x)]
  The metavariable α is any expression containing call(x) as a subexpression.

Combinatorial semantics

When two signs combine,

1. all constraints on the combining signs' semantic representation also apply to the resulting combination, and
2. additional constraints may be added through principles of grammar.

Internal content

internal content: The scopally lowest contributed element, marked in curly brackets: {φ}

• call: {call(x)}
• everyone: ∀x({person(x)} : β[x])

External content

external content: The semantic representation of a complete sign, marked by underlining: φ

• everyone: ∀x({person(x)} : β[x])

Combinatorially added constraints

Internal content raisers

When an auxiliary combines with its VP complement, the internal content of the complement must be the internal content of the auxiliary.

Example:

• call: {call(x)}
• didn't: ¬α[{α'}]
• didn't call: β[call(x), ¬α[{α'}, α'≡call(x) ]
  This constraint can be expressed more compactly as: β[¬α[{call(x)}]]

Determiner-noun combinations

When a determiner combines with a head, the head's internal content is a subexpression of the determiner's restrictor.

Example:

• book: {book(x)}
• every: {∀}x(φ[x] : ψ[x])
• every book: α[{book(x)}, {∀}x(φ[x] : ψ[x]), φ[book(x)]
  This constraint can be expressed more compactly as: α[{∀}x(φ[x, {book(x)}] : ψ[x])]

Quantified NPs as non-heads

When a quantified NP combines with a head, the head's internal content is a subexpression of the quantifier's restrictor.

Example:

• called: {call(x)}
• someone: ∃x({person(x)} : ψ[x])
• Someone called.: α[{call(x)}, ∃x({person(x)} : ψ[x]), ψ[call(x)]]
  This constraint can be expressed more compactly as: α[∃x({person(x)} : ψ[x,{call(x)}])]

External content principle

The semantic representation of an utterance

1. can only contain the constants, variables, and operators that occur in the constraints contributed by lexical items and
2. it must respect all constraints contributed by the lexical and non-lexical items contained in the utterance

Modality

Basic assumptions

We assume:

• Our model contains a set of possible worlds W, and a set of accessiblity relations ACC, i.e. M = < U, I, W, ACC >
• There is a special variable, w₀ (abbreviated as ω or @) that refers to the current world: [[w₀]]^M,w = w
• There are special predicates for the different accessibility relations (modal bases):
  • Alethic: ALETH_sst
  • Deontic: DEON1_sst and DEON2_esst
  • Epistemic: EPIST_esst
  • Doxastic: DOX_esst
  • Dynamic: DYNA_esst
  • Bouletic: BOUL_esst

Example formulae

Sentences are translated as formulae with a free occurrence of the world variable ω.

Alex called. call(ω,alex) - also written as call_ω(alex)

Modal expressions introduce an explicit quantification over worlds (modal force), where ACC is any of the accessibility relations in ACC.

Necessity modality:

• Without cognitive agent: □φ ≡ ∃w (w=ω : ∀ω (ACC(w,ω) : φ))
• With cognitive agent a: □_aφ ≡ ∃w (w=ω : ∀@ (ACC(a,w,ω) : φ))

Possibility modality:

• Without cognitive agent: ⋄φ ≡ ∃w (w=@ : ∃ω (ACC(w,ω) : φ))
• With cognitive agent a: ⋄_aφ ≡ ∃w (w=ω : ∃ω (ACC(a,w,ω) : φ))

Example sentences:

A linguist necessarily likes logic. (deontic necessity, without cognitive agent)

• □ ∃x(linguist(x) : like-logic(x))

≡ ∃w (w=ω: ∀ω(DEON1(w,ω): ∃x(linguist(ω,x) : like-logic(ω,x))))

• ∃x(linguist(x) : □(like-logic(x))

≡ ∃x(linguist(ω,x) : ∃w (w=ω: ∀ω(DEON1(w,ω): like-logic(ω,x))))

Lexical constraints

Note: Non-contributed parts of formulae are marked in brown.

• Modal auxiliaries:
  • Alethic must: ∃w (w=ω : ∀ω (ALETH(w,ω) : α[{α'[ω]}]))

• • Deontic must with cognitive agent: ∃w (w=ω : ∀ω (DEON2(a,w,ω) : α[{α'[ω,a]}]))

• Verbs: All verbs contribute the world variable ω in their lexical entry.
  • call: call(ω,x)

• Nouns: Nouns don't contribute their world variable. There is a global constraint that their world variable must be identical with @ or with an other world variable in whose scope it is in the utterance.
  • linguist: Qx(α[{linguist(w,x)}]:β[x])