WiSe22/23: Constraint-based Semantics 2: Difference between revisions
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We assume: | We assume: | ||
* Our model contains a set of possible worlds ''W'', and a set of accessiblity relations ''ACC'' | * 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, ''w''<sub>0</sub> or ''@'' that refers to the current world: [[''w''<sub>0</sub>]]<sup>M,w</sup> = w | * There is a special variable, ''w''<sub>0</sub> (abbreviated as ''ω'' or ''@'') that refers to the current world: [[''w''<sub>0</sub>]]<sup>M,w</sup> = w | ||
* There are special predicates for the different accessibility relations (modal bases): | * There are special predicates for the different accessibility relations (modal bases): | ||
** Alethic: '''ALETH'''<sub>sst</sub> | ** Alethic: '''ALETH'''<sub>sst</sub> | ||
** Deontic: '''DEON1'''<sub>sst</sub> and '''DEON2'''<sub>esst</sub> | ** Deontic: '''DEON1'''<sub>sst</sub> and '''DEON2'''<sub>esst</sub> | ||
** Epistemic: '''EPIST'''<sub> | ** Epistemic: '''EPIST'''<sub>esst</sub> | ||
** Dynamic: ''' | ** Doxastic: '''DOX'''<sub>esst</sub> | ||
** Dynamic: '''DYNA'''<sub>esst</sub> | |||
** Bouletic: '''BOUL'''<sub>esst</sub> | ** Bouletic: '''BOUL'''<sub>esst</sub> | ||
== Example formulae == | == Example formulae == | ||
Sentences are translated as formulae with a free occurrence of the world variable '' | Sentences are translated as formulae with a free occurrence of the world variable ''ω''. | ||
''Alex called.'' '''call'''( | ''Alex called.'' '''call'''(''ω'','''alex''') - also written as '''call'''<sub>''ω''</sub>('''alex''') | ||
Modal expressions introduce an explicit quantification over worlds (modal force) | Modal expressions introduce an explicit quantification over worlds (modal force), where '''ACC''' is any of the accessibility relations in ''ACC''. | ||
Necessity modality: □φ ≡ ∃''w'' (''w''='' | Necessity modality: | ||
* Without cognitive agent: □φ ≡ ∃''w'' (''w''=''ω'' : ∀''ω'' ('''ACC'''(''w'',''ω'') : φ)) | |||
* With cognitive agent ''a'': □<sub>''a''</sub>φ ≡ ∃''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'': ⋄<sub>''a''</sub>φ ≡ ∃''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 == | == Lexical constraints == | ||
Alethic ''must'': ∃w (w= | '''Note:''' Non-contributed parts of formulae are marked in <span style="color:brown">brown</span>. | ||
* 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'',<span style="color:brown">''x''</span>) | |||
* 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'': <u><span style="color:brown">Q</span>''x''(α[{'''linguist'''(<span style="color:brown">''w''</span>,x)}]:β[x])</u> |
Latest revision as of 18:57, 11 November 2022
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,
- all constraints on the combining signs' semantic representation also apply to the resulting combination, and
- 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
- can only contain the constants, variables, and operators that occur in the constraints contributed by lexical items and
- 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: ALETHsst
- Deontic: DEON1sst and DEON2esst
- Epistemic: EPISTesst
- Doxastic: DOXesst
- Dynamic: DYNAesst
- Bouletic: BOULesst
Example formulae
Sentences are translated as formulae with a free occurrence of the world variable ω.
Alex called. call(ω,alex) - also written as callω(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])