Semantics
and some syntax, math, and computational linguistics too
LING 001 - October 16, 2006
Joshua Tauberer
Semantics
• Why does a sentence mean what it
means?
• What are the meanings of words and how
do they come together to make larger
meanings (i.e. phrases, sentences)?
• Perhaps the only level of linguistic description actually
needed for there to be language…?
Overview
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Machine Translation
Quantifier Scope Ambiguity
Negative Polarity Items
Object Language vs Meta Language
Compositionality
Idioms
Presupposition
Formal Semantics (Propositional Logic, etc.)
…….
Machine Translation
• Can we make a computer program to
translate text between languages
automatically?
MT: Morphological Analysis
• Direct word-to-word mapping
Billy eats the cake quickly.
Billy come la torta rápidamente.
(Spanish)
MT: Morphological Analysis
• Word-to-word mapping doesn’t work well.
Billy ate the cake quickly.
Billy keki çabukça yedi.
(Turkish (I hope))
MT: Morphological Analysis
• Word-to-word mapping doesn’t work well.
What did Billy eat quickly?
Billy neyi çabukça yedi?
(Turkish (I hope))
MT: Morphological Analysis
• Word-to-word mapping doesn’t work well.
Wawirri kapi-rna panti-rni
“Kangaroo
will-I
spear
yalumpu.
that.” .
I will spear that kangaroo.
(Warlpiri, from Hale (1983) via Legate (2002)).
MT: Syntactic Analysis
• Tree-to-tree mapping:
MT: The Pyramid
Interlingua
Syntactic
Structure
tree-to-tree
translation
Syntactic
Structure
actual MT systems today
Morphological word-to-word translation
Structure
Input Language
Morphological
Structure
Output Language
MT: Syntactic Analysis
• Even syntactic MT runs into trouble.
• Let’s take a brief trip into quantifier scope
ambiguity…
Quantifier Scope Ambiguity
Two students met with every teacher.
•
•
(Syntactically unambiguous.)
Semantically ambiguous.
1. Two particular students each met all of the
teachers.
2. Each teacher was visited by two students,
but possibly different students meeting with
each.
Quantifier Scope Ambiguity
1
2
Quantifier Scope & MT
• Unfortunately, not all languages have the
same quantifier scope ambiguities.
• Proper translation requires recognition (&
maybe resolution) of ambiguity, and then
selection of appropriate form in the target
language.
Quantifier Scope & MT
• English:
Everyone loves someone.
– Ambiguous.
• Japanese: Daremo-ga dareka-o aisite-iru.
everyone-NOM someone-ACC love
– Unambiguous. “Everyone loves someone or other.”
– Using this translation would be wrong unless the computer has
resolved the ambiguity, i.e. if it knows what the speaker intended.
• Japanese: Dareka-o daremo-ga aisite-iru.
– Ambiguous.
– Close to English “Someone, everyone loves.”
– A (potentially) awkward translation if the other one would work.
(source: Kuno, Takami, and Wu 1999)
MT: Semantic Analysis
• The holy grail of MT.
• Obviously a computer cannot truly
understand anything, but it has to have a
symbolic representation of the meaning.
– Translate the input sentence into the
‘interlingua’ which represents the full original
meaning.
– Translate ‘interlingua’ into the target
language.
Other Practical Applications
• Question-Answering
• Automated Summarization
• Existing solutions don’t use any
sophisticated syntax or semantics.
– Because when they try…
Negative Polarity Items
• NPIs are words that seem to only be allowed in
negative contexts.
I did not see anything/any books at the store.
I didn’t get paid a red cent for my trouble.
I have not ever been to Mexico.
I don’t give a damn about the homework.
* I saw any book at the store.
* I got paid a red cent for my trouble.
* I have ever been to Mexico.
* I give a damn about the homework.
Negative Polarity Items
• What constituents a negative context?
I didn’t see anyone at the store.
I never see anyone at the store.
I rarely see anyone at the store.
* I saw anyone at the store.
* I always see anyone at the store.
* I sometimes see anyone at the store.
Negative Polarity Items
• But there are other licensing contexts too:
If I see anyone at the store after hours . . .
Students who bought anything from the
bookstore . . .
• What do these have in common?
– Negation
– The antecedent of a conditional
– Relative clauses
Negative Polarity Items
• This is an upward-entailing context:
I saw something in the fishbowl.
more
general
entails
I saw a fish in the fishbowl.
entails
I saw a goldfish in the fishbowl.
more
specific
Negative Polarity Items
• This is a downward-entailing context:
I didn’t see a thing in the fishbowl.
more
general
entails
I didn’t see a fish in the fishbowl.
entails
I didn’t see a goldfish in the fishbowl.
more
specific
Negative Polarity Items
If I find a fish in the fishbowl, I will feed it.
• Is fish in an upward-entailing or
downward-entailing context?
Negative Polarity Items
If I find a fish in the fishbowl, I will feed it.
Situation
I found a worm (an animal).
I found a goldfish.
Feed it?
NO
YES
• So the conditional above entails:
If I find a goldfish in the fishbowl, I will feed it
• Goldfish is more specific.
• It is downward entailing.
Negative Polarity Items
Students who bought a book will get a
rebate.
Situation
I bought merchandise.
I bought a textbook.
Rebate?
NO
YES
• This is also downward-entailing.
Negative Polarity Items
If Clinton wins in ’08, some politicians will be happy.
• Clinton wins. Let’s see who is happy.
Group
some people
Republicans
Happy?
YES
NO
• This is upward entailing.
• The antecedent of a conditional is downward-entailing,
but the consequent is upward-entailing.
Negative Polarity Items
• Licit only in downward-entailing contexts.
– Where replacement with a more specific term
yields a sentence entailed by the original.
• NPIs also have a syntactic requirement.
– “c-command” under the standard generative
model of sentence structure
• There are also positive-polarity items.
Object vs. Meta Language
• When describing meaning, it doesn’t help
to use the words we’re trying to define.
• The quick brown fox jumped.
– What does this mean?
– It doesn’t help to just repeat the sentence.
– We need a controlled vocabulary that we can
agree on to describe language.
Object vs. Meta Language
• I will use italics for utterances of English,
our object language.
– The quick brown fox jumped.
• I will use CAPITALS for the metalanguage, the language to talk about
language.
Object vs. Meta Language
deep blue oceans
• What does this mean? I think it means things
that are…
– OCEANS
– AND DEEP
– AND BLUE
• Reduction of meaning into smaller pieces:
– AND , OCEANS , DEEP , BLUE
Object vs. Meta Language
• We can’t possibly list the meaning of every
phrase. (Is there a longest phrase?)
• But we can list the meaning of every word.
– “oceans” “deep” “blue”
• And we can add a little bit of glue and
some rules for putting the meanings
together.
Object vs. Meta Language
deep blue oceans
ADJ ADJ …. N
• The meaning 〚…〛 of a noun phrase of the
form above is the conjunction of the meaning of
its parts.
〚ADJ1 ADJ2 ADJ3 . . . N〛
= things that are〚ADJ1〛 AND〚ADJ2〛AND
〚ADJ3〛AND〚N〛
Compositionality
• The meaning of a constituent is
determined by
– The meaning of its parts
– The way the parts are put together
– (And nothing else.)
• It seems obvious, but there are some
complications.
Compositionality Complications:
Idioms
• Idioms
– Phrases that defy compositionality
– Meaning of the whole must be listed lexically
a red cent (‘nothing’)
give a damn (‘care’)
kick the bucket (‘die’)
sleeping with the fishes (‘killed’)
the cat has got your tongue (‘speechless’)
Compositionality Complications:
Idioms
• Are they just multi-word words?
• Idioms differ in their rigidity...
Compositionality Complications:
Idioms
• In most idioms, one cannot replace any
words and retain the idiomatic meaning:
– a red cent / *penny / *coin
– *punch/*tap the bucket
• But some have replaceable parts:
– the cat got my/your/the teacher’s tongue
Compositionality Complications:
Idioms
• Some but not all idioms can be syntactically
shuffled around (here, passivized):
Keep tabs on Henry. (‘track his whereabouts’)
Tabs were kept on Henry for three days.
Don’t spill the beans. (‘don’t give up the secret’)
The beans were spilled already.
* The bucket was kicked by the old man.
* His tongue has been gotten by the cat.
Compositionality Complications:
Idioms
• This suggests idioms have internal
syntactic structure, but perhaps no internal
semantic structure.
Compositionality Complications:
Idioms
• This suggests idioms have internal
syntactic structure, but perhaps no internal
semantic structure.
Compositionality Complications:
Non-Intersective Adjectives
• We previously saw ‘intersective’
adjectives:
– A hungry alligator is something that is both
hungry and an alligator.
– Something that is a hungry alligator comes
from the intersection of the set of hungry
things and the set of alligators.
– 〚ADJ N〛= 〚ADJ〛∩ 〚N〛
Compositionality Complications:
Non-Intersective Adjectives
• There are also non-intersective adjectives:
– a good plumber is not someone who is both good (in
general) and a plumber. He only has to be good at
plumbing.
– a proud father is not necessarily a proud person
– 〚ADJ N〛= 〚ADJ〛∩ 〚N〛
– At least a good plumber is a plumber and a proud
father is a father. These are called ‘subsective’
because it still finds a subset.
• 〚ADJ N〛⊆ 〚N〛
Compositionality Complications:
Non-Intersective Adjectives
• Then there are non-intersective, nonsubsective adjectives:
– a former student is not even a student (let
alone ‘former’, cf. ‘blue’)
• The whale is blue.
• *John is former.
– an alleged criminal is not (by necessity) a
criminal.
– counterfeit money is not money (arguably, but
certainly not the way we usually use money).
Compositionality Complications:
Non-Intersective Adjectives
• How to reconcile non-intersective
adjectives with compositionality?
• If 〚former student〛≠ 〚former〛∩ 〚student〛
then we have to give up either:
– Compositionality
– Intersection ∩
Brief Interlude:
Functions
A FUNCTION
FROM GREYBROWN COGS TO
RED/YELLOW COGS
Brief Interlude:
Functions
FORMER
(the notion of a student)
(the notion of a
former student)
Brief Interlude:
Functions
• Notation:
– SQRT(100) = 10
– FORMER(〚student〛) = 〚former student〛
=〚former〛(〚student〛)
Compositionality Complications:
Non-Intersective Adjectives
• By treating the meaning of former as a
function from one notion to another, we
can have a compositional account of
former X.
• For non-intersective adjectives:
– 〚ADJ N〛= 〚ADJ〛(〚N〛)
– Treat the meaning of ADJ as a function and apply it to
the meaning of N.
Compositionality
• Meanings can be compositional in two ways:
– By conjunction/intersection:
〚X Y〛= things that are both〚X〛and〚Y〛
〚X Y〛= 〚X〛∩〚Y〛
– By function-application:
〚X Y〛= 〚X〛(〚Y〛)
Presupposition
A man sat in the witness chair awaiting the next
question from the attorney….
When did you stop beating your wife?
The jury gasps, but the man is simply confused.
He responds:
But I never beat my wife!
Presupposition
The King of France is bald.
• Huh?
• It’s not false, per se. It’s just weird.
Presupposition
• Compare:
I don’t think that the Earth is flat.
(a true statement)
I don’t know that the Earth is flat.
(presupposition failure)
Presupposition
• If an utterance has a presupposition π,
then π must be true in order for the
utterance to be ‘OK’.
• Further, π must be established as common
ground in the discourse.
• (Unless the presupposition is
‘accommodated’.)
Presupposition
• The hallmark of presupposition is that it
remains despite negation.
• Thus we can separate an utterance into
two parts:
– the assertion, which is affected by negation
– the presupposition, which is not
Presuppositions Under Negation
• I think the Earth is flat.
– Assertion: I believe the Earth is flat.
– Presupposition: None
– Sentence is false (i.e. a lie), but otherwise OK.
• I know the Earth is flat.
– Assertion: I believe the Earth is flat.
– Presupposition: The Earth is flat.
– Presupposition is not true, therefore sentence is
weird.
Presuppositions Under Negation
• I didn’t think the Earth is flat.
– Assertion: I didn’t believe the Earth is flat.
– Presupposition: None
– Sentence is true.
• I didn’t know the Earth is flat.
– Assertion: I didn’t believe the Earth is flat.
– Presupposition: The Earth is flat.
– Presupposition is still not true, therefore sentence is
still weird.
Presupposition Triggers
• definite descriptions (‘the King of France’)
π = ‘there is a King of France’
• quantificational NPs (‘every cat I own’)
π = ‘I own at least one cat’
• factive verbs (‘regret’, ‘know’, ‘discover’)
π = the proposition regretted/known/discovered
• aspectual verbs/adverbs (‘stop’, ‘still’)
π = the action was happening previously
• questions (‘who stole the cookies?’)
π = ‘someone stole the cookies’
Presupposition Projection
• Presuppositions can ‘project’ or percolate
up recursively embedded sentences.
I think [John knows [the Earth is flat.]]
If [John knows the Earth is flat] then . . .
• Even though ‘think’/‘if’ are not a p-triggers,
‘know’ is, and its presupposition passes
through ‘think’/‘if’.
Presupposition Filters
• On the other hand, presuppositions can be
blocked.
If the Earth is flat, then a good scientist probably
would know the Earth is flat.
• There is no presupposition here.
• If π, a presupposition of the consequent, is
asserted in the antecedent, it is not a
presupposition of the whole sentence.
Presupposition Filters
If France had a King, the King of France
would be a very powerful man.
Presupposition Accommodation
• Usually presuppositions have to be established:
– A man off the street walks up to you and says:
I regret that I didn’t buy the tomato.
• You say: “Oh. You were going to buy a tomato?”
• The presupposition was not a part of the
common ground.
Presupposition Accommodation
• But sometimes we accept sentences with
presuppositions not already established:
If the North Korean ambassador turned up, then it
is amazing that both the North and South
Korean ambassadors are here.
(Beaver 2002)
• π = the S.K. ambassador is here
• π is ‘accommodated’
Formal Semantics
• Not just what things mean,
• but representing meaning & composition in
precise logical terms
• Hashing out the meta language.
Propositional Logic
• Mathematical representation of meaning.
• Symbols like p, q stand in for propositions about
what is true in the world. Propositions can be
either true or false.
• Let p = ‘It is raining.’
• p is true iif it is raining.
– If p is true, it must be raining.
– If it is raining, p must be true.
Propositional Logic: Connectives
• Propositions can be combined into
formulas using special connectives:
and: ∧
or: ∨
not: ¬
if: → (aka implies, conditional)
iif: ↔ (aka if and only if, biconditional)
Propositional Logic: Connectives
• Let p = ‘It is raining.’
• Let q = ‘It is snowing.’
• Let r = ‘I will play outside.’
• (p ∨ q) → ¬ r
• ‘If it is raining or snowing, then I will not
play outside.’
Predicate Logic
• Predicate logic adds names and
predicates on top of propositional logic.
KNOWS(JOHN, MARY)
the predicate
capitals for the
meta language
the arguments (also names)
• Let KNOWS be the predicate that is true
just when the first argument knows the
second argument.
Predicate Logic: Examples
If John meets Mary, then he will know her.
MEETS(JOHN, MARY)
→ KNOWS(JOHN, MARY)
Predicate Logic: Examples
On days without a cloud in the sky,
whenever my dog Sparky barks, and only
when he barks, I take him for a walk.
¬CLOUDY → [BARKS(SPARKY) ↔
WALK(ME, SPARKY)]
Predicate Logic & Natl. Language
• 〚John〛= JOHN
• 〚Mary〛= MARY
• 〚knows〛= KNOWS( … , … )
• 〚John knows Mary〛= some combination of
〚John〛〚Mary〛and 〚knows〛with either
conjunction/intersection or function application
Predicate Logic & Compositionality
• Formal semantics starts where generative
syntax ends.
= KNOWS(JOHN, MARY)
= JOHN
= KNOWS(…, …)
= MARY
Predicate Logic & Compositionality
Syntax
S → NP1 V NP2
Semantics
〚S〛=〚V〛(〚NP1〛,〚NP2〛)
S → John knows Mary 〚S〛=〚knows〛(〚John〛,〚Mary〛)
S → John knows Mary 〚S〛= KNOWS(JOHN, MARY)
= KNOWS(JOHN, MARY)
= JOHN
= MARY
= KNOWS(…, …)
Predicate Logic & Compositionality
Syntax
Semantics
CP → if S1 then S2
〚CP〛=〚S1〛 → 〚S2〛
(roughly)
= MEETS(JOHN, MARY) → KNOWS(JOHN, MARY)
MET(JOHN, MARY)
KNOWS(JOHN, MARY)
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