The logical representation of semantic structure in MeaningText Theory

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The logical representation of semantic structure
in Meaning-Text Theory

• P. Boldini
• CAMS-EHESS, Université Paris IV

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Graph representation of Ssem (Melcuk )

• I think that Leo knows that Alan is in love with
  Helen

think
know
I
be in love
Leo
Alan
Helen
THINK(I, KNOW(LEO, BE-IN-LOVE(ALAN, HELEN)))
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Meaning and reference in classical logic
meaning
The snow is white
truth conditions
reference
truth value True/False
Universe
snow
white
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Propositional attitudes
Leo knows that Alan is in love with Helen
True/False
KLeo logical constant
KLeo(KLeo p) ? KLeo p , etc
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No need to say more
 (...) le terme sens doit être interprété ici de
la façon la plus étroite possible. Il ne sagit
aucunement du sens que nous obtenons comme
résultat dune bonne compréhension dun énoncé
quelconque, que nous en dégageons grâce à la
logique, à nos connaissances extralinguistiques
etc., (...) nous ne visons que le sens purement
langagier  le plus superficiel, le plus
littéral, celui qui est accessible uniquement
grâce à la maîtrise de la langue en cause. 
(Melcuk, Cours de morphologie générale, Vol. 1
1993), p. 42.
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DRT-like representation (Kahane)

•  Zoé essaye de manger la soupe 

essayer
x Zoé y soupe e manger(x,y) e
essayer(x,e)
Zoé
manger
soupe
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Propositions as sets (Curry-Howard)

• Formation

Martin-Löf Intuitionistic Type Theory
N Set
x N
Introduction
0 N
Sx N
Elimination/computation

• SS0 SSS0 N

• S(SS0 SS0) N
• SS(SS0 S0) N
• SSS(SS0 0) N
• SSS(SS0) N
• SSSSS0 N

Equality (referential)
23 5 N
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Propositions as sets
S - types

• Introduction
• a A b B(a)
• (a,b) (?xA)B(x)
• Elimination
• c (? xA)B(x)
• p(c) A q(c) B(p(c))
• Equality
• c (p(c),q(c)) (? xA)B(x)

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Meaning Sense - Reference
Formal language
Language
Set A
Sentence A
Meaning
What?
Statement a A
Expression a A
Sense
How?
Sense
Reference a A
Truth-makers
Reference a A
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Communication
Speaker  Somebody called up for John 
a Person b called(a,John)
Hearer  Somebody called up for John
(?xPerson)called(x,John)
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Compositionality of meaning
Common Noun
Set Prop
Adjective
(A)Prop
Verb1
(A)Prop
Verb2
(A)(B)Prop
Adverbs
x walks(John) quickly(x) Prop quickly
(xPerson)(walk(x))Prop
John walks quickly
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Proof-objects
Definite descriptions
Johns wife
(?x Person)(female(x)?married(x,John))

a Person b female(a)?married(a,John)
(a,b)
Propositional attitudes
Leo knows that Alan loves Helen
a loves(Alan, Helen) b knows(Leo,a)
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Algorithm
I think that Leo knows that Alan is in love with
Helen
z think(I,y)
y know(Leo,x)
I
x be_in_love(Alan,Helen)
Leo
Alan
Helen

z(?y(? xbe_in_love(Alan,Helen))know(Leo,x))thin
k(I,q(y))