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Logic, Representation and
Inference
Introduction to Semantics
• What is semantics for?
• Role of FOL
• Montague Approach
February 2009
Introduction to Semantics
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Semantics
• Semantics is the study of the meaning
of NL expressions
• Expressions include sentences,
phrases, and sentences.
• What is the goal of such study?
– Provide a workable definition of meaning.
– Explain semantic relations between
expressions.
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Examples of Semantic Relations
• Synonymy
– John killed Mary
– John caused Mary to die
• Entailment
– John fed his cat
– John has a cat
• Consistency
– John is very sick
– John is not feeling well
– John is very healthy
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Different Kinds of Meaning
X means Y
• Meaning as definition:
– a bachelor means an unmarried man
• Meaning as intention:
– What did John mean by waving?
• Meaning as reference:
"Eiffel Tower " means
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Workable Definition of Meaning
• Restrict the scope of semantics.
• Ignore irony, metaphor etc.
• Stick to the literal interpretations of
expressions rather than metaphorical
ones. (My car drinks petrol).
• Assume that meaning is understood in
terms of something concrete.
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Concrete Semantics
• Procedural semantics: the meaning of a
phrase or sentence is a procedure:
“Pick up a big red block”
(Winograd 1972)
• Object–Oriented Semantics: meaning is
an instance of a class.
• Truth-Conditional Semantics
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Truth Conditional Semantics
• Key Claim: the meaning of a sentence is
identical to the conditions under which it is
true.
• Know the meaning of "Ġianni ate fish for tea"
= know exactly how to apply it to the real
world and decide whether it is true or false.
• On this view, one task of semantic theory is to
provide a system for identifying the truth
conditions of sentences.
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TCS and Semantic Relations
• TCS provides a precise account of semantic
relations between sentences.
• Examples:
–
–
–
–
S1 is synonymous with S2.
S1 entails S2
S1 is consistent with S2.
S1 is inconsistent with S2.
• Just like logic!
• Which logic?
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NL Semantics: Two Basic Issues
• How can we automate the process of
associating semantic representations
with expressions of natural language?
• How can we use semantic
representations of NL expressions to
automate the process of drawing
inferences?
• We will focus mainly on first issue.
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Associating Semantic
Representations Automatically
•
•
•
Design a semantic representation
language.
Figure out how to compute the
semantic representation of sentences
Link this computation to the grammar
and lexicon.
February 2009
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Semantic Representation
Language
• Logical form (LF) is the name used by
logicians (Russell, Carnap etc) to talk
about the representation of contextindependent meaning.
• Semantic representation language has
to encode the LF.
• One concrete representation for logical
form is first order logic (FOL)
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Why is FOL a good thing?
• Has a precise, model-theoretic semantics.
• If we can translate a NL sentence S into a
sentence of FOL, then we have a precise
grasp on at least part of the meaning of S.
• Important inference problems have been
studied for FOL. Computational solutions
exist for some of them.
• Hence the strategy of translating into FOL
also gives us a handle on inference.
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Anatomy of FOL
• Symbols of different types
–
–
–
–
–
–
–
constant symbols:
variable symbols:
function symbols:
predicate symbols:
connectives: &, v, 
quantifiers: , 
punctuation: ), (, “,”
February 2009
a,b,c
x, y, z
f,g,h
p,q,r
Introduction to Semantics
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Anatomy of FOL
• Symbols of different types
–
–
–
–
–
–
–
constant symbols: csa3180, nlp, mike, alan, rachel, csai
variable symbols: x, y, z
function symbols:
lecturerOf, subjectOf
predicate symbols:
studies, likes
connectives: &, v, 
quantifiers: , 
punctuation: ), (, “,”
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Anatomy of FOL
With these symbols we can make expressions of
different types
– Expressions for referring to things
• constant:
• variable:
• term:
alan, nlp
x
subject(csa3180)
– Expressions for stating facts
• atomic formula: study(alan,csa3180)
• complex formula:
study(alan,csa3180) & teach(mike, csa3180)
• quantified expression:
xy teaches(lecturer(x),x) & studies(y,subject(x))
xy likes(x,subjectOf(y))  studies(x,y)
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Logical Form of Phrases
word
POS
Logic
Representation
csai
proper noun
individual
constant
csai
student
common noun
1 place
predicate
student(x)
easy
adjective
1 place
predicate
easy(x)
easy interesting
course
adj/noun
1 place
predicate
easy(x) &
interesting(x) &
course(x)
snores
intrans verb
1 place
predicate
snore (x)
studies
trans. verb
2 place
predicate
study(x,,y)
gives
February 2009
ditrans verb
3 place pred
Introduction to Semantics
give(x,y,z)
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Logical Forms of Sentences
• John kicks Fido:
kick(john, fido)
• Every student wrote a program
xy( stud(x)  prog(y) & write(x,y))
yx(stud(x)  prog(y) & write(x,y))
• Semantic ambiguity related to quantifier
scope
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Building Logical Form
Frege’s Principle of Compositionality
• The POC states that the LF of a complex
phrase can be built out of the LFs of the
constituent parts.
• An everyday example of compositionality is
the way in which the “meaning” of arithmetic
expressions is computed
(2+3) * (4/2) =
(5 * 2) =
10
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Compositionality for NL
• The LF of the whole sentence can be
computed from the LF of the subphrases, i.e.
• Given the syntactic rule X  Y Z.
• Suppose [Y], [Z] are the LFs of Y, and Z
respectively.
• Then [X] = ([Y],[Z]) where  is some function
for semantic combination
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Claims of Richard Montague:
• Each syntax rule is associated with a
semantic rule that describes how the LF of
the LHS category is composed from the LF of
its subconstituents
• 1:1 correspondence between syntax and
semantics (rule-to-rule hypothesis)
• Functional composition proposed for
combining semantic forms.
• Lambda calculus proposed as the
mechanism for describing functions for
semantic combination.
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Sentence Rule
• Syntactic Rule:
S  NP VP
• Semantic Rule:
[S] = [VP]([NP])
i.e. the LF of S is obtained by "applying" the
LF of VP to the LF of NP.
• For this to be possible [VP] must be a
function, and [NP] the argument to the
function.
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Parse Tree with Logical Forms
S
write(bertrand,principia)
VP
y.write(y,principia)
NP
bertrand
bertrand
V
x.y.write(y,x)
writes
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Introduction to Semantics
NP
principia
principia
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