Transcript Memory-Based Learning Instance-Based Learning K

(SYNTAX-DRIVEN)
SEMANTICS
Yılmaz Kılıçaslan
Outline
 Semiotic Triangle
– Syntax
– Semantics
– Pragmatics
 (Syntax-driven) Semantics
– Semantics via languages of logic
– Model-based semantics
– Lambda-calculus
 Computational Issues
– Coding the semantic value of basic expressions
– Handling the lambda operator
– Implementing compositionality with lambda calculus
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Semiotic Triangle
The three branches of semiotics:
Ogden and Richard’s Semiotic Triangle
syntactics (syntax)—how signs relate to
other signs (example: how the word “dog”
relates to the other words in the sentence
“The dog ate my homework.”)
semantics—study of how signs relate to
things (example: how the word “dog”
relates to an actual dog)
pragmatics—actual use of codes in
everyday life; effects of signs on human
behavior and how people mold signs and
meanings in their actual interaction
(example: How would the sentence “The
dog ate my homework” be used in
everyday life? How would my teacher
react to it?)
A Formal Language for First-Order Logic
SYNTAX:
SEMANTICS: (Ignoring Quantification)
Con  
Quant   | 
Term  Name
Term  Var
Name  j | k | v | m
Var  x | y | z | x1 | ...
Pred1 S
0
1
[]M =
A = {a, b, c}
0
0
1
1
0
1
0
1
[] =
Pred2  L
M = <A, F>
1
0
0
1
M
For  Pred1(Term)
[Pred1(α)]M = F(Pred1)(F(Term))
For  Pred2(Term, Term)
[Pred2(α,β)]M = [F(Pred2)(F(β))](F(α))
For  [For Con For]
[F1  F2]M = [[]M([F1]M)]([F2]M)
For   For
[ F]M = []M([F]M)
For  Quant Var For
MODEL:
Example:
[[L(j, k)]]M = [] M([L(j, k)]]M)
= [] M([F(L)(F(k))](F(j)))
= [] M([F(L)(b)](a)) = [] M(0) = 1
F(j) = a
F(k) = b
F(v) = c
F(m) = c
a
b
c
F(S) =
a
F(L) =
b
c
0
1
a
b
c
0
a
b
c
0
a
b
c
0
1
1
1
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Natural Language Semantics in FOL
Compositional Translation:
Example: ̏John loves Katy̋
S
N
John
L(j,k)
VP
j
V
loves
xL(x,k)
N
yxL(x,y)
Katy
k
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A Fatal Problem with Semantics in FOL
Referentially Opaque Constructions:
i.e. constructions creating contexts where Leibnitz’ Law fails.
The result of substituting in any formula
one name for another name denoting the
same individual results in a formula that
is true iff the original formula was true
Example:
(a) ̏John believes that the Morning Star is the Morning Star.̋
(b) ̏John believes that the Morning Star is Venus.̋
[m]M = [v]M = c
⇒
B(j, m = m)
B(j, m = v)
[B(j, m = m)]M = [B(j, m = v)]M
But, (b) might be false while (a) is true.
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Intensional Logic (IL)
A Platonic View of Reality
Plato’s
Ideas
Frege’s
Senses
SYNTAX:
Carnap’s
Intensions
Kripke’s
Possible Worlds
SEMANTICS:
MODEL:
...
For  (Term = Term)
M = <A, W, F>
[α = β]M,w = 1 ⇔ [α]M,w = [β]M,w
...
Expr  ^Expr
Montague’s
Intensional Logic
w1
...
wN
[^α]M =
...
[α]M,w
1
A = {a,b,c,d} W = {w1,w2}
w
F(j) = w1
2
[α]M,w
N
F(v) =
w1
w2
c
c
F(m) =
w1
w2
c
d
Example:
[^(m =
m)]M
w1
= w
2
1
1
w1
w2
1
0
[^(m = v)]M =
a
a
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Natural Language Semantics in IL
Translations into IL:
De Dicto Readings:
A De Re Reading:
(a) J̏ ohn believes that the Morning Star is
the Morning Star.̋
B(j, ^(m = m))
J̏ ohn believes that the Morning Star is
Venus.̋
x[B(j, ^(x = v))](m)
(b) J̏ ohn believes that the Morning Star is
Venus.̋
B(j, ^(m = v))
As
[^(m = m)]M
≠
[^(m = v)]M
(b) can be false while (a) is true.
Problems with Possible Worlds Semantics
Logically Equivalent Sentences :
If two sentences  and  are logically equivalent, then it will always be a
logically valid inference from B(α, ^) to B(α, ^), for any believer
denoted by α.
However, a person might believe a sentence without believing its
equivalent.
Example: Necessarily True Sentences
For every rational believer denoted by α, the following would hold:
B(α, ^(m = m))
But,
𝑛
B(α,
^(ex =
∞ 𝑥
𝑛=0 𝑛!))
could still be false.
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Conclusion
Using formal languages of logic as a bridge, we can
develop a truth-conditional, model-theoretic
natural language semantics adhering to the
principle of compositionality.
However, this approach ultimately fails (e.g. in the
case of logically equivalent sentences in opaque
contexts).
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