Agenda
Part Ⅰ: 10.1 Introduction 10.2 Model-Theoretical Semantics Part Ⅱ: 10.3 Translating English into a Logical Metalanguage
10.4 The Semantics of the Logical Metalanguage
Individual Constant
(2)Mulan is NOT asleep.
m
A
predicate letter
Am
Am
10.3 Translating English into a Logical Metalanguage
10.3.2 Simple Statements in Predicate Logic
Nobody visited Liverpool. Not everybody visited Liverpool. ∀x¬ (V(x,l)) ¬ ∀x(V(x,l))
10.4 The Semantics of the Logical Metalanguage
10.4.1ter
Pfbh
10.3 Translating English into a Logical Metalanguage
10.3.2 Simple Statements in Predicate Logic
If we negate a statement:
(1)Mulan is asleep.
10.3 Translating English into a Logical Metalanguage
10.3.3 Quantifiers in Predicate Logic
Quantification
one, some, a few, many, a lot, most, every and all.
Denotational Approach
Model-Theoretic Semantics
Truth Conditional Semantics Montague Grammar
Logical Semantics
10.1 Introduction
10.1.2 Denotational Approach
10.3.2 Simple Statements in Predicate Logic
If we begin with the two simple statements below:
(1)Mary is asleep.
Individual Constant
(2)Bill smokes.
predicate letter
X wrote a paper. Wxp
A few/Many/Most/All /Every student(s) wrote a paper.
?
Predicate Logic
10.3 Translating English into a Logical Metalanguage
10.3.3 Quantifiers in Predicate Logic
Model
10.4 The Semantics of the Logical Metalanguage
10.4.2 The semantic interpretation of predicate logic
Correspondence Theory
P
Situation
v
Matched Unmatched
10.2 Model-Theoretical Semantics (Montague Grammar)
Step 1. Translate a Natural Language into Logical Language (10.3)
Step 2. Establish a Model (10.4-10.7)
Other relational sentences will be represented in the similar way:
(2)Fatima prefers Bill to Henry.
Individual Constant
f
P
b
h
Key: P: f: b: h:
prefer Fatima Bill Henry
10.3 Translating English into a Logical Metalanguage
10.3.4 Some advantages of predicate logic translation
Scope Negation “Everybody didn't visit Liverpool.” can be interpreted in English as follows:
10.1 Introduction
10.1.3 Logical Semantics
English
French
Chinese
…
RealWorld Situations
Circularity
English
Logical Language
Precision
Using Logic as a Semantic Metalanguage
Matched
Correspondence Theory
True False
Unmatched
Correspondence Theory says that a preposition, P, is true if and only if the state of affairs is as P asserts it to be.
10.3 Translating English into a Logical Metalanguage
10.3.3 Quantifiers in Predicate Logic
Now, you try: Every student knows the professor.
The professor know every student.
(1) Bill killed Mike.
(2)Peter is crazier than Ryan.
b
K
m
p
C
r
Kbm
Cpr
10.3 Translating English into a Logical Metalanguage
10.3.2 Simple Statements in Predicate Logic
Step 3. Check the Mapping between LL and Modeled situation.
10.3 Translating English into a Logical Metalanguage
10.3.1 Introduction
10.3 Translating English into a Logical Metalanguage
10.5 Checking the Truth-Value of Sentences
10.6 Word Meaning: Meaning Postulates
10.7 Intentionality Part Ⅲ: 10.8 Summary
10.1 Introduction
10.1.1 Formal Semantics
Other compound sentences: Fred smokes and Kate drinks: Sf ∧ Dk If Bill drinks, Jenny gets angry: Db Aj
We can even translate sentences containing relative clauses: Wendy, who is a mother, is an professor: Mw ∧ Pw Tony is a cat that doesn’t purr: ? Ct ∧ ¬Pt
m
A
b
S
Am
Sb
10.3 Translating English into a Logical Metalanguage
10.3.2 Simple Statements in Predicate Logic
If we want to leave the identity of the subject unspecified: (1) X is asleep.
Variable
(2)Y smokes.
predicate letter
x
A
y
S
Ax
Sy
10.3 Translating English into a Logical Metalanguage
10.3.2 Simple Statements in Predicate Logic
Other relational sentences will be represented in the similar way:
True False
v [p] = 1
v [p] = 0
10.4 The Semantics of the Logical Metalanguage
10.4.2 The semantic interpretation of predicate logic
Individual Constant terms: Benjamin → b Predicate Constant: Smoke → S One-place predicate {x:
There is at least one thing x, such that x is a student and x wrote a paper.