2. INTRODUCTION
In this chapter we look at the approach known as formal
semantics. Although any approach might be formalized, this
label is usually used for a family of denotational theories
which use logic in semantic analysis. Other names which
focus on particular aspects or versions of this general
approach include truth conditional semantics, model-
theoretic semantics, and Montague Grammar. 1 As we shall
see, another possible label might be logical semantics
3. INTRODUCTION
• In discussing formal semantics we touch on
an important philosophical divide in
semantics: between representational and
denotational approaches to meaning.
4. Model-Theoretical Semantics
Much of the investigation of logic and natural language
semantics has been conducted by philosophers,
logicians and mathematicians
5. Model-Theoretical Semantics
• for example the predicate logic we describe in this
chapter derives largely from the work of the logician
and mathematician Gottlob Frege,5 the notion of
truth owes much to Alfred Tarski (1944, 1956), and
much of the recent and contemporary debate has
been
6. Translating English into a Logical Metalanguage
• As we have said, the first stage of this semantic
analysis consists of translation. The basic idea is that
we can translate from a sentence in an individual
language like English into an expression in a universal
metalanguage
7. its syntax, that is, how it combines with
sentence constants p, q, and so on, and an
approximate English equivalent:
2 Connectives in propositional logic Connective Syntax
English ¬ ¬p it is not the case that p ∧ p ∧ q p and q ∨ p
∨ q p and/or q ∨e p ∨e q p or q but not both → p → q if
p, then q ≡ p ≡ q p if and only if
8. Simple statements in predicate logic
• If we begin with simple statements like 10.3 and 10.4
below:
• Mulligan is sleeping
• Bill smokes.
9. • we can identify a subject-predicate structure where
the subject is a referring expression (Mulligan, Bill)
and the predicate tells us something about the
subject (is asleep, smokes).
10. • The subject argument can be represented by a lower-
case letter (usually chosen from a to t and called an
individual constant), for example:
• Mulligan: m Bill: b
11. Quantifiers in predicate logic
• One important feature of natural languages that
formal semanticists have to deal with in their
translation into logical form is quantification. All
languages have strategies for allowing a proposition
to be generalized over ranges or sets of individuals.
In English for example quantifiers include words like
one, some, a few, many, a lot, most, and all.
12. Some advantages of predicate logic translation
• The predicate logic we have been looking at is used
by logicians to demonstrate the validity of arguments
and reasoning.
13. • We can take as an example the way that the
representation of quantifiers, as introduced above,
clarifies some ambiguities found in natural
languages. One of these is scope ambiguity, which
can occur when there is more than one quantifier in
a sentence. For example the English sentence 10.32a
below has the two interpretations paraphrased in
10.32b and c:
