class: center, middle ### PLIN3004/PLING218 # Advanced Semantic Theory ### Week 2 ### 13 October, 2016 --- # Compositionality Principle
Goal
: Want to build a finite semantic system that computes the truth-conditions of any grammatical declarative sentence in English.
Fact
: There are infinitely many grammatical declarative sentences in English, but native speakers know their truth-conditions.
Hypothesis
: There are general ways of combining meanings. In particular, we assume that natural language semantics obeys the **compositionality principle**. > **The Compositionality Principle**
The meaning of a complex phrase is determined solely by the meanings of its parts and the syntactic structure. --- # Compositionality Principle (cont.) Example: - We know the truth-condition of any grammatical declarative sentence, e.g. '
A man saw Mary
'. - According to the compositionality principle, the truth-condition of the sentence is determined by the meaning of '
a man
', the meaning of '
saw Mary
', and how they are combined. - Furthermore, the meaning of '
a man
' is determined by the meaning of '
a
', the meaning of '
man
', and how they are combined. Then, all we need to know is the meanings of individual words/morphemes, '
a
', '
man
', etc. and how to combine them to construct meanings of larger phrases. --- # Two Finite Components Our semantics will have two finite components. - **Lexicon**: List of meanings of morphemes/terminal nodes - **Compositional Rules**: Instructions about how to combine meanings To compute the meaning of an arbitrary phrase, we start with the meanings of morphemes in the lexicon, and apply the compositional rules to derive the meanings of component phrases and ultimately the meaning of the whole phrase. --- # The Role of Syntax How the words are combined (i.e. the syntax) is important here, because '
A man saw Mary
' and '
Mary saw a man
' have different truth-conditions, despite the fact that the these sentences are made up of the same ingredients.
Toy Syntax (Phrase Structure Grammar) -
S
→
DP VP
-
DP
→
John
|
Mary
-
VP
→
smokes
|
left
This grammar generates 4 sentences:
[
S
[
DP
John] [
VP
smokes]]
[
S
[
DP
John] [
VP
left]]
[
S
[
DP
Mary] [
VP
smokes]]
[
S
[
DP
Mary] [
VP
left]]
--- # Goals for Today We'll develop a compositional semantics for this small language with four sentences today and next week. Then we'll enrich it for the rest of the course. Today, we analyze: - The meanings/denotations of sentences = truth-values - The meanings/denotations of proper names = entities/individuals Based on these, we'll go on to analyze the meanings of VPs as functions in Lecture 3. --- class: center, middle # Sentences Denote Truth-Values --- # Models What we know so far is the truth-conditions of the sentences, e.g. '
John smokes
' is true iff John smokes. When a particular situation (with enough information) is given, the sentence becomes either true or false. To determine whether it's true or false, you need to know: - Which person '
John
' refers to; and - Whether that person smokes or not. We represent such information in a mathematical structure called a **model**. A model is a mathematical representation of a particular situation/state of affairs. --- # Denotation of Sentences For any model M and for any sentence
φ
, M tells you whether
φ
is true or false. It is common to use numbers, **1** and **0**, instead of 'true' and 'false'. They are called **truth-values**. 1 means true, and 0 means false. We say
φ
**denotes** 1 or 0 in M, or the **denotation** of
φ
in M is true or false. >
Notation
: ⟦
α
⟧
M
= the denotation of expression
α
in M. E.g. for any model M and for any sentence
φ
, ⟦
φ
⟧
M
=1 or ⟦
φ
⟧
M
=0. --- # Truth-Conditions vs. Truth-Values Importantly, truth-values and truth-conditions are different things, although both are meanings of sentences in some sense. When a situation is given, the truth-condition yields a particular truth-value (0 or 1). E.g. In the following figure the sentence "
There is a circle in a square
" is true. If M1 represents this situation, ⟦
There is a circle in a square
⟧
M1
= 1 .
--- # Summary so far A model is a mathematical structure that represnts a particular state of affairs. - The truth-condition of a sentence is a semantic property of the sentence. - The denotation of a sentence in a model is a truth-value (1 or 0). E.g.
= 0 or 1, depending on what M is --- # Summary so far (cont.) The truth-condition of this sentence can be stated as: > For any model M,
= 1 iff John smokes in M Accoding to the compositionality principle:
is determined by
and
We discuss ⟦
[
DP
John]
⟧
M
today and ⟦
[
VP
smokes]
⟧
M
next time. --- class: center, middle # Proper Names Denote Individuals --- # Referential Intuitions We have intuitions about what
should be. Using a proper name, we **refer** to a particular individual/entity. Exactly who we refer to by
[
DP
John]
varies across situations (e.g. we might be referring to John Harris or John Lennon), but in each situation we refer to a particular person. Note that who the referent is matters for the truth-condition of the sentence. If we refer to John Harris by
[
DP
John]
, the sentence
John smokes
is about John Harris. --- # Proper Names Denote Individuals Just like M fixes the denotation of a sentence to a particular truth-value, M fixes the denotation of
[
DP
John]
to a particular individual. > For any model M,
is some individual in the model For this course, we abstract away from how exactly the referent is determined, and simply assume that the model tells you which individual John should be. E.g. ⟦
[
DP
John]
⟧
M1
=John Harris; ⟦
[
DP
John]
⟧
M2
=John Lennon, etc. --- # Non-Branching Node According to the compositionality principle,
is determined by ⟦
John
⟧
M
It seems natural to assume that simply projecting a non-branchnig XP from X does not change the meaning. So, we assume: >
= ⟦
John
⟧
M
--- # Summary > **The Compositionality Principle**
The meaning of a complex phrase is determined solely by the meanings of its parts and the syntactic structure. Relative to a model M, which represents a particular situation: - (Declarative) sentences denote truth-values (1 or 0). - Proper names (and their DP projections) denote individuals. In the next lecture we will analyze the meanings of
[
VP
smokes]
and other verb phrases as **functions** of a particular kind. --- class: center, middle # Mathematical Concepts: Functions --- # Functions **Functions** are especially important for formal semantics. This will become clear in the next lecture, but it is useful to review the concept of functions at this point. Functions are a special type of mapping where for each input there is a unique output. You can think of any function F to be something that takes an input A, and returns a value as its output. This returned value is written as **F(A)** (read 'F applied to A' or 'F of A'). --- # Examples It's important that if F is a function, then for any argument A, the output value F(A) is **unique**. That is, every time you apply F to A, you get the same result. - Suppose that a mapping named S maps any positive or negative integer n to n
2
. This mapping is a function, because for each input value, there is a unique output value. - Notice that there can be multiple inputs that have the same output, as long as for each input, there is a unique result. In the case of S, S(2) = S(-2), but this doesn't make S a non-function. --- # Examples (cont.) - Similarly suppose that a mapping C maps any country to its capital. This is also a function, because for any country, there is a unique capital. - However, a mapping G that maps any person to the languages they speak is not a function, because there are people who speak several languages. For example, G(Andrew Nevins) is not uniquely determined. --- # Defining a function In order to define a function, you need to specify the following three things: 1. **Domain**: The set of objects the function can take as inputs. 2. **Range**: The set of objects the function returns as outputs. 3. What happens to each input. Let take the case of the squaring function S: 1. The domain of S, dom(S), is the set of all integers, ℤ. 2. The range of S, ran(S), is the set of all natural numbers (including 0), ℕ. 3. For any integer n, S(n) = n
2
. The first two pieces of information are sometimes written together as 'S: ℤ→ℕ' (read 'S is from ℤ to ℕ'). --- # Defining a function (cont.) Instead of saying 'For any integer n, S(n) = n
2
', you can represent the function by enumerating what happens to each input, e.g.:
Of course, if the domain is infinitely large, as in the case of S, this is not effective. But if it is finite, we can enumerate all of them, e.g.:
--- # Exercise Which of the following are functions?
--- # For Next Week - Do Assignment 2. - Read Lecture Notes for Week 3.