Formal Semantics - Week 1: IntroductionSWU LI713 M .Louie August 2015

1 Introduction

• Semanticists study meaning

1. But how exactly do we formalize “meaning”?

2. What sort of data do we aim to account for?

3. What sort of model can we use to represent how meaning isexpressed and combined in natural language?

2 Defining Meaning

• Two Theories of Meaning

1. Meaning as truth-conditions

2. Meaning as use-conditions

• We’ll focus on the first theory1

• Two main concepts regarding how to formalize “meaning”:

1. truth

2. reference

Basic Intuition: The Meaning of a Sentence

If you know what a sentence means, you know. what the world would have to be like. in order for that sentence to be true.

1Note that these two theories are not necessarily incompatible (despite many oldphilosophers arguing about these theories as if they were incompatible). The secondtheory is the basis of formal pragmatics.

(1) Harry Potter is wearing glasses

• (1) is true in World 1, 3, and 4!

(2) Harry Potter is wearing round glasses

• (2) is only true in World 1!

• But how do you know that?

• You know what some of the words/morphemes in (1) refer to...

– Harry Potter

– glasses

– wear

• These words represent some individual, or set of individuals, orrelation between individuals, in the world

• ...and you know how other words/morphemes modify what thosewords refer to

– is, -ing

– round

• So when we talk about what words “mean,” we’re talking aboutwhat words refer to, and how words modify truth-conditions

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2.1 Semantics VS Syntactic Data

• So what does a semanticist’s data look like?

• One piece of data is a 〈context,utterance, judgement〉

• A minimal pair should minimally differ only in. (i) the context, or. (ii) the utterance (not both!)

• Examples of minimal pairs probing the truth-conditional effects oftense in English:

Example I: Semantic Minimal Pair

(3) Context: I was working hard on preparing class notes, butnow I’m taking a facebook break. My supervisor walks upand asks me what I’m doing. She can’t see my laptop screen,so I tell her:

a. I am working on my class notes! False!

b. I was working on my class notes! True!

Example II: Semantic Minimal Pair

(4) a. Context: I was working hard on preparing class notes,but now I’m taking a facebook break. My supervisorwalks up and asks me what I’m doing. She can’t see mylaptop screen, so I tell her:

I am working on my class notes! False!

b. Context: I was taking a facebook break, but now I’mworking hard on preparing class notes. My supervisorwalks up and asks me what I’m doing.

I am working on my class notes! True!

3 Introduction to Model Theoretic Semantics

→ If you know what a sentence means, you know. what the world would have to look like. . in order for that sentence to be true

Frege’s Principle of Compositionality

The meaning of a sentence is determined by. (i) the meaning of its parts, and. (ii) the way that those parts combine

(5) Round glasses are wearing Harry Potter.

• We can formalize all of these intuitions with a semantic system

• There are three major components to the semantic system:

1. The Model/Ontology2

2. Lexical Entries

3. Compositional Rules

• The ontology consists of the elements we’ll use to represent thingsin the real world (and other things we think and talk about)

• Lexical entriesare the ‘meaning’ component of each word/morpheme- the lexical entries “point” to (simple or complex) elements in theontology

• The compositional rules express how the meanings of the variouslexical entries combine to form new/modified meanings

2Many formal semanticists don’t consider the model/ontology to be an important partof a semantic model. They have their lexical entries directly pointing to things in the realworld. For example, Heim & Kratzer (1998). But I like models.

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3.1 The Model/Ontology

• If you want to illustrate the way the world must be, in order fora sentence to be true, you need elements to refer to things in theworld:

• For a semantic system, we usually just use letter symbols, eg.,. a = Harry Potter,. b = Hermione Granger,. c = Ron Weasley,. d = Draco Malfoy,. e = Buckbeak the Hippogriff. f = Luna Lovegood,. g = Ginny Weasley,. h = Hedwig the owl. i = Hagrid. q = quidditch

• The elements in the ontology are organized into domains:

1. De, Individuals = {a, b, c, d, ...}2. ...3. Dt, Truth Values = {1, 0}

• The domain of truth-values:

– True=1– False=0

• Elements in the ontology are often structured in a way that willlet us account for various kinds of entailment patterns - hopefullywe’ll return to this later!

a+b+c

a+c

ba

b+ca+b

c

Figure 1: Lattice-Structure of Individuals

3.2 Lexical Entries

• Lexical entries represent the ‘meaning’ component of words andmorphemes

• Every word/morpheme is associated with a lexical entry that either

– refers to an element in the ontology, or– refers to an complex element that is based on elements in the

ontology (eg., sets, ordered pairs,...)

• Here’s one way of setting up lexical entries:

1. Names refer directly to individuals in the ontology,eg., Harry Potter= a, Draco Malfoy = d

2. Common Nouns refer to sets of individuals,eg., girl = {b, f, g}, animal={e, h}

3. Adjectives refer to sets of individuals,eg., red-haired = {c, g}, blond = {d, f}

4. Verbsintransitive verbs: refer to sets of individualseg., flies3 = {h,e,...}transitive verbs: refer to sets of ordered pairs of individualseg., love = {〈g,a〉, 〈i,e〉, 〈a,q〉, 〈c,q〉, 〈d, q〉, 〈g, q...〉 }

3Flies without a broomstick or flying motorcycle.

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Note on some Formal Notions

• The curly brackets ({a, b, c, ...}) are used to indicate a set:

. A set is a collection of (any number of) entities

. The empty set, ∅, is the only set with no members.

• The arrowed brackets (〈a,b,c, ...〉) are used to indicate a tuple

. An n-tuple is an ordered collection of (n) entities

. (A pair is a tuple with two members, i.e., n=2)

• The difference between sets and n-tuples: ordering

. The order of elements in a set doesn’t matter

. {a, b, c} = {b, c, a} = {c, a, b} = {b, a, c}...

. The order of elements in a tuple matters

. 〈a, b, c〉 , 〈b, c, a〉 , 〈c, a, b〉 , 〈b, a, c〉...

My Particular Notation:

• I use Boldfont to refer to elements in the ontology

• And Italics to refer to lexical elements in the language underinvestigation

3.3 Compositional Rules

• Compositional Rules are how we bring everything, so that we canuse the ontology and lexical entries to represent truth-conditions

• Syntax lets us combine words:

. S→ DP VP

S

VP

V

flies

DP

Hedwig

• But the syntax doesn’t tell us how to interpret these combinations

• This is what the compositional rules do4

• The rule predicationmakes a claim about set-membership

PREDICATION: Interpreting S→ DP VP

S =1 (i.e., S is true), iff. The individual referred to by the DP,. is a member of the set referred to by the VP.

(6) Derivation of Truth-Conditions

. Hedwig = h

. flies = {h, e}

. S=1 iff h∈{h, e}

• There are also syntactic rules to combine adjectives with nouns:

. NP→ Adj N

NP

N

girl

Adj

red-haired

4These are just temporary rules! I’m setting up a very simple system to show youhow a semantic system can work. We’ll set up a semantic system more suited to thecomplexity of natural language over the next few weeks.

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• The rule modification interprets this with set-intersection

MODIFICATION: Interpreting NP→ ADJ N

The NP refers to the intersection of. the set referred to by the ADJ, and. the set referred to by the N.

c, g g, b, f

The set denoted by red-haired The set denoted by girl

Figure 2: Sets

c g b, f

The set denoted by red-haired girl

Figure 3: Set Intersection

(7) Derivation of Reference. The Adj red-haired refers to the set {c,g}

. The N girl refers to the set {g,b,f}

. The NP red-haired girl referes to their intersection, {g}

• The syntax can also combine determiners with NPs to form DPs:

. DP→ Det NP

DP

NP

N

girl

Adj

red-haired

D

The

• We can posit a compositional rule for this too:5

SELECT: Interpreting DP→ Det NP

When a determiner like the. combines with a singleton set NP,. The DP refers to the member of that singleton set

(8) The NP red-haired girl = {g},so the DP the red-haired girl = g

• The syntax can also combine DPs with Vs to form VPs:

VP

DP

quidditch

V

love

5This rule for the is just for explanatory purposes. It isn’t actually a very good repre-sentation of what the means. My favourite analysis of the is one where it applies to a setand the resulting DP refers to the maximal member of that set. In the case of a singletonset, it is that sole member. In the case of a set with plural indidividuals in it, it takes thelargest plural individual - eg., a+b+c in the lattice-structure for individuals.

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Saturate: Interpreting VP→ V DP

When a transitive verb, V, (which is a set of 〈x,y〉 pairs)combines with a DP, which refers to an entity w,The VP refers to the set of individuals, x, that form 〈x,w〉 pairsin V.

Assume a transitive verb like. love = {〈g,a〉, 〈i,e〉, 〈a,q〉, 〈c,q〉, 〈d, q〉, 〈g, q〉...}

If this combines with the DP quidditch, q, then

The VP love quidditch refers to the set of individuals that form a 〈x, q〉pair above - i.e., love quidditch={a, c, d, g}

3.4 Example Problems

(9) Derive truth-conditions for the following, assuming that:. loves quidditch = {a, c, d, g}. blond = {d, f}. girl = {b, f, g}

S

VP

loves quidditch

DP

NP

N

girl

Adj

red-haired

D

The

S

VP

loves quidditch

DP

NP

N

girl

Adj

blond

D

The

3.5 Interpretation Brackets

• It starts to get tedious to say. “the X that the NP/DP/VP refers to,”

• This is why semanticists use the formal notation ~.�

• These are “interpretation brackets”

• ~DP� can be read as “the interpretation of DP”. - i.e., “the individual that the DP refers to”

• ~V� can be read as “the interpretation of V”. - i.e., “the set of 〈x, y〉 pairs that the V refers to”

• This allows us state the compositional rules in a more succinct way:

1. Predication: S→ DP VP. ~S� =1 iff, ~DP� ∈ ~VP�,. ~S�=0 otherwise

2. Modification: NP→ Adj N. ~NP� = ~Adj� ∩ ~N�

3. Select: DP→ D NP. ~DP� = σ(~NP�)

4. Saturate: VP→ V DP. ~VP� = {x: 〈x,~DP�〉∈ ~V�}

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Formal Set Notation

• x∈Y is read as “x is a member of Y,” where Y is a set

• X∩Y is read as “the intersection of X and Y,” where X and Yare both sets

• X∪Y is read as “the union of X and Y,” where X and Y areboth sets

• The notation {x: x such that P} defines a set consisting of allelements x, such that x satisfies the description P

• The σ is not part of formal set notation.

• The σ represents that maximality operator.

• This indicates an operation that applies to a set and yieldsthe maximal member of that set. (Landman (1989))

4 Summary and Conclusion

• Formal semantics is the study of meaning

• One way of formalizing meaning is with truth-conditions andreference

• A core intuition about meaning is the principalof compositionality

. - i.e., that the meaning of complex things are derived from

. . (i) the meaning of their simple parts, and

. . (ii) the way they are combined

• The toy system set up here allows us to refer to things includingthe notion of true and false (with the ontology and lexical entries)

• It also captures compositionality (with compositional rules)

• A formal system like this allows us to objectively calculate truth-conditions, which is crucial for making predictions and doing se-mantic research!

• Reading: Heim & Kratzer (1998), Ch. 1 (useful information on sets,functions and ordered tuples)

References

Heim, Irene & Angelika Kratzer. 1998. Semantics in generative grammarBlackwell Textbooks in Linguistics. Blackwell Oxford.

Landman, Fred. 1989. Groups I. Linguistics and Philosophy 12. 559–605.

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