Semantics and Logical Form

Reference: Chapter 8 of Allen

To describe a language for representing logical forms - that is, intermediate representations on the way to transforming a parse tree into the final meaning representation. Logical forms must be able to encode possible ambiguities of meaning of a particular parse of a sentence.
Keywords: co-agent, compositional semantics, exists, experiencer, failure of substitutivity, FOPC, forall, instrument, logical form, logical operator, modal, MOST1, patient, PLUR, predicate operator, semantics, substitutivity, term, THE, thematic role, theme, victim
  • Definition of compositional semantics
  • Word senses and ambiguity
  • Logical form language - terms, predicates, propositions, logical operators, quantifiers, predicate operators, modal operators.
  • Ambiguity in logical forms
  • Verbs and states in logical forms - thematics roles
  • Logical forms for speech acts and for embedded sentences


Facets of Meaning

Logical Form

Model-based Semantics

Word Senses and Ambiguity

Word Senses and Ambiguity 2

Word Senses and Ambiguity 3

The Basic Logical Form Language

This section defines a formal language of logical forms, resembling FOPC (first order predicate calculus):

constants or expressions that describe objects: FIDO1, JACK1

constants or expressions that describe relations or properties: BITES1.

Each predicate has an associated number of arguments - BITES1 is binary (unary = 1 argument; ternary = 3 arguments; n-ary = n arguments).

Logical Form Language 2

a predicate followed by the appropriate number of arguments: (BITES1 FIDO1 JACK1), (DOG1  FIDO1) - Fido is a dog.

More complex propositions can be constructed using logical operators:


Note that and does not always "translate" as logical & - e.g. it may suggest temporal sequence: I went home and had a drink compared to I had a drink and went home.

Logical Form Language Part 3

In FOPC, only forall and exists.
English has vaguer quantifiers, too: most, many, a few, a, the, ...

Variables are introduced here, as in FOPC.

However variables in logical form language persist beyond the "scope" of the quantifier.

A man came in. He went to the table.

The first sentence introduces a new object of type MAN. The He in the second sentence refers to this object.

Quantifiers continued


NL quantifiers are typically restricted in the range of objects that the variable ranges over. In Most dogs bark the variable in the MOST1 quantifier is restricted to DOG1 objects:
(MOST1 d1 : (DOG1 d1) (BARKS1 d1))

the and a give rise to important NL quantifiers - the dog barks has logical form

(THE x : (DOG1 x) (BARKS1 x))

which would be true only if we can determine a unique dog in context, and that dog barks. How we find the unique dog is discussed in the section on Reference.

Logical Form Language Part 4

predicate operator
We also need a way to handle plurals as in the dogs bark.

A new type of thing called a predicate operatoris introduced that takes a predicate as an argument and produces a new predicate.

For plurals, PLUR: if DOG1 is true of any dog, then (PLUR DOG1) is true of any set of dogs with more than one member:

(THE x : ((PLUR DOG1) x) (BARKS1 x)).

Logical Form Language Part 5

modal operator
used for verbs like believe, know, want, for tense, and other purposes. Sue believes Jack is happy becomes


Modal operators may exhibit failure of substitutivity:

JACK1 may = JOHN22 (i.e. the individual known as Jack may also be called John, e.g. by other people)

However, Sue believes John is happy may not be true, e.g. because Sue may not know that JACK1 = JOHN22.

Logical Form Language Part 6


Tenses: use modals PRES, PAST, FUT:


Again, substitutivity may fail.

JOHN1 may = MINISTER1, and (PAST (OWNS1 JOHN1 FIDO1)) may be true, but (PAST (OWNS1 MINISTER1 FIDO1)) can still be false because John was not minister when he owned Fido.

Converting Logical Forms to a Prolog-type Notation

Converting Logical Forms to Prolog 2

Encoding Ambiguity in the Logical Form

Encoding Ambiguity in the Logical Form 2

Abbreviation: <EVERY c1 CHILD1>= (EVERY c1 : (CHILD1 c1))

Encoding Ambiguity in the Logical Form 3

Proper Names and Pronouns

Proper Names and Pronouns 2

Verbs and States in Logical Form

Verbs and States in Logical Form 2

Verbs and States in Logical Form 3

Verbs and States in Logical Form 4

Verbs and States in Logical Form 5

Thematic Roles

Thematic Roles 2

Thematic Roles 3

Preposition and Thematic Roles

Speech Acts and Embedded Sentences

Speech Acts and Embedded Sentences 2

Wh-Questions and Beyond

Wh-Questions and Beyond 2

Summary: Semantics and Logical Form
We have described a logical form language that includes terms, predicates, propositions, logical operators, quantifiers (including special NL quantifiers such as THE), and shown how this language can be used to represent ambiguous sentences.

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Copyright (C) Bill Wilson, 2006, except where another source is acknowledged.