Semantics and Logical Form

Reference: Chapter 8 of Allen

Aim:
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
Plan:
  • 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


Semantics


Facets of Meaning


Logical Form


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):

terms
constants or expressions that describe objects: fido1, jack1

The main purpose of the lower case, and the number "1", in fido1, jack1, and bites1, is to distinguish between the words "Fido", "Jack", and "bites" and the semantic symbols fido1, jack1, and bites1. If we needed to distinguish between different word senses, as we might with the word "dog", we would use dog1, dog2, dog3, etc.

predicates
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

propositions
a predicate followed by the appropriate number of arguments:
bites1(fido1, jack1) - Fido bites Jack
dog1(fido1) - Fido is a dog.

More complex propositions can be constructed using logical operators:

not(loves1(sue1, jack1))
(& bites1(fido1, jack1), dog1(fido1))

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

quantifiers
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 man1. 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))

Note that D1 is a variable.

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

the(X, dog1(X), barks1(X))

The schema for this is: the(X, Restriction on X, Proposition about 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 the predicate 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

believe1(sue1, happy(jack1)).

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

Tense operators: use modals pres, past, fut:

pres(sees1(john1, fido1)) - John sees Fido
past(sees1(john1, fido1)) - John saw Fido
fut(sees1(john1, fido1)) - John will see Fido

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 between Allen's notation and our Prolog-type Notation


Converting LISP-style 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


Term Constructors: Proper Names


Term Constructors: Pronouns


Verbs and States in Logical Form

  1. John broke it
  2. John broke it with the hammer

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, 2009, except where another source is acknowledged.