LIN241H1 Lecture Notes - Lecture 5: If And Only If, Ditransitive Verb, Computer-Aided Technologies
5 – Compositionality I
Principle of Compositionality (of meaning):
The meaning of a composite expression is a function of the meaning of its immediate
constituents and the way these constituents are put together.
Principle of Compositionality (of extensions):
The extension of a composite expression is a function of the principle of its immediate
constituents and the way these constituents are put together.
Compositionality
• The extension of a composite expression depends on two types of factors:
o The extension of its constituents
o The way these constituents are put together
• Express the extension of sentences in terms of the extensions of their constituents
o E.g. [[Every boy is happy]] = 1 iff [[boy]] ⊆ [[happy]]
• Extension of Every boy is happy is a function of:
o The extension of boy, i.e., [[boy]]
o The extension of is happy, i.e., [[happy]]
o The relation of inclusion contributed by [[every]]
• Still missing an account of how syntax is used to "glue together" these meanings
o E.g. [[Every boy is happy]] = 1 iff [[happy]] ⊆ [[boy]] -> Every happy person is a boy,
which is wrong
• Theory needs to be able to compute the truth-conditions of a sentence automatically,
given a specification of:
o The extensions of its words
o Its syntactic structure
Truth-conditions and word meaning
Putting extensions together
• Truth-conditions notation:
o [[John smokes]] = 1 iff John smokes
• Notation of extensions of different kinds of words
o [[John]] = John
o [[smokes]] = {x: x smokes}
• Combining these two pieces of information together
E.g., John smokes.
• Three linguistic expressions:
o Sentence: John smokes.
o Subject: John
o Predicate: smokes
• Notation that allows us to refer to the extension of
o [[John smokes]] is the extension of John smokes
o [[John]] is the extension of John
o [[smokes]] is the extension of smokes
• Trying to find the relation R that captures the correct truth-conditions for the sentence
o [[John smokes]] = 1 iff R([[John]], [[smokes]])
o Preserve information of John and smokes
find more resources at oneclass.com
find more resources at oneclass.com
o Must find R
Finding R
• R must meet the following condition:
o R([[John]], [[smokes]]) iff John smokes
• The relation must take into account the type of denotation of John and smokes
o [[John]] is an individual
o [[smokes]] is a set of individuals
• Solution: R is the relationship of membership
o [[John smokes]] = 1 iff [[John]] ∈ [[smokes]]
o E.g., not inclusion (only holds for pairs of sets)
• Check that the solution respects the types of [[John]] and [[smokes]]
o The relation ∈ holds between sets and their members
o [[smokes]] is a set of individuals; [[John]] is an individual
o Therefore, [[John]] ∈ [[smokes]] is a meaningful statement
• Check that it captures the right truth-conditions
o [[John smokes]] = 1 iff [[John]] ∈ [[smokes]]
o [[John smokes]] = 1 iff [[John]] ∈ {x: x smokes}
o [[John smokes]] = 1 iff John smokes.
Note: should be able to follow this process for:
• Intransitive, transitive, ditransitive sentences
• Sentences with only proper names as arguments of verbs
• Sentences with at most one quantifier
Transitive sentences; e.g., Sacha likes Denis.
• Building blocks (words) are: Sacha, likes, Denis
• Sacha and Denis denote the individuals [[Sacha]] and [[Denis]]
• [[likes]] is the set of pairs of individuals <x,y> such that x likes y
• Find R such that: [[Sacha likes Denis]] = 1 iff R([[Sacha]], [[Denis]], [[likes]])
o Relates the three words
• [[Sacha likes Denis]] = 1 iff <[[Sacha]], [[Denis]]> ∈ [[likes]]
o <[[Sacha]], [[Denis]]> is a pair of individuals
o [[likes]] is a set of pairs of individuals <x,y> such that x likes y
• Sanity check
o <[[Sacha]],[[Denis]]> is a pair of individuals
o [[likes]] is a set of pairs of individuals
o <[[Sacha]], [[Denis]]> ∈ [[likes]] is a meaningful statement
• Check that we captured the right truth-conditions
o [[Sacha likes Denis]] = 1 iff <[[Sacha]], [[Denis]]> ∈ [[likes]]
o [[Sacha likes Denis]] = 1 iff <Sacha, Denis> ∈ { <x,y>: x likes y
Ditransitive sentences; e.g., Sacha sold Fluffy to Denis.
• Treat [[sold]] as a verb with three nominal arguments
o Triples
• [[Sacha sold Fluffy to Denis]] = 1 iff <[[Sacha]], [[Fluffy]], [[Denis]]> ∈ [[sold]]
• Check that these are the correct truth-conditions
find more resources at oneclass.com
find more resources at oneclass.com
Document Summary
The meaning of a composite expression is a function of the meaning of its immediate constituents and the way these constituents are put together. The extension of a composite expression is a function of the principle of its immediate constituents and the way these constituents are put together. [[every boy is happy]] = 1 iff [[boy]] [[happy]: e. g. [[john smokes]] = 1 iff john smokes: notation of extensions of different kinds of words. [[smokes]] = {x: x smokes: combining these two pieces of information together. E. g. , john smokes: three linguistic expressions, sentence: john smokes, subject: john, predicate: smokes, notation that allows us to refer to the extension of. [[john smokes]] is the extension of john smokes. [[smokes]] is the extension of smokes: trying to find the relation r that captures the correct truth-conditions for the sentence. [[john smokes]] = 1 iff r([[john]], [[smokes]]: preserve information of john and smokes, must find r.
