Not all linguistic expressions
are compositional e.g. idioms
Local and structure-dependent: determiners
and head nouns need to be able to compose
together, just like verbs and their arguments
Based on syntax
Meaning of linguistic expression is
determined by the meaning of its subparts,
and the rules that combine them
DENOTATIONS
Ostensive definition:
what it points to
Denotations pick
out relevant sets
Bare nouns are predicates -
they denote properties and can
occur in the copula construction
Predicates denote sets of
individuals that have the property
ascribed
TRUTH CONDITIONS
What the world would have to
be like in order for the
proposition to be true
Intensions are a more
accurate
representation of word
meaning
Presuppositions: A proposition X
presupposes Y if Y is assumed to be
true by anyone who expresses X
OPPOSITES
Extension - thing picked out in actual world
Intension - thing picked out in all possible worlds
Literal (sentence) - derived from linguistic
expression, independent of context.
Inferred (utterance) - everything else
Tautology - necessarily true in all worlds
Contradiction - necessarily false in all worlds
ENTAILMENTS
Useful to make reference to
truth relations between
statements, like the relation of
entailment
Part of its encoded
meaning, but not encoded
directly
Uttering a sentence
makes you committed to
the truth of the sentence
and the entailment
A statement A entails a
statement B if wherever A is
true, B must also be true
Upward
Entailment towards the
superset, e.g. John likes blue
cheese entails that he likes
cheese. (towards the
GENERAL)
Downward
Entailment towards the subset, e.g. Every
dog barked entails that every small dog
barked. (towards the SPECIFIC)
Logic:
Lectures 3-6
PROPOSITIONAL
Treats propositions as atoms and
provides a way of representing
the constructing of complex
propositions with connectives
Conjunction ^ - typically
represented by 'and'. Order doesn't
matter, only true when T+T, all
others false
Loss of meaning in
translation: sequential
elements, and the
distinction of 'but'
Disjunction v - typically
represented by 'or', only
true if p is true.
Negation ¬ -
Opposites apply
Material implication -> - Partially
resembles 'if...then', order matters,
the antecedent 'if' must occur first.
Only false if when T+F
Loss of meaning in translation:
logic doesn't represent the
causation implication
Equivalence <-> -
Expressed as 'if and only
if', order is irrelevant, true
when T+T and F+F
Equivalences in logic
are known as De
Morgan's Law
Exclusive disjunction - Often
what is expressed by 'or', but
exclusivity may be an
implicature
Can also be derived by
adding 'but not both'
The Key: Match key to
variables, propositions must
not decompose any operators
(negation), full propositions
only, order implications
correctly
PREDICATE
Represents the inner
structure of propositions in
terms of arguments,
predicates and quantifiers
Form: Melissa is tall = TALL (m)
Order of Obliqueness: Subject >
Indirect Object > Direct Object
Prepositions left out, which
alters meaning, e.g. the
distinction between in/on
Quantifiers
Universal - variable bound by the universal
quantifier e.g. ∀x(ILLUMINATED(x))
Existential - variable bound by the existential
quantifier e.g. ∃x(ILLUMINATED(x))
Scope is read from the formula
left-to-right, and is typically
reflected by hierarchical relations
Restricted quantification - quantificational determiner exists with a
noun that restricts it Universals require the implication symbol, and
existentials require conjunctions e.g. ∀x(BOOK(x) -> READ(s,x)) and
∃x(BOOK(x) ^ READ(s,x))
Argument positions can be occupied
by complex expressions like
embedded sentences e.g.
SAY(s(BARK(m)))
GENERALIZED
QUANTIFIER THEORY
(GQT)
Need to generalize quantifiers because a
number of quantifiers fail to translate
into the predicate logic system
All quantifiers can be recast as
relations between sets e.g.
[Every x: CIRCLE(x)] RED(x)
These are sensible ostensive definitions, they
point to two sets [the main predicate and the
nominal restriction], and say something
numerical about the relation between them
Quantifiers:
Lectures 7-9
SET THEORY
Sets - collections of things
Set theory - algebras of set
things
Notation
1. A=B: A is identical to B 2. A⊂B: A is a proper subset of B 3. A⊆B: A is a subset of B
4. |A|=2: The cardinality of A is 2; A has 2 members 5. |B>A|: B has more members
than A 6. |B∩A|: The cardinality of the intersection of B and A is two 7.
|B-A|>|A∩B|: The number of members in B that are not also in A is greater than
the number of elements in both
Asymmetric
Quantifiers
Most, few, every
Asymmetric quantifiers describe
what proportion of the nominal
restriction is in the intersection
Most circles are red ≠ Most red things
are circles
PROPORTIONAL QUANTIFIERS
Symmetric
Quantifiers
Numbers
Symmetric quantifiers
just count subsets
CARDINAL QUANTIFIERS (do
not come with an existence
presupposition)
NEGATIVE POLARITY ITEMS
Words that seem to need to be in the
scope of a negative element e.g. ever and anything
NPIs are licensed in non-negative
sentences with PROPORTIONAL
quantifiers that have downward entailments
CARDINAL quantifiers
do not license NPIs at all
No: downward entailing on restriction and
main predicate so NPIs licensed in both
positions = neg + NPI
Every: downward
entailing only on
restriction, only
licenses NPIs here
Three: not downward
entailing on either
argument, so NPIs not
licensed anywhere
EXISTENTIAL SENTENCES
THERE+BE+NP
Only some quantifiers can
be in the subject of such
sentences, e.g. some,
many, four
Proportional quantifiers resist these positions because
they presuppose the existence of a background set, and
it would sound odd to use them in an existential
sentence, because existence is already ASSERTED by the quantifiers
Only propositions can give
rise to entailments, but
lexical items can give rise to
presuppositions
Tests
Negation test: presuppositions are
retained under negation, but
entailments are not
Interruption test: "I didn't know
P" is an appropriate interruption
Kinds of presuppositions
Existential: presupposes
the existence of a given
identity
Factive: presupposes the
truth of a following
proposition
Counterfactual:
presupposes the falsity of
a following proposition
Lexical/aspectual: presupposes another
concept with the use of a given
expression, typically related to structured
sequence of events
DEFINITE DESCRIPTIONS
Refer to a single individual in the actual
world, but definite descriptions are not rigid
designators, because the intension of a
definite description is richer than the name
Russell's proposal
Definite descriptions assert existence (there
must be an individual accurately described by
the expression), and uniqueness (there must be
exactly one individual satisfying the
descriptions)
Formula in classical predicate logic, using ∃ for
existence and ∀ for uniqueness e.g.
∃x(KING-OF-FRANCE(x) & ∀y(KING-OF-FRANCE(y) ->
y=x) & BALD (x)))
Alternative: GQT + Benefits
Makes 'the' a
proportional
quantifier,
ensures it comes
with an existence
presupposition
Accounts for
familiarity effect
with 'the'
Easily
amended to
account for
plural
definites
OPACITY
Principle of
Substitutivity
Identical expressions have the
same truth value
Does not hold opaque
sentences like "often
been an Italian"
Opaque contexts: modalized sentences; the
complement of propositional attitude
predicates like 'want'
Adverbs
Adverbs like often are not just
simple adjuncts, they are
propositional operators that take
scope, the issue is the scope of the
adverb with respect to the definite
expression
Weird scope reading: One
individual cannot have a property
like Italian-ness on and off, it's an
individual level predicate which
must hold through all contexts
4 logical states of modality:
necessarily true/false ◻
(non-contingent) & possibly
true/false ⋄ (contingent)
EPISTEMIC MODALITY
What is necessary/possible, given
WHAT IS KNOWN (context-dependent).
Involves quantification over worlds
which are compatible with what we
know (epistemically possible worlds)
Necessary
Must/can't
∀we
Possible
Might/could be
∃we
DEONTIC MODALITY
What is MORALLY/LEGALLY necessary/possible, according to
some set of rules (contextually given). Involves quantification
over the set of worlds in which the relevant codes of behaviour
are adhered to (perfect obedience worlds)
Necessary
Must/required to
∀wpo
Possible
Can/may/allowed to
∃wpo
POSSIBLE WORLDS
Modalized sentences don't
assert that the core proposition
is true in the actual world, they
say something about the set if
worlds in which it would be true
Possible to quantify over these worlds
using ∀w - all possible worlds & ∃w - at
least one possible world
RESTRICTION
If modals quantify over possible
worlds, the modal flavour is a
restriction of this quantification.
This is know as the modal base
Modal base is characterised as being accessible to
the actual word, the restriction is specifying an
accessible relation between the actual world and the
set of possible worlds (epistemically/deontically
accessible).
Approaches
VARIABLE STRENGTH APPROACH:
different strengths require
different quantifications over
possible worlds. Stronger the modal
- the more accessible the world is
VARIABLE RESTRICTION APPROACH:
concentrating on different worlds. The
narrower the restriction, the more
accessible
Pragmatics
Modal "must" cannot be used when we
have direct evidence, because it comes
with an evidential presupposition.
Ambiguities in specific/non-specific
interpretations