Summary• Knowledge of meaning is knowledge of entailment

patterns:To grasp the meaning of S is to grasp what it entails(and, of course, act accordingly)

• We have applied this idea to sentential connectives• Result: A recursive, compositional characterization of how

truth-conditions are channeled (‘project’) through syntactic structure using connectives

• We now know how an infinite set of entailments can be in principle captured through a finite, in fact, small machinery

A note on the bad and the ugly.

• In developing our theory of connectives, we have implicitly adopted the good (= classical logic/set theory).

What about the bad and the ugly?

• The bad:

i. Not (||IP||t) = 1- ||IP||t

ii. And (||IP1||t, ||IP2||t) = MIN (||IP1||t, ||IP1||t)

iii. Or(||IP1||t, ||IP2||t) = MAX (||IP1||t, ||IP1||t)

Multivalued logics

The ugly

i. Not(||IP||t) = 1, if ||IP||t = 0; Not(||IP||t) = 0, if ||IP||t = 0;

Not(||IP||t) is undefined otherwise

ii. And (||IP1||t, ||IP2||t) = 1 if both ||IP1||t, ||IP2||t are defined and true; And (||IP1||t, ||IP2||t) = 0 if one of ||IP1||t, ||IP2||t is defined and false; And (||IP1||t, ||IP2||t) is undefined otherwise

iii. Or (||IP1||t, ||IP2||t) = 1 if one of ||IP1||t, ||IP2||t is defined and true; And (||IP1||t, ||IP2||t) = 0 if both of ||IP1||t, ||IP2||t are defined and false; And (||IP1||t, ||IP2||t) is undefined otherwise.

Strong Kleene logic (information based)

Which of these modification of the good predicts better how truth conditions project (= how complex concepts behave)?

Are we really getting it right?

• Suppose (a) is true at some time t. Our approach predicts that sentences (b)-(c) would be true at any such time:

(a) John doesn’t smoke

(b) If John smokes, he is happy

(c) John drinks or doesn’t smoke

• Yet, we certainly would not use (b) or (c) to describe what is going on in (a). Why?

Close encounters of the 4th kind:after truth conditions, entailments, (and

presuppositions),Implicatures

Words/sentences have an indefinite number of meaningsa. The ham sandwich wants his billb. Who stole the steak? The dog looks happyc. The man with the Martini is a bore[Context: you are at a party; a man is sipping a drink, which you know to be Vodka; nobody else is drinking:What did the speaker mean?Would you understand what was meant?]

How is all this possible?

Meaning is composite. But it has a precise structure

• Some information comes directly from ‘core’ meaning.

• Other information rides on reasoning about the speakers intentions and what is known in the context.

• Both aspects need computation and logic

… including jokes, lies, poems, etc.

In the beginning there is syntax…

• Sounds are arranged into patterns by syntax

• Words/morphemes are anchored to data points/information structures.

• Sentences have truth conditions that depend ultimately on what words refer to and how they are syntactically arranged

[Compositionality]

Then, there is core meaning(which may not be what you think)

||John||t = j

||snores||t = s, where for any u, s(u) = 1 iff u does whatever it is that amunts to snoring

|| doesn’t ||t = Neg’

|| [John doesn’t snore] ||t =

[Neg’(||snore||t)] (j) = Neg(s(j))

Then, comes use (Thanks to Paul Grice)

It is ultimately least effortful to cooperate, when speaking.

Be relevant

Be truthful

Do not under- or over-inform

Be orderly

Both core meaning and conditions of use are not

‘prescriptions’. We spontaneously conform to them

Layers of meaning

• Who stole the steak?

The dog looks happy

entails recursive semantics

• Some pet looks happy

implicates maxims

• The dog stole the steak

Example 1: Quantity

General effect of quantity

• If IP contains IP’ but doesn’t entail it, the speaker isn’t sure that IP’ is true

• If John is home, he is watchning TV

• The speaker isn’t sure that John is home or that is watching TV

• Mary or Sue will bring the pizza

Etc.

Example 2: inclusive vs. exclusive or.

(a) If we get the money, we’ll hire either Mary or Sue

(b) If we hire either Mary or Sue, we’ll be fine

Against the ambiguity thesis• No language has different words for exclusive vs.

inclusive or.• Cf: bank ambiguous in English but not in Italian

(banca vs. banco)• Piantina ambiguous in I but not in E (small plant vs

map)

Exclusiveness as an implicature: Quantity again

You’ll understand either Jesse or GennaroEntailment: You’ll understand at least one of J and G

Implicature: You won’t understand bothGricean reasoning:The speaker, who is well informed and opinionated, might have said You will understand both J and GHe hasn’t; hence, he doesn’t believe it to be true

Relevance Implicatues

A: Shall we smoke?B: There is a no smoking sign right there.

Implicature: we should not smoke here.

But specific properties of the context matter in a crucial way.

Relevance Implicature

A:Shall we smoke?B: There is a no smoking sign right there.[New context: A and B are members of a civil disobedience group for the liberalization of Cannabis. They are cancer patients who want to get themselves arrested in order to call public attention to the issue.]Implicature: we should smoke here.

The mechanics of relevance impicatures

A:Shall we smoke?B: There is a no smoking sign right there.B is apparently not addressing A’s question. B’s answer seems irrelevant (in violation to what normally happens)But that can’t really be…What can B be meaning, given what we know about the current situation?

The secret of temporal and:Manner Implicatures

John met the love of his life and got marriedImplicature: He first met the love of his life and then got marriedBe orderly!Cancellability: …But not in that order.Compare:John met the love of his life and then got married. But not in that order.

Metaphors

John is a riot John is a lot of funTime flies Time goes by quicklyMetaphors as implicatures: they involve a blatantviolation of quality. We ‘restore’ it by transfering some properties of literal field (e.g. the speed of flying) to the target field (e.g. the flow of time).Metaphors of this kind are ‘frozen’ (perhaps stored in your mental lexicon with their non literal meaning, or something).

Irony

A: Its pouring outside

What a nice day!

B: How was Gennaro’s reading of Leopardi’s

poem?

He got the spelling perfectly.

A is a blatant violation of quality

B is a blatant violation of quantity

The big picture

The multiple meanings of any single expression are computed from

• Reference of non logical words

• The reference of function words

(‘logical component’)

• Truth conditions

• Use conditions (= Gricean maxims)

Implicatures are computed through:

• Literal (‘core’) meaning

• Conditions on use (maxims)

• Contextually salient information

• Background knowledge

• The typical trigger: some apparent violation of the maxims, which we refuse to take as such.

The special power of small words

• Reasoning uses truth functions: and, or, if, not…

• Reasoning can be machine executed• The meaning of logical words in English does match pretty well their logical

meaning… once conditions on use are factored in• Logical words play multiple roles:

– They glue together what we say = Project truth-conditions of atomic sentences to molecular ones

– They are necessary to draw implicatures– They are probably necessary also in the understanding of non logical

words [Test: try describing what snoring is without or’s, and’s, not’s, and if ‘s]

Where we stand

• Knowledge of meaning is knowledge of entailment/implicatures patterns

• We have applied this idea to sentential connectives (a subset of the functional lexicon)

• Result: A recursive, compositional characterization of how truth is channeled through syntactic structure

• A theory of how meanings goes beyond its core through conditions on use (/maxims)

More of the functional lexicon:Quantifiers in natural language.

a. Every man smokes

b. No man is an island

c. The boy is tired

• Det: some, no, the, many, most, no more than three,...• The category Det expresses quantifiers• Quantifiers enable us to formulate general statements• They enable us to talk about quantities• They enable us to say things about individuals

without having to name them

The components of a quantified sentence

IPNP I’

Det N’ every man smokes

Functions vs. sets:i. || man ||t (u) = 1 iff u is a man is tii. || smoke ||t (u) = 1 iff u smokes in tiii. || man ||t (u) = 1 iff u {x: x is a man in t} iv. || smoke ||t (u) = 1 iff u {x: x smokes in t}v. || man ||t ≈ {x: x is a man in t} iv. || smoke ||t ≈ {x: x smokes in t}

every

||every man smokes||t = 1 iff i. all the members of the class of men are also members of the class of smokers (the class of men is a subset of the class of

smokers)ii. {x: x is a man in t } {x: x smokes in t}iii. || man ||t || smokes ||t iv. || every ||t =

How it all clicks together

IPNP I’

Det N’ every man smokes

Functions vs. sets:i. || every||t (X)(Y) = 1 iff {u: X(u) = 1} {u: Y(u) = 1}ii. || every N’ I’||t = || every||t(||N’||t)(||I’||t) = iii. || every man smokes||t = || every||t (m)(s) iv. || every||t (m)(s) = 1 iffv. {u: m(u) = 1} {u: s(u) = 1}iv. {u: u is a man at t} {u: u smokes at t}

some

|| some man smokes||t = T iff• some member of the class of man is also a

member of the class of smokers

• the intersection of the class of men with the class of smokers is non empty

• {x: x is a man in t } {x: x smokes in t} ≠

no

|| no man smokes||t = T iff• no member of the class of man is also a member

of the class of smokers

• the intersection of the class of men with the class of smokers is empty

• {x: x is a man in t } {x: x smokes in t} =

two

|| two men smoke||t = T iff• two members of the class of man are also

members of the class of smokers• the intersection of the class of men with the class

of smokers contains two members• ‘Exactly’ interpretation:

i. |{x: x is a man in t } {x: x smokes in t}| = 2• ‘At least’ interpretation

ii. |{x: x is a man in t } {x: x smokes in t}| ≥ 2|A| = n , where n is the number of members of A

most

|| most men smoke||t = T iff• the majority of the members of the class of man

are also members of the class of smokers

• the intersection of the class of men with the class of smokers contains more than one half of the class of men

• |{x: x is a man in t } {x: x smokes in t}| > 1/2 |{x: x is a man in t }|

many

|| many men smoke||t = T iff• the members of the class of man that are also

members of the class of smokers are ‘many’ (more than n)

• |{x: x is a man in t } {x: x smokes in t}| > n

Q1. Does every man smoke entail many men smoke?

Q2. Does many men smoke entail some men smoke?

few

|| few men smoke||t = T iff• the members of the class of men that are also

members of the class of smokers are ‘few’ (less than n)

• |{x: x is a man in t } {x: x smokes in t}| < n

For any N and any I’, there is no t such that || many N I’||t = || few N I’||t = T

at most two

|| at most two men smoke||t = T

• the members of the class of man that are also members of the class of smokers are equal to or less than 2

• |{x: x is a man in t} {x: x smokes in t}| ≤ 2

the

|| the man smokes||t has a value only if

{x: x is a man in t } is a singleton;

when that happens, then

|| the man smokes||t = T iff

{x: x is a man in t } {x: x smokes in t}

The and presuppositions

• ||the||t (||man||t)(||snores||t) has a value iff

||man||t (u) = 1 for just one u

• Whenever the(X)(Y) is defined,

the(X)(Y) = every(X)(Y)

i.e. true, if X Y; and false if X Y

• IP1 presupposes IP2 =

||IP1||t = 1 or 0 iff ||IP2||t = 1

Further presuppositions of quantifiers

(a) Every cat is purring but there are no cats

(b) Not every cat is purring but there are no cats

(c) Many cats are purringbut there are no cats

(d) Not many cats are meowing because there are

no cats

• Every (A)(B) presupposes There are As

• Many(A)(B) does not.

A host of new predictions

(a) Every cat purrs

(b) Some cat purrs

(c) Not every cat purrs

(d) Not every cat doesn’t purr

(e) No cat purrs

(f) No cat doesn’t purr(a)entails (b) but not viceversa; (a) and (c) entail each other; (b)

and (d) entail each other; (e) entails (d), if there are cats, etc.

Wonders of quantification

• Numbers and quantifiers are part of the functional vocabulary

• They emancipates us from having to name things

• They add more ‘glue’ to our thoughts

• They expose more of the logical structure of language

DP's in object position

a. Lee loves every cat

b. every cat Lee loves

c. every cati [Lee loves ei]

d. every’(cat’, {xi : Lee loves xi })

e. cat’ {xi: Lee loves xi }

The invisible movement hypothesis

IP every’(cat’, {x i : love (l, x i )})

NP i IP love (l, x i)

Det N NP VP

N V NPEvery, cat, cat’

Lee , l loves, love’ e i , x i