Semantic competence: summaryFor any sentence S of our language, we know:

1. What the world would have to be like for it to be true (truth conditions)

2. What it entails

3. What it presupposes

4. What it implicates

These are the semantic counterparts of intuitions about well

formedness; they constitute our primary semantic data.

We want a computationally tractable theory of (1)-(4)

WE WILL MOSTLY FOCUS ON (1), (2) and eventually (4)

Getting started: Truth conditions

Gennaro swims Gennaro is a linguist This is red

Interpretation:- A mapping of expressions into meanings- Meanings: ways of getting from situations of

use/times/contexts to informational values/data points

Gennaro swims Gennaro is a linguist This is red

Gennaro swims Gennaro is a linguist This is red

Composition rules:

i. || [TPNP T’]||t = ||T’||(||NP||)

ii.||Gennaro runs||t = ||runs||t (||Gennaro||t) = r(GC)

which is 1 iff GC runs at t

Truth conditions: a specification of how a

sentence leads us from times/contexts t to

whether that sentence holds in t

Meaning is compositional

TP

NP T’

N T AP

This is A

red||this is red||t = ||is red||t (||this||t) = ||is||t (||red||t)(||this||t) =

what the speaker is pointing at at t is red

||is||t (At) = At

||was||t (At) = At’, where t’ is some t that precedes t

Variations on the semantics of predicates and theories of concepts

The Classical strategy (‘the good’):

||red||t (u) = 1 iff the necessary and sufficient conditions for

being red are met by u. Under this view, the function ||red||t

corresponds to/determines a set { x: ||red||t(x) = 1}

The Fuzzy strategy (‘the bad’):

||red||t (u) = n, where n [1,0] depending how close u is to

focal red

The Supervaluation strategy (‘the ugly’)

||red||t(u) = 1, if u is certainly red, ||red||t(u) = 0, if u is

certainly not red; if u is neither, then ||red||t(u) lacks a value

Comparatives:this is more red than that

• The Fuzzy strategy/ (the bad): r(u) > r(u’)• The Supervaluation strategy (the ugly):

It is impossible to extend r to r+ so as to make

r+(u’) = 1, without also making r+(u) = 1• The Classical strategy (the good)

This is red = rd (u) = u is red to degree d

This is more red than that = the degree d such

that rd(u) = 1 is higher than the degree d’ such

that rd’(u’) = 1

Where we stand

The beginning of a theory of Truth Conditions, in which they depend on the denotation of predicates and nouns.

• Our current model of model of the semantics of predicates:

Functions from times and individuals into truth values (in three variants – the good, the bad, and the ugly)

Three questions that this approach raises:

i. Which predicate is associated with which function?

ii. How does one determine whether ||red||t(u) = 1 (what makes something red/a cat, etc.?)

iii. What does a normally competent speaker know about the red-function?

Our three questions

Referential DPs: t individuals

Predicates: t characteristic functions from

individuals into truth values

i. Which predicate is associated with which function?

This is determined by some historically established link between a phonological form and a particular function, sustained and transmitted by a spontaneous social convention

• This first question is relatively uncontroversial

Two more controversial questions

ii. How do you determine whether ||cat||t(u) = 1?

= What makes something a cat?

To be addressed in terms of the best theory of what cats are (a certain genetic template?); but also a matter of social practices/ecology (when are people willing to call a cat-embryo a cat? When does a dead cat cease to be a cat?)

iii. What does a normally competent speaker know about the cat-function? What concept does the competent speaker associate with being a cat?

Some way of computing a cat-function causally linked to cats and reliable enough for successful communication/survival

Two positions• Externalism: only extra-mental cat-functions matter to language• Internalism: only mind-internal cat-functions matter to language

An internalist needs some story of how concepts are linked to their extra-mental manifestations. Not trivial.

• A classic externalist argument:

Let w be your water-function; applied to some quantity x of clear liquid it returns 1; we then discover that x is not H2O.

Do we still want to say that x is water?

Internalist prediction: yes (x fits with the procedure)

Externalist prediction: no (x is not a sample of what water is causally linked to)

In conclusion, what do we know about the meaning of content words?

• Something about:

- their ‘logical type’

- how to link them up with the appropriate data set in our

environment and/or to the corresponding cognitive

structures

Perhaps this isn’t all that much.• But wait until we get to function words…

(Which is what linguists have been doing anyhow):

- how content words contribute to entailment, presuppositions, etc. and viceversa

Good news: we don’t have to decide between exernalism and internalism to keep going

The power of little words

a. not (doesn’t, isn’t), if, and, or

b. i. John doesn’t smoke

ii. it isn’t true that Lee or Kim smoke

c. If Kim smokes, Lee smokes

d. If Kim smokes and Lee sings, Sue isn’t happy

You will find these words in every language

And their syntax is quite rich…

A highly simplified starting point

con1 TP TP con1 ifa. TP

TP con2 TP con2 and, or

• TP con3 TP con3 not TP

it is not true that TP

Lee smokes

How this syntax will need to be changed:Make it more X’-theoretic

Similarly, for other binary connectives

Mary likes John or Mary likes Bill

TP

TP Con TP

M likes J or M likes BProbably or is a head and the structure is binary:

XP

TP X’

X TP

M likes J or M likes B

Constituent Coordination

Constituent (VP-) negation

Structural properties of connectivesa. Structural ambiguity

i. it is not true that Lee smokes and Kim smokesii. it is not true that Lee smokes and Kim smokesiii.it is not true that Lee smokes and Kim smokesiv. There were…

[[smart women] and men] vs [smart [women and men]]

b. Recursiveness TP

TP con TP TP con TP ........................

[Lee smokes and [Sue smokes and …]

Truth conditions for negative sentences1 0

a. Lexicon: ||not||t = 0 1b. || not TP ||t = || not ||t (|| TP ||t)c. It is not true that Leo smokes = 1 iff Leo smokes = 0Consequencesa. It is not true that it is not true that John is Italianb. John is Italianc. ||[not [not J is Italian]]||t = ||not||t (||[not J is Italian]]||t ) = ||not||(||not||( J is Italian))||t = ||J is Italian||t d. ||[not [not [not J is Italian]]]||t = ||[not [J is Italian]]||t

Double (and triple) negations in real life

A1: Nobody will come

B: I doubt it it = that nobody will come

A2: I don’t [doubt it]

B = I doubt that nobody will come

= (I believe that) it is false that nobody will come

= (I believe that) somebody will come

A2: I don’t doubt that nobody will come

= I do not believe that it is not the case that

nobody will come = nobody will come

All of a sudden: entailment

Cosmetics: VP negation

• Neg’ maps functions from individuals into truths values into their negative counterparts

(concepts negative concepts)• For any t, and any u,

[Neg’(||VP||t)](u) = Neg (||VP||t (u))

(a) John doesn’t run

(b) [||doesn’t||t (||run||t)](j) = [Neg’(||run||t)](j) =

Neg(||run||t(j))

So, ||John doesn’t run||t = ||it is not true that John runs||t

Truth conditions for connectives: conjunction

1 1 11 0 0

||and||t = 0 1 00 0 0

a. || TP1 and TP2 ||t = ||and||t (||TP||t, ||TP||t)

b. ||TP1 and TP2 ||t = 1 iff || TP1 ||t = || TP2 ||t = 1c. i. Leo is Italian and Lee is American

ii. Lee is Americaniii. Lee is American and Leo is Italian(i) entails (ii) and it also entails (iii)

A problem: temporal interpretations of conjunctions

Truth conditions for connectives: disjunction

1 1 1

||or||t = 1 0 1 INCLUSIVE

0 1 1

0 0 0

a. || TP1 or TP2 ||t = ||or||t (||TP1||, || TP2 ||t)

b. || TP1 or TP2 ||t = 1 iff

one of the following conditions holds:

|| TP1||t = 1 and || TP2 ||t = 0

|| TP1||t = 0 and || TP2 ||t = 1

[|| TP1||t = 1 and || TP2 ||t = 1]

John or Bill could lend us the money. Not Paul

Is or ambiguous?

1 1 0

||or||t = 1 0 1 exclusive

0 1 1

0 0 0

a. || TP1 or TP2 ||t = ||or||t (||TP1||, || TP2 ||t)

b. || TP1 or TP2 ||t = 1 iff

one of the following conditions holds:

|| TP1||t = 1 and || TP2 ||t = 0

|| TP1||t = 0 and || TP2 ||t = 1

They hired (either) Mary or Sue [false, if they hired both]

A complex intuition

(a) Mary doesn’t like (both) Sue and Bill

(b)Mary doesn’t like Sue or she doesn’t like Bill (or possibly she doesn’t like either)

(c) It is not true that Mary likes Sue and that she (also) likes Bill

(c) Neg (And (||M likes S||t, ||M likes B||t))

Is (a) [ = (c)] predicted to entail (b)?

Yes!(a)Neg (And (||M likes S||, ||M likes B||)) =

(b) Or (Neg(||M likes S||), Neg(||M likes B||))

Proof that (a) entails (b): [by reductio/contraposition]

For any t, assume ||(b)||t = 0; then Neg(||M likes S||t) = 0 and Neg(||M likes B||t) = 0; if so, ||M likes S||t = 1 and ||M likes B||t = 1. But then, ||(a)||t = 1. So there can’t be any t such that ||(a)||t = 1 and ||(b)||t = 0.

Proof that (a) entails (b): [direct]

For any t, assume ||(a)||t = 1; then And(||M likes S||t, ||M likes B||t)) = 0; if so ||M likes S||t = 0, or ||M likes B||t = 0, or both. But any of these conditions suffices for ||(b)||t = 1 (for if, e.g.,

||M likes S||t = 0, then Neg(||M likes ||t) = 1, etc.)

Truth conditions for connectives: conditionals

Intuition: if A, B is true iff you can rule out with certainty that

A = 1 and B = 0

E.g.: If Lee is happy, Kim is happy

The speaker excludes that Lee is happy and Kim isn’t (though

she may not know whether Lee is in fact happy)

1 1 1

||if||t = 1 0 0

0 1 1

0 0 1

|| if TP1 TP2 ||t = ||if||(||TP1||, ||TP2 ||t ) iff

it is not the case that: || TP1||t = 1 and || TP2 ||t = 0

A calculus of entailmenta. TP1

if TP2 TP3

TP4 TP5

Leo comes or Liz comes we will have fun

b. TP6

if TP7 TP8

Leo comes we will have funb. Assumption ad absurdum: for some t, ||TP1||t = 1 and ||TP6||t = 0• ||TP7||t = 1 and ||TP8||t = 0, from (i) and sem. If

b. ||TP4||t = 1 and ||TP3||t = 0, from (ii) and syntactic identity• ||TP2||t = 1, from (iii) and sem. Or

• ||TP1||t = 0, from (iii),(ii) and sem. If. BUT: this contradicts (i).

Further predictions

(a) John won’t hire Mary or Sue

(b) John wont hire Mary and won’t hire Sue

(c) If John has enough money he will hire Mary and Sue

(d) If John has enough money he will hire Sue

(e) If John is in a bad mood, he is taciturn

(f) If John is not taciturn, he is not in a bad mood

(a) and (b) entail each other; (c) entails (d) but not

viceversa; (e) and (f) entail each other.

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