Meaning as Sense (Frege 1892)

The meaning of an expression is its sense, an

abstract object that determines its reference.

Senses are functions that determine reference.

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Functions: a Reminder

• A function is an abstract (mathematical) object

which, given an appropriate input, specifies a unique

output.

The square function (fsq) outputs a unique num-

ber, given any number as input: fsq(2) = 4, fsq(111)

12,321

Two inputs could be mapped onto the same out-

put: fsq(−2) = 4, fsq(−111) = 12,321

2

Senses as Functions 1

• A sense is a function which, given a particular lo-

cation (a point in space-time), specifies a unique

individual or set as its reference.

A sense is an abstract (mathematical) object, not

a psychological object. Why?

Given a particular location, two expressions with

different senses can have the same reference.

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Senses as Functions 2

The meaning of the expression the POTUS: Given

any location, this sense gives back the unique in-

dividual who at this location holds the office of

president of the USA.

The meaning of the expression GWB: Given any

location, this sense gives back the individual born

in New Haven, CT to George H.W. and Barbara

Bush, the brother of Jeb,...

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Given the location Hallowe’en 2007 in 380-380C,

the reference of the POTUS is identical to the

reference of GWB.

Given the location Hallowe’en 1996 in 380-380C,

the reference of the POTUS is Bill Clinton - the

individual born in 1946 in Hope, AK, the husband

of Hilary...

Given the location Hallowe’en 1996 in 380-380C,

the reference of GWB is George W. Bush - the

individual born in New Haven, ...

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Some expressions, like proper names, have the

same reference at any location.

Other expressions, in fact most expressions, have

a reference that changes from location to loca-

tion.

the POTUS, republican, the World Series cham-

pion, admires,...

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So what is the meaning of a sentence of the form:

NP1 is NP2

for example:

The POTUS is the POTUS.

GWB is the POTUS.

GWB is GWB.

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Proposition: The sense of a declarative sentence

The POTUS is the POTUS:

The proposition that the unique individual who at

that location holds the office of president of the

USA is the unique individual who at this location

holds the office of president of the USA.

GWB is GWB:

The proposition that the individual born in New

Haven, CT to George H.W. and Barbara Bush,...

is the individual born in New Haven, CT to George

H.W. and Barbara Bush,...

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An Insight of Frege’s

The meaning (sense) of a declarative clause is the

relevant proposition.

The reference of a sentence is a truth value (true

or false).

So the meaning of GWB is GWB is a function

that, given any location as input, gives true as

output just in case the individual born in New

Haven, CT to George H.W. and Barbara Bush,

the brother of Jeb,... is the individual born in New

Haven, CT to George H.W. and Barbara Bush, the

brother of Jeb,..., and false otherwise.

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Two Kinds of Proposition

Analytic proposition: a proposition that is al-

ways true, because of its meaning.

Synthetic proposition: a proposition that is true

or false, depending on the way the world is.

The POTUS is GWB.

The proposition that, given any location, gives

true just in case the the unique individual who at

that location holds the office of president of the

USA is the individual born in New Haven, CT to

George H.W. and Barbara Bush, the brother of

Jeb,...

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Truth Conditions

• Two words or sentences ‘mean the same thing’

just in case their truth conditions are the same.

• This term refers to the conditions in the world

necessary to determine whether a proposition is

true or false, i.e. what its truth value is.

• The meaning of The POTUS is the POTUS is

an analytic proposition – a tautology (a proposi-

tion that’s always true).

• The meaning of GWB is the POTUS is a syn-

thetic proposition, one that is sometimes true and

sometimes false.

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Substitution 1: Expressions with the same mean-

ing

The bottle contains water.

The bottle contains H2O.

The gazelle was killed by a lion.

The gazelle was killed by a Panthera leo.

Chris just bought a new car.

Chris just bought a new automobile.

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Substitution 2

Eadem sunt quae mutuo substitui possunt

salva veritate. [G.W. Leibniz]

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Substitution 2: Expressions with the Same Ref-

erence

Eadem sunt quae mutuo substitui possunt salva

veritate. [Leibniz]

Those [expressions that are] identical [in refer-

ence] can be substituted for one another, preserv-

ing truth.

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Substitution 3

• The POTUS got up this morning.

• GWB got up this morning.

• Mark Twain wrote Tom Sawyer.

• Samuel Clemens wrote Tom Sawyer.

• Every creature with a heart has a kidney and vice

versa.

• There’s a creature with a heart in the cellar.

• There’s a creature with a kidney in the cellar.

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Inference 1

• Leibniz’ Law (the principle of substitutivity) is a

rule of inference (A valid inference pattern)

• Goals of semantic theory:

to understand the role of linguistic meaning in

inference (argument validity).

to understand the role of linguistic entailment.

. . .

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Inference 2

• The lexical semantic issues we started with can

be seen in terms of entailment and inference:

Synonyms: ‘Vakie was a cat’ entails ‘Vakie

was a feline’ (and vice versa).

Complementary Antonyms: ‘Pat is alive’ en-

tails ‘Pat is not dead’ (and vice versa).

Hyponyms/Hypernyms: ‘Kim has a car’ en-

tails ‘Kim has a vehicle’ (but not vice versa).

. . .

Entailments and inference can be studied em-

pirically.

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• The sense of a declarative sentence:

a proposition

The proposition that the son of GHWB and BB...

nominated the man born in 1955 in San Antonio,

TX, husband of Rebecca Turner Gonzales...

• The sense of an interrogative sentence:

a question

The question of whether the son of GHWB and

BB... nominated the man born in 1955 in San

Antonio, TX, husband of Rebecca Turner Gonza-

les...

• The sense of an imperative sentence:

a directive

The directive to nominate the man born in 1955

in San Antonio, TX, husband of Rebecca Turner

Gonzales...

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• The president of the US is Texan.

• George W. Bush is Texan.

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• ‘The president of the US’ and ‘George W. Bush’

have different meanings (senses).

• ‘The president of the US’ and ‘George W. Bush’

have the same reference (denotation).

20

S

NP

the N

POTUS

VP

is A

Texan

21

S

NP

PN

GWB

VP

is A

Texan

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‘The POTUS is Texan’ is true at l just in case

the unique individual who is the president of the

US at l is in the set of Texans at l.

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M(the POTUS) = a function that, given a loca-

tion as input, gives as output: the unique presi-

dent of the US at that location.

M(GWB) = a function that, given a location as

input, gives as output: George W. Bush.

M(Texan) = a function that, given a location as

input, gives as output: the set of Texans at that

location.

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The ‘Rule-to-Rule Hypothesis’

• NP → PN

M(NP) = M(PN)

• VP → be A

M(VP) = M(A)

• S → NP VP

M(S) = a function that, given a location as input,

gives the output true just in case the reference of

NP at that location is an element of the reference

of the VP at that location.

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S:true

NP:George...

PN:George...

GWB

VP:the Texans at l

is A:the Texans at l

Texan

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1 + 2 x 3

(1 + 2) x 3

1 + (2 x 3)

(1 + (2 x 3)); (1 + (2 x 3))

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Compositionality

S:9

NP:3

NP:1

1

+:plus NP:2

2

x:times NP:3

3

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Compositionality

S:7

NP:1

1

+:plus NP:6

NP:2

2

x:times NP:3

3

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