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(Pre-)Algebras for Linguistics7. Modelling Meaning and Reference
Carl Pollard
Linguistics 680:Formal Foundations
Autumn 2010
Carl Pollard (Pre-)Algebras for Linguistics
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Expression, Meaning, and Reference
Following Frege (1892), semanticists distinguish betweenthe meaning (or sense) of a linguistic expression and itsreference (or denotation).
We say an expression expresses its meaning, and refersto, or denotes, its reference.
A source of confusion: the terms Frege used were Sinn andBedeutung, usually glossed by German-English dictionariesas ‘sense’ and ‘meaning’.
In general, the reference of an expression can becontingent (depend on how things are), while the meaningis independent of how things are (examples coming soon).
Carl Pollard (Pre-)Algebras for Linguistics
![Page 3: (Pre-)Algebras for Linguisticsis independent of how things are (examples coming soon). Carl Pollard (Pre-)Algebras for Linguistics. Expression, Meaning, and Reference Following Frege](https://reader031.vdocument.in/reader031/viewer/2022041112/5f1c68f6418eef585f42aa89/html5/thumbnails/3.jpg)
Expression, Meaning, and Reference
Following Frege (1892), semanticists distinguish betweenthe meaning (or sense) of a linguistic expression and itsreference (or denotation).
We say an expression expresses its meaning, and refersto, or denotes, its reference.
A source of confusion: the terms Frege used were Sinn andBedeutung, usually glossed by German-English dictionariesas ‘sense’ and ‘meaning’.
In general, the reference of an expression can becontingent (depend on how things are), while the meaningis independent of how things are (examples coming soon).
Carl Pollard (Pre-)Algebras for Linguistics
![Page 4: (Pre-)Algebras for Linguisticsis independent of how things are (examples coming soon). Carl Pollard (Pre-)Algebras for Linguistics. Expression, Meaning, and Reference Following Frege](https://reader031.vdocument.in/reader031/viewer/2022041112/5f1c68f6418eef585f42aa89/html5/thumbnails/4.jpg)
Expression, Meaning, and Reference
Following Frege (1892), semanticists distinguish betweenthe meaning (or sense) of a linguistic expression and itsreference (or denotation).
We say an expression expresses its meaning, and refersto, or denotes, its reference.
A source of confusion: the terms Frege used were Sinn andBedeutung, usually glossed by German-English dictionariesas ‘sense’ and ‘meaning’.
In general, the reference of an expression can becontingent (depend on how things are), while the meaningis independent of how things are (examples coming soon).
Carl Pollard (Pre-)Algebras for Linguistics
![Page 5: (Pre-)Algebras for Linguisticsis independent of how things are (examples coming soon). Carl Pollard (Pre-)Algebras for Linguistics. Expression, Meaning, and Reference Following Frege](https://reader031.vdocument.in/reader031/viewer/2022041112/5f1c68f6418eef585f42aa89/html5/thumbnails/5.jpg)
Expression, Meaning, and Reference
Following Frege (1892), semanticists distinguish betweenthe meaning (or sense) of a linguistic expression and itsreference (or denotation).
We say an expression expresses its meaning, and refersto, or denotes, its reference.
A source of confusion: the terms Frege used were Sinn andBedeutung, usually glossed by German-English dictionariesas ‘sense’ and ‘meaning’.
In general, the reference of an expression can becontingent (depend on how things are), while the meaningis independent of how things are (examples coming soon).
Carl Pollard (Pre-)Algebras for Linguistics
![Page 6: (Pre-)Algebras for Linguisticsis independent of how things are (examples coming soon). Carl Pollard (Pre-)Algebras for Linguistics. Expression, Meaning, and Reference Following Frege](https://reader031.vdocument.in/reader031/viewer/2022041112/5f1c68f6418eef585f42aa89/html5/thumbnails/6.jpg)
Expression vs. Utterance
In this course, we are ignoring the distinction between anexpression and an utterance of an expression.
But that distinction can no longer be ignored when oneexamines the interdependence between the meaning of anexpression and the context in which it is uttered.
This interdependence is the topic of the Winter/Spring2011 Interdisciplinary Seminar on theSyntax/Semantics/Pragmatics Interface (Linguistics 812).
Carl Pollard (Pre-)Algebras for Linguistics
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Expression vs. Utterance
In this course, we are ignoring the distinction between anexpression and an utterance of an expression.
But that distinction can no longer be ignored when oneexamines the interdependence between the meaning of anexpression and the context in which it is uttered.
This interdependence is the topic of the Winter/Spring2011 Interdisciplinary Seminar on theSyntax/Semantics/Pragmatics Interface (Linguistics 812).
Carl Pollard (Pre-)Algebras for Linguistics
![Page 8: (Pre-)Algebras for Linguisticsis independent of how things are (examples coming soon). Carl Pollard (Pre-)Algebras for Linguistics. Expression, Meaning, and Reference Following Frege](https://reader031.vdocument.in/reader031/viewer/2022041112/5f1c68f6418eef585f42aa89/html5/thumbnails/8.jpg)
Expression vs. Utterance
In this course, we are ignoring the distinction between anexpression and an utterance of an expression.
But that distinction can no longer be ignored when oneexamines the interdependence between the meaning of anexpression and the context in which it is uttered.
This interdependence is the topic of the Winter/Spring2011 Interdisciplinary Seminar on theSyntax/Semantics/Pragmatics Interface (Linguistics 812).
Carl Pollard (Pre-)Algebras for Linguistics
![Page 9: (Pre-)Algebras for Linguisticsis independent of how things are (examples coming soon). Carl Pollard (Pre-)Algebras for Linguistics. Expression, Meaning, and Reference Following Frege](https://reader031.vdocument.in/reader031/viewer/2022041112/5f1c68f6418eef585f42aa89/html5/thumbnails/9.jpg)
Examples
The meaning of a declarative sentence is a proposition,while its reference is the truth value of that proposition.
The meaning of a common noun (e.g. donkey) or anintransitivc verb (e.g. brays), is a property, while itsreference is the set of things that have that property.
Names are controversial! Vastly oversimplifying:
Descriptivism (Frege, Russell) the meaning of a name is adescription associated with the name by speakers; thereference is what satisfies the description.Direct Reference Theory (Mill, Kripke) the meaning ofa name is its reference, so names are rigid (their referenceis independent of how things are.)
Carl Pollard (Pre-)Algebras for Linguistics
![Page 10: (Pre-)Algebras for Linguisticsis independent of how things are (examples coming soon). Carl Pollard (Pre-)Algebras for Linguistics. Expression, Meaning, and Reference Following Frege](https://reader031.vdocument.in/reader031/viewer/2022041112/5f1c68f6418eef585f42aa89/html5/thumbnails/10.jpg)
Examples
The meaning of a declarative sentence is a proposition,while its reference is the truth value of that proposition.
The meaning of a common noun (e.g. donkey) or anintransitivc verb (e.g. brays), is a property, while itsreference is the set of things that have that property.
Names are controversial! Vastly oversimplifying:
Descriptivism (Frege, Russell) the meaning of a name is adescription associated with the name by speakers; thereference is what satisfies the description.Direct Reference Theory (Mill, Kripke) the meaning ofa name is its reference, so names are rigid (their referenceis independent of how things are.)
Carl Pollard (Pre-)Algebras for Linguistics
![Page 11: (Pre-)Algebras for Linguisticsis independent of how things are (examples coming soon). Carl Pollard (Pre-)Algebras for Linguistics. Expression, Meaning, and Reference Following Frege](https://reader031.vdocument.in/reader031/viewer/2022041112/5f1c68f6418eef585f42aa89/html5/thumbnails/11.jpg)
Examples
The meaning of a declarative sentence is a proposition,while its reference is the truth value of that proposition.
The meaning of a common noun (e.g. donkey) or anintransitivc verb (e.g. brays), is a property, while itsreference is the set of things that have that property.
Names are controversial! Vastly oversimplifying:
Descriptivism (Frege, Russell) the meaning of a name is adescription associated with the name by speakers; thereference is what satisfies the description.
Direct Reference Theory (Mill, Kripke) the meaning ofa name is its reference, so names are rigid (their referenceis independent of how things are.)
Carl Pollard (Pre-)Algebras for Linguistics
![Page 12: (Pre-)Algebras for Linguisticsis independent of how things are (examples coming soon). Carl Pollard (Pre-)Algebras for Linguistics. Expression, Meaning, and Reference Following Frege](https://reader031.vdocument.in/reader031/viewer/2022041112/5f1c68f6418eef585f42aa89/html5/thumbnails/12.jpg)
Examples
The meaning of a declarative sentence is a proposition,while its reference is the truth value of that proposition.
The meaning of a common noun (e.g. donkey) or anintransitivc verb (e.g. brays), is a property, while itsreference is the set of things that have that property.
Names are controversial! Vastly oversimplifying:
Descriptivism (Frege, Russell) the meaning of a name is adescription associated with the name by speakers; thereference is what satisfies the description.Direct Reference Theory (Mill, Kripke) the meaning ofa name is its reference, so names are rigid (their referenceis independent of how things are.)
Carl Pollard (Pre-)Algebras for Linguistics
![Page 13: (Pre-)Algebras for Linguisticsis independent of how things are (examples coming soon). Carl Pollard (Pre-)Algebras for Linguistics. Expression, Meaning, and Reference Following Frege](https://reader031.vdocument.in/reader031/viewer/2022041112/5f1c68f6418eef585f42aa89/html5/thumbnails/13.jpg)
Dividing the Labor
The grammar of a language not only assigns expressions tosyntactic categories but also specifies their meanings.
Grammar says nothing about reference.
Instead, a separate, nonlinguistic, theory tells how theextension of a meaning depends on how things are.
An expression’s reference is just its meaning’s extension.
So reference also depends on how things are.
Carl Pollard (Pre-)Algebras for Linguistics
![Page 14: (Pre-)Algebras for Linguisticsis independent of how things are (examples coming soon). Carl Pollard (Pre-)Algebras for Linguistics. Expression, Meaning, and Reference Following Frege](https://reader031.vdocument.in/reader031/viewer/2022041112/5f1c68f6418eef585f42aa89/html5/thumbnails/14.jpg)
Dividing the Labor
The grammar of a language not only assigns expressions tosyntactic categories but also specifies their meanings.
Grammar says nothing about reference.
Instead, a separate, nonlinguistic, theory tells how theextension of a meaning depends on how things are.
An expression’s reference is just its meaning’s extension.
So reference also depends on how things are.
Carl Pollard (Pre-)Algebras for Linguistics
![Page 15: (Pre-)Algebras for Linguisticsis independent of how things are (examples coming soon). Carl Pollard (Pre-)Algebras for Linguistics. Expression, Meaning, and Reference Following Frege](https://reader031.vdocument.in/reader031/viewer/2022041112/5f1c68f6418eef585f42aa89/html5/thumbnails/15.jpg)
Dividing the Labor
The grammar of a language not only assigns expressions tosyntactic categories but also specifies their meanings.
Grammar says nothing about reference.
Instead, a separate, nonlinguistic, theory tells how theextension of a meaning depends on how things are.
An expression’s reference is just its meaning’s extension.
So reference also depends on how things are.
Carl Pollard (Pre-)Algebras for Linguistics
![Page 16: (Pre-)Algebras for Linguisticsis independent of how things are (examples coming soon). Carl Pollard (Pre-)Algebras for Linguistics. Expression, Meaning, and Reference Following Frege](https://reader031.vdocument.in/reader031/viewer/2022041112/5f1c68f6418eef585f42aa89/html5/thumbnails/16.jpg)
Dividing the Labor
The grammar of a language not only assigns expressions tosyntactic categories but also specifies their meanings.
Grammar says nothing about reference.
Instead, a separate, nonlinguistic, theory tells how theextension of a meaning depends on how things are.
An expression’s reference is just its meaning’s extension.
So reference also depends on how things are.
Carl Pollard (Pre-)Algebras for Linguistics
![Page 17: (Pre-)Algebras for Linguisticsis independent of how things are (examples coming soon). Carl Pollard (Pre-)Algebras for Linguistics. Expression, Meaning, and Reference Following Frege](https://reader031.vdocument.in/reader031/viewer/2022041112/5f1c68f6418eef585f42aa89/html5/thumbnails/17.jpg)
Dividing the Labor
The grammar of a language not only assigns expressions tosyntactic categories but also specifies their meanings.
Grammar says nothing about reference.
Instead, a separate, nonlinguistic, theory tells how theextension of a meaning depends on how things are.
An expression’s reference is just its meaning’s extension.
So reference also depends on how things are.
Carl Pollard (Pre-)Algebras for Linguistics
![Page 18: (Pre-)Algebras for Linguisticsis independent of how things are (examples coming soon). Carl Pollard (Pre-)Algebras for Linguistics. Expression, Meaning, and Reference Following Frege](https://reader031.vdocument.in/reader031/viewer/2022041112/5f1c68f6418eef585f42aa89/html5/thumbnails/18.jpg)
Review of Propositions
We have a set P of propositions with the entailmentpreorder v and the following operations:
u a glb operation, the meaning of and
→ a residual operation for u, the meaning of implies
t a lub operation, the meaning of or
¬ a complement operation, the meaning of no way
> a top, a necessary truth
⊥ a bottom, a necessary falsehood
Carl Pollard (Pre-)Algebras for Linguistics
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Review of Truth Values (aka Booleans)
There is a boolean algebra with:
a. B = 2 (= {0, 1}) as the underlying set (in this context 1and 0 are usually called t and f respectively)
b. ≤ as the order
c. t and f as top and bottom respectively
d. operations given by the usual truth tables: ∧ (glb), ∨(lub), ⊃ (residual of ∧), and ∼ (complement).
Carl Pollard (Pre-)Algebras for Linguistics
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Review of Truth Values (aka Booleans)
There is a boolean algebra with:
a. B = 2 (= {0, 1}) as the underlying set (in this context 1and 0 are usually called t and f respectively)
b. ≤ as the order
c. t and f as top and bottom respectively
d. operations given by the usual truth tables: ∧ (glb), ∨(lub), ⊃ (residual of ∧), and ∼ (complement).
Carl Pollard (Pre-)Algebras for Linguistics
![Page 21: (Pre-)Algebras for Linguisticsis independent of how things are (examples coming soon). Carl Pollard (Pre-)Algebras for Linguistics. Expression, Meaning, and Reference Following Frege](https://reader031.vdocument.in/reader031/viewer/2022041112/5f1c68f6418eef585f42aa89/html5/thumbnails/21.jpg)
Review of Truth Values (aka Booleans)
There is a boolean algebra with:
a. B = 2 (= {0, 1}) as the underlying set (in this context 1and 0 are usually called t and f respectively)
b. ≤ as the order
c. t and f as top and bottom respectively
d. operations given by the usual truth tables: ∧ (glb), ∨(lub), ⊃ (residual of ∧), and ∼ (complement).
Carl Pollard (Pre-)Algebras for Linguistics
![Page 22: (Pre-)Algebras for Linguisticsis independent of how things are (examples coming soon). Carl Pollard (Pre-)Algebras for Linguistics. Expression, Meaning, and Reference Following Frege](https://reader031.vdocument.in/reader031/viewer/2022041112/5f1c68f6418eef585f42aa89/html5/thumbnails/22.jpg)
Review of Truth Values (aka Booleans)
There is a boolean algebra with:
a. B = 2 (= {0, 1}) as the underlying set (in this context 1and 0 are usually called t and f respectively)
b. ≤ as the order
c. t and f as top and bottom respectively
d. operations given by the usual truth tables: ∧ (glb), ∨(lub), ⊃ (residual of ∧), and ∼ (complement).
Carl Pollard (Pre-)Algebras for Linguistics
![Page 23: (Pre-)Algebras for Linguisticsis independent of how things are (examples coming soon). Carl Pollard (Pre-)Algebras for Linguistics. Expression, Meaning, and Reference Following Frege](https://reader031.vdocument.in/reader031/viewer/2022041112/5f1c68f6418eef585f42aa89/html5/thumbnails/23.jpg)
A Theory of Meanings and Extensions (1/5)
Our theory will use the following sets as building blocks:
P The propositions (things that can be sentence meanings)
B The truth values (things that can be extensions ofpropositions)
I The individuals (things that can be meanings of names).
W The worlds (ultrafilters of propositions)
1 The unit set {0}.It’s conventional to call the member of this set ∗, ratherthan 0, since the important thing about it is that it is asingleton and not what its member is.
Carl Pollard (Pre-)Algebras for Linguistics
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A Theory of Meanings and Extensions (1/5)
Our theory will use the following sets as building blocks:
P The propositions (things that can be sentence meanings)
B The truth values (things that can be extensions ofpropositions)
I The individuals (things that can be meanings of names).
W The worlds (ultrafilters of propositions)
1 The unit set {0}.It’s conventional to call the member of this set ∗, ratherthan 0, since the important thing about it is that it is asingleton and not what its member is.
Carl Pollard (Pre-)Algebras for Linguistics
![Page 25: (Pre-)Algebras for Linguisticsis independent of how things are (examples coming soon). Carl Pollard (Pre-)Algebras for Linguistics. Expression, Meaning, and Reference Following Frege](https://reader031.vdocument.in/reader031/viewer/2022041112/5f1c68f6418eef585f42aa89/html5/thumbnails/25.jpg)
A Theory of Meanings and Extensions (1/5)
Our theory will use the following sets as building blocks:
P The propositions (things that can be sentence meanings)
B The truth values (things that can be extensions ofpropositions)
I The individuals (things that can be meanings of names).
W The worlds (ultrafilters of propositions)
1 The unit set {0}.It’s conventional to call the member of this set ∗, ratherthan 0, since the important thing about it is that it is asingleton and not what its member is.
Carl Pollard (Pre-)Algebras for Linguistics
![Page 26: (Pre-)Algebras for Linguisticsis independent of how things are (examples coming soon). Carl Pollard (Pre-)Algebras for Linguistics. Expression, Meaning, and Reference Following Frege](https://reader031.vdocument.in/reader031/viewer/2022041112/5f1c68f6418eef585f42aa89/html5/thumbnails/26.jpg)
A Theory of Meanings and Extensions (1/5)
Our theory will use the following sets as building blocks:
P The propositions (things that can be sentence meanings)
B The truth values (things that can be extensions ofpropositions)
I The individuals (things that can be meanings of names).
W The worlds (ultrafilters of propositions)
1 The unit set {0}.It’s conventional to call the member of this set ∗, ratherthan 0, since the important thing about it is that it is asingleton and not what its member is.
Carl Pollard (Pre-)Algebras for Linguistics
![Page 27: (Pre-)Algebras for Linguisticsis independent of how things are (examples coming soon). Carl Pollard (Pre-)Algebras for Linguistics. Expression, Meaning, and Reference Following Frege](https://reader031.vdocument.in/reader031/viewer/2022041112/5f1c68f6418eef585f42aa89/html5/thumbnails/27.jpg)
A Theory of Meanings and Extensions (1/5)
Our theory will use the following sets as building blocks:
P The propositions (things that can be sentence meanings)
B The truth values (things that can be extensions ofpropositions)
I The individuals (things that can be meanings of names).
W The worlds (ultrafilters of propositions)
1 The unit set {0}.It’s conventional to call the member of this set ∗, ratherthan 0, since the important thing about it is that it is asingleton and not what its member is.
Carl Pollard (Pre-)Algebras for Linguistics
![Page 28: (Pre-)Algebras for Linguisticsis independent of how things are (examples coming soon). Carl Pollard (Pre-)Algebras for Linguistics. Expression, Meaning, and Reference Following Frege](https://reader031.vdocument.in/reader031/viewer/2022041112/5f1c68f6418eef585f42aa89/html5/thumbnails/28.jpg)
A Theory of Meanings and Extensions (2/5)
The set of semantic domains is defined as follows:
a. P, I, and 1 are semantic domains.
b. If A and B are semantic domains, so is A×B.
c. If A and B are semantic domains, so is A→ B, the set offunctions (arrows) with domain A and codomain B.
d. Nothing else is a semantic domain. (In particular, W andB are not involved in the definition of semantic domains.)
Later we will see that an expression meaning is always amember of a semantic domain (which one depending on thesyntactic category of the expression).
Carl Pollard (Pre-)Algebras for Linguistics
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A Theory of Meanings and Extensions (2/5)
The set of semantic domains is defined as follows:
a. P, I, and 1 are semantic domains.
b. If A and B are semantic domains, so is A×B.
c. If A and B are semantic domains, so is A→ B, the set offunctions (arrows) with domain A and codomain B.
d. Nothing else is a semantic domain. (In particular, W andB are not involved in the definition of semantic domains.)
Later we will see that an expression meaning is always amember of a semantic domain (which one depending on thesyntactic category of the expression).
Carl Pollard (Pre-)Algebras for Linguistics
![Page 30: (Pre-)Algebras for Linguisticsis independent of how things are (examples coming soon). Carl Pollard (Pre-)Algebras for Linguistics. Expression, Meaning, and Reference Following Frege](https://reader031.vdocument.in/reader031/viewer/2022041112/5f1c68f6418eef585f42aa89/html5/thumbnails/30.jpg)
A Theory of Meanings and Extensions (2/5)
The set of semantic domains is defined as follows:
a. P, I, and 1 are semantic domains.
b. If A and B are semantic domains, so is A×B.
c. If A and B are semantic domains, so is A→ B, the set offunctions (arrows) with domain A and codomain B.
d. Nothing else is a semantic domain. (In particular, W andB are not involved in the definition of semantic domains.)
Later we will see that an expression meaning is always amember of a semantic domain (which one depending on thesyntactic category of the expression).
Carl Pollard (Pre-)Algebras for Linguistics
![Page 31: (Pre-)Algebras for Linguisticsis independent of how things are (examples coming soon). Carl Pollard (Pre-)Algebras for Linguistics. Expression, Meaning, and Reference Following Frege](https://reader031.vdocument.in/reader031/viewer/2022041112/5f1c68f6418eef585f42aa89/html5/thumbnails/31.jpg)
A Theory of Meanings and Extensions (2/5)
The set of semantic domains is defined as follows:
a. P, I, and 1 are semantic domains.
b. If A and B are semantic domains, so is A×B.
c. If A and B are semantic domains, so is A→ B, the set offunctions (arrows) with domain A and codomain B.
d. Nothing else is a semantic domain. (In particular, W andB are not involved in the definition of semantic domains.)
Later we will see that an expression meaning is always amember of a semantic domain (which one depending on thesyntactic category of the expression).
Carl Pollard (Pre-)Algebras for Linguistics
![Page 32: (Pre-)Algebras for Linguisticsis independent of how things are (examples coming soon). Carl Pollard (Pre-)Algebras for Linguistics. Expression, Meaning, and Reference Following Frege](https://reader031.vdocument.in/reader031/viewer/2022041112/5f1c68f6418eef585f42aa89/html5/thumbnails/32.jpg)
A Theory of Meanings and Extensions (3/5)
Examples of word meanings:
The meaning of Chiquita will be an individual Chiquita′ ∈ I.
The meaning of the dummy pronoun itd (as in It is obviousthat Chiquita is a donkey) will be ∗ ∈ 1.
The meaning of the common noun donkey will be a functiondonkey′ : I→ P. For each individual i, we think of donkey′(i)as the proposition that i is a donkey.
The meaning of the sentential adverb obviously will be afunction obvious′ : P→ P. For each proposition p, we think ofobvious′(p) as the proposition that p is obvious.
Carl Pollard (Pre-)Algebras for Linguistics
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A Theory of Meanings and Extensions (3/5)
Examples of word meanings:
The meaning of Chiquita will be an individual Chiquita′ ∈ I.
The meaning of the dummy pronoun itd (as in It is obviousthat Chiquita is a donkey) will be ∗ ∈ 1.
The meaning of the common noun donkey will be a functiondonkey′ : I→ P. For each individual i, we think of donkey′(i)as the proposition that i is a donkey.
The meaning of the sentential adverb obviously will be afunction obvious′ : P→ P. For each proposition p, we think ofobvious′(p) as the proposition that p is obvious.
Carl Pollard (Pre-)Algebras for Linguistics
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A Theory of Meanings and Extensions (3/5)
Examples of word meanings:
The meaning of Chiquita will be an individual Chiquita′ ∈ I.
The meaning of the dummy pronoun itd (as in It is obviousthat Chiquita is a donkey) will be ∗ ∈ 1.
The meaning of the common noun donkey will be a functiondonkey′ : I→ P. For each individual i, we think of donkey′(i)as the proposition that i is a donkey.
The meaning of the sentential adverb obviously will be afunction obvious′ : P→ P. For each proposition p, we think ofobvious′(p) as the proposition that p is obvious.
Carl Pollard (Pre-)Algebras for Linguistics
![Page 35: (Pre-)Algebras for Linguisticsis independent of how things are (examples coming soon). Carl Pollard (Pre-)Algebras for Linguistics. Expression, Meaning, and Reference Following Frege](https://reader031.vdocument.in/reader031/viewer/2022041112/5f1c68f6418eef585f42aa89/html5/thumbnails/35.jpg)
A Theory of Meanings and Extensions (3/5)
Examples of word meanings:
The meaning of Chiquita will be an individual Chiquita′ ∈ I.
The meaning of the dummy pronoun itd (as in It is obviousthat Chiquita is a donkey) will be ∗ ∈ 1.
The meaning of the common noun donkey will be a functiondonkey′ : I→ P. For each individual i, we think of donkey′(i)as the proposition that i is a donkey.
The meaning of the sentential adverb obviously will be afunction obvious′ : P→ P. For each proposition p, we think ofobvious′(p) as the proposition that p is obvious.
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A Theory of Meanings and Extensions (4/5)
For each semantic domain, there is a corresponding extensiondomain, recursively defined as follows:
a. Ext(I) = I.
b. Ext(1) = 1.
c. Ext(P) = B.
d. If A and B are semantic domains, thenExt(A×B) = Ext(A)× Ext(B).
e. If A and B are semantic domains, thenExt(A→ B) = A→ Ext(B).
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A Theory of Meanings and Extensions (4/5)
For each semantic domain, there is a corresponding extensiondomain, recursively defined as follows:
a. Ext(I) = I.
b. Ext(1) = 1.
c. Ext(P) = B.
d. If A and B are semantic domains, thenExt(A×B) = Ext(A)× Ext(B).
e. If A and B are semantic domains, thenExt(A→ B) = A→ Ext(B).
Carl Pollard (Pre-)Algebras for Linguistics
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A Theory of Meanings and Extensions (4/5)
For each semantic domain, there is a corresponding extensiondomain, recursively defined as follows:
a. Ext(I) = I.
b. Ext(1) = 1.
c. Ext(P) = B.
d. If A and B are semantic domains, thenExt(A×B) = Ext(A)× Ext(B).
e. If A and B are semantic domains, thenExt(A→ B) = A→ Ext(B).
Carl Pollard (Pre-)Algebras for Linguistics
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A Theory of Meanings and Extensions (4/5)
For each semantic domain, there is a corresponding extensiondomain, recursively defined as follows:
a. Ext(I) = I.
b. Ext(1) = 1.
c. Ext(P) = B.
d. If A and B are semantic domains, thenExt(A×B) = Ext(A)× Ext(B).
e. If A and B are semantic domains, thenExt(A→ B) = A→ Ext(B).
Carl Pollard (Pre-)Algebras for Linguistics
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A Theory of Meanings and Extensions (4/5)
For each semantic domain, there is a corresponding extensiondomain, recursively defined as follows:
a. Ext(I) = I.
b. Ext(1) = 1.
c. Ext(P) = B.
d. If A and B are semantic domains, thenExt(A×B) = Ext(A)× Ext(B).
e. If A and B are semantic domains, thenExt(A→ B) = A→ Ext(B).
Carl Pollard (Pre-)Algebras for Linguistics
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A Theory of Meanings and Extensions (5/5)
We recursively define, for each semantic domain A, a functionextA : (A×W)→ Ext(A).
We abbreviate Ext(a,w) as a@w, read ‘the extension of a at w’.
a. For all w ∈W, ∗@w = ∗b. For all i ∈ I and w ∈W, i@w = i.
c. For all p ∈ P and w ∈W, p@w is t if p ∈ w and f otherwise.
d. For all a ∈ A, b ∈ B, and w ∈W , 〈a, b〉@w = 〈a@w, b@w〉.e. For all f ∈ A→ B and w ∈W, f@w is the function from A
to Ext(B) that maps each a ∈ A to f(a)@w.
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A Theory of Meanings and Extensions (5/5)
We recursively define, for each semantic domain A, a functionextA : (A×W)→ Ext(A).
We abbreviate Ext(a,w) as a@w, read ‘the extension of a at w’.
a. For all w ∈W, ∗@w = ∗b. For all i ∈ I and w ∈W, i@w = i.
c. For all p ∈ P and w ∈W, p@w is t if p ∈ w and f otherwise.
d. For all a ∈ A, b ∈ B, and w ∈W , 〈a, b〉@w = 〈a@w, b@w〉.e. For all f ∈ A→ B and w ∈W, f@w is the function from A
to Ext(B) that maps each a ∈ A to f(a)@w.
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A Theory of Meanings and Extensions (5/5)
We recursively define, for each semantic domain A, a functionextA : (A×W)→ Ext(A).
We abbreviate Ext(a,w) as a@w, read ‘the extension of a at w’.
a. For all w ∈W, ∗@w = ∗
b. For all i ∈ I and w ∈W, i@w = i.
c. For all p ∈ P and w ∈W, p@w is t if p ∈ w and f otherwise.
d. For all a ∈ A, b ∈ B, and w ∈W , 〈a, b〉@w = 〈a@w, b@w〉.e. For all f ∈ A→ B and w ∈W, f@w is the function from A
to Ext(B) that maps each a ∈ A to f(a)@w.
Carl Pollard (Pre-)Algebras for Linguistics
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A Theory of Meanings and Extensions (5/5)
We recursively define, for each semantic domain A, a functionextA : (A×W)→ Ext(A).
We abbreviate Ext(a,w) as a@w, read ‘the extension of a at w’.
a. For all w ∈W, ∗@w = ∗b. For all i ∈ I and w ∈W, i@w = i.
c. For all p ∈ P and w ∈W, p@w is t if p ∈ w and f otherwise.
d. For all a ∈ A, b ∈ B, and w ∈W , 〈a, b〉@w = 〈a@w, b@w〉.e. For all f ∈ A→ B and w ∈W, f@w is the function from A
to Ext(B) that maps each a ∈ A to f(a)@w.
Carl Pollard (Pre-)Algebras for Linguistics
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A Theory of Meanings and Extensions (5/5)
We recursively define, for each semantic domain A, a functionextA : (A×W)→ Ext(A).
We abbreviate Ext(a,w) as a@w, read ‘the extension of a at w’.
a. For all w ∈W, ∗@w = ∗b. For all i ∈ I and w ∈W, i@w = i.
c. For all p ∈ P and w ∈W, p@w is t if p ∈ w and f otherwise.
d. For all a ∈ A, b ∈ B, and w ∈W , 〈a, b〉@w = 〈a@w, b@w〉.e. For all f ∈ A→ B and w ∈W, f@w is the function from A
to Ext(B) that maps each a ∈ A to f(a)@w.
Carl Pollard (Pre-)Algebras for Linguistics
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A Theory of Meanings and Extensions (5/5)
We recursively define, for each semantic domain A, a functionextA : (A×W)→ Ext(A).
We abbreviate Ext(a,w) as a@w, read ‘the extension of a at w’.
a. For all w ∈W, ∗@w = ∗b. For all i ∈ I and w ∈W, i@w = i.
c. For all p ∈ P and w ∈W, p@w is t if p ∈ w and f otherwise.
d. For all a ∈ A, b ∈ B, and w ∈W , 〈a, b〉@w = 〈a@w, b@w〉.
e. For all f ∈ A→ B and w ∈W, f@w is the function from Ato Ext(B) that maps each a ∈ A to f(a)@w.
Carl Pollard (Pre-)Algebras for Linguistics
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A Theory of Meanings and Extensions (5/5)
We recursively define, for each semantic domain A, a functionextA : (A×W)→ Ext(A).
We abbreviate Ext(a,w) as a@w, read ‘the extension of a at w’.
a. For all w ∈W, ∗@w = ∗b. For all i ∈ I and w ∈W, i@w = i.
c. For all p ∈ P and w ∈W, p@w is t if p ∈ w and f otherwise.
d. For all a ∈ A, b ∈ B, and w ∈W , 〈a, b〉@w = 〈a@w, b@w〉.e. For all f ∈ A→ B and w ∈W, f@w is the function from A
to Ext(B) that maps each a ∈ A to f(a)@w.
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The Reference of an Expression at a World
We define the reference of an expression e at a world w to bethe extension at w of e’s meaning.
The assignment of meanings to expressions is done by thegrammar (next lecture).
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The Reference of an Expression at a World
We define the reference of an expression e at a world w to bethe extension at w of e’s meaning.
The assignment of meanings to expressions is done by thegrammar (next lecture).
Carl Pollard (Pre-)Algebras for Linguistics
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Examples of Word Reference
At any world w, the reference at w of:
Chiquita is Chiquita′ (cf. the direct reference theory ofnames)
itd is ∗ (vacuous reference)
donkey is the function from individuals to truth values thatmaps each individual i to t if the proposition donkey′(i) is inw, and to f otherwise. (Informally speaking, this is (thecharacteristic function of) the set of individuals that aredonkeys at w.)
obviously is the function from propositions to truth valuesthat maps each proposition p to t if the propositionobvious′(p) is in w, and to f otherwise. (Informally speaking,this is (the characteristic function of) the set of all propositionswhich are obvious at w.)
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Examples of Word Reference
At any world w, the reference at w of:
Chiquita is Chiquita′ (cf. the direct reference theory ofnames)
itd is ∗ (vacuous reference)
donkey is the function from individuals to truth values thatmaps each individual i to t if the proposition donkey′(i) is inw, and to f otherwise. (Informally speaking, this is (thecharacteristic function of) the set of individuals that aredonkeys at w.)
obviously is the function from propositions to truth valuesthat maps each proposition p to t if the propositionobvious′(p) is in w, and to f otherwise. (Informally speaking,this is (the characteristic function of) the set of all propositionswhich are obvious at w.)
Carl Pollard (Pre-)Algebras for Linguistics
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Examples of Word Reference
At any world w, the reference at w of:
Chiquita is Chiquita′ (cf. the direct reference theory ofnames)
itd is ∗ (vacuous reference)
donkey is the function from individuals to truth values thatmaps each individual i to t if the proposition donkey′(i) is inw, and to f otherwise. (Informally speaking, this is (thecharacteristic function of) the set of individuals that aredonkeys at w.)
obviously is the function from propositions to truth valuesthat maps each proposition p to t if the propositionobvious′(p) is in w, and to f otherwise. (Informally speaking,this is (the characteristic function of) the set of all propositionswhich are obvious at w.)
Carl Pollard (Pre-)Algebras for Linguistics
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Examples of Word Reference
At any world w, the reference at w of:
Chiquita is Chiquita′ (cf. the direct reference theory ofnames)
itd is ∗ (vacuous reference)
donkey is the function from individuals to truth values thatmaps each individual i to t if the proposition donkey′(i) is inw, and to f otherwise. (Informally speaking, this is (thecharacteristic function of) the set of individuals that aredonkeys at w.)
obviously is the function from propositions to truth valuesthat maps each proposition p to t if the propositionobvious′(p) is in w, and to f otherwise. (Informally speaking,this is (the characteristic function of) the set of all propositionswhich are obvious at w.)
Carl Pollard (Pre-)Algebras for Linguistics
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Properties and their Extensions
For any semantic domain A, the functions in A→ P are calledA-properties.
Since Ext(A→ P) = A→ B, the extension (at any world) ofan A-property is (the characteristic function of) a set of A’s.
Example: the extension of an individual property(e.g. donkey′) is (the characteristic function of) a set ofindividuals.
Example: the extension of a property of propositions(e.g.obvious′) is (the characteristic function of) a set ofpropositions.
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Properties and their Extensions
For any semantic domain A, the functions in A→ P are calledA-properties.
Since Ext(A→ P) = A→ B, the extension (at any world) ofan A-property is (the characteristic function of) a set of A’s.
Example: the extension of an individual property(e.g. donkey′) is (the characteristic function of) a set ofindividuals.
Example: the extension of a property of propositions(e.g.obvious′) is (the characteristic function of) a set ofpropositions.
Carl Pollard (Pre-)Algebras for Linguistics
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Properties and their Extensions
For any semantic domain A, the functions in A→ P are calledA-properties.
Since Ext(A→ P) = A→ B, the extension (at any world) ofan A-property is (the characteristic function of) a set of A’s.
Example: the extension of an individual property(e.g. donkey′) is (the characteristic function of) a set ofindividuals.
Example: the extension of a property of propositions(e.g.obvious′) is (the characteristic function of) a set ofpropositions.
Carl Pollard (Pre-)Algebras for Linguistics
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Properties and their Extensions
For any semantic domain A, the functions in A→ P are calledA-properties.
Since Ext(A→ P) = A→ B, the extension (at any world) ofan A-property is (the characteristic function of) a set of A’s.
Example: the extension of an individual property(e.g. donkey′) is (the characteristic function of) a set ofindividuals.
Example: the extension of a property of propositions(e.g.obvious′) is (the characteristic function of) a set ofpropositions.
Carl Pollard (Pre-)Algebras for Linguistics