Compositionality Tutorial1 The Basic Idea

Dag WesterstahlStockholm University and University of Gothenburg

Tsinghua University, BeijingJune 17, 2011

Dag Westerstahl () Compositionality 1Tsinghua University, Beijing June 17, 2011 1

/ 23

An old idea

An old idea

Very roughly:

Sentences have structure, they are composed of parts.

The meanings of sentences are composed in a similar way.

This is an essential feature of language.

1 of 23

An old idea

An old idea

Very roughly:

Sentences have structure, they are composed of parts.

The meanings of sentences are composed in a similar way.

This is an essential feature of language.

1 of 23

An old idea

An old idea

Very roughly:

Sentences have structure, they are composed of parts.

The meanings of sentences are composed in a similar way.

This is an essential feature of language.

1 of 23

An old idea

An old idea

Very roughly:

Sentences have structure, they are composed of parts.

The meanings of sentences are composed in a similar way.

This is an essential feature of language.

1 of 23

An old idea

Plato

In Plato’s Sophist, an argument from a sophist is presented, according towhich it is impossible to say anything false: for if you say something false,you attempt to speak of not-being, but nothing can be said about not-being.

The Eleatic Stranger replies that a sentence is made up of a noun and averb. It is true if the predicate (verb) attributes to what the subject (noun)signifies ‘things that are’ (actions the subject is performing), e.g.

(1) Theaetetus is sitting.

(since Theaetetus is currently sitting).

2 of 23

An old idea

Plato

In Plato’s Sophist, an argument from a sophist is presented, according towhich it is impossible to say anything false: for if you say something false,you attempt to speak of not-being, but nothing can be said about not-being.

The Eleatic Stranger replies that a sentence is made up of a noun and averb. It is true if the predicate (verb) attributes to what the subject (noun)signifies ‘things that are’ (actions the subject is performing), e.g.

(1) Theaetetus is sitting.

(since Theaetetus is currently sitting).

2 of 23

An old idea

Plato, cont.

It is false if the predicate attributes to what the subject signifies ‘things thatare not’ (actions that are different from what the subject is doing), e.g.

(2) Theaetetus is flying.

(So, the Stranger implies, there is nothing mysterious about sayingsomething false.)

Thus, sentences have structure, and their parts perform different functions.

So the truth value of the sentence is a function of what the predicateattributes and the noun signifies. When these are taken as meanings ofsentences, verbs, and nouns, we have a compositional analysis.

3 of 23

An old idea

Plato, cont.

It is false if the predicate attributes to what the subject signifies ‘things thatare not’ (actions that are different from what the subject is doing), e.g.

(2) Theaetetus is flying.

(So, the Stranger implies, there is nothing mysterious about sayingsomething false.)

Thus, sentences have structure, and their parts perform different functions.

So the truth value of the sentence is a function of what the predicateattributes and the noun signifies. When these are taken as meanings ofsentences, verbs, and nouns, we have a compositional analysis.

3 of 23

An old idea

Plato, cont.

It is false if the predicate attributes to what the subject signifies ‘things thatare not’ (actions that are different from what the subject is doing), e.g.

(2) Theaetetus is flying.

(So, the Stranger implies, there is nothing mysterious about sayingsomething false.)

Thus, sentences have structure, and their parts perform different functions.

So the truth value of the sentence is a function of what the predicateattributes and the noun signifies. When these are taken as meanings ofsentences, verbs, and nouns, we have a compositional analysis.

3 of 23

An old idea

Plato, cont.

It is false if the predicate attributes to what the subject signifies ‘things thatare not’ (actions that are different from what the subject is doing), e.g.

(2) Theaetetus is flying.

(So, the Stranger implies, there is nothing mysterious about sayingsomething false.)

Thus, sentences have structure, and their parts perform different functions.

So the truth value of the sentence is a function of what the predicateattributes and the noun signifies. When these are taken as meanings ofsentences, verbs, and nouns, we have a compositional analysis.

3 of 23

An old idea

Al-Farabı

Al-Farabı, ca. 870–950 (from Hodges):

“. . . the imitation of the composition of meanings by thecomposition of expressions is by [linguistic] convention . . .”

Here meanings are thoughts or ideas in the mind.

For Al-Farabı, linguistic expressions reflect (imitate) thoughts.

4 of 23

An old idea

Al-Farabı

Al-Farabı, ca. 870–950 (from Hodges):

“. . . the imitation of the composition of meanings by thecomposition of expressions is by [linguistic] convention . . .”

Here meanings are thoughts or ideas in the mind.

For Al-Farabı, linguistic expressions reflect (imitate) thoughts.

4 of 23

An old idea

Al-Farabı

Al-Farabı, ca. 870–950 (from Hodges):

“. . . the imitation of the composition of meanings by thecomposition of expressions is by [linguistic] convention . . .”

Here meanings are thoughts or ideas in the mind.

For Al-Farabı, linguistic expressions reflect (imitate) thoughts.

4 of 23

An old idea

Abelard

The first to propose a general principle of this nature in the Westerntradition seems to have been Peter Abelard in the first half of the 12th century:

Just as a sentence materially consists in a noun and a verb, so toothe understanding of it is put together from the understandings ofits parts.

(Logica ‘ingredientibus’. 3. Commentary on Aristoteles DeInterpretatione, transl. Peter King)

5 of 23

An old idea

Buridan

The Additive Principle, which was common in late middle ages, is morespecific, e.g. in the formulation by John Buridan:

The signification of a complex expression is the sum of thesignification of its non-logical terms.

In 1372, Pierre d’Ailly refers to the common view that it “belongs to the[very] notion of an expression that every expression has parts each one ofwhich, when separated, signifies something of what is signified by thewhole”.

Still, ‘adding’ is a too unspecific way to put together meanings: it doesn’tdistinguish between e.g. John likes Mary and Mary likes John.

6 of 23

An old idea

Buridan

The Additive Principle, which was common in late middle ages, is morespecific, e.g. in the formulation by John Buridan:

The signification of a complex expression is the sum of thesignification of its non-logical terms.

In 1372, Pierre d’Ailly refers to the common view that it “belongs to the[very] notion of an expression that every expression has parts each one ofwhich, when separated, signifies something of what is signified by thewhole”.

Still, ‘adding’ is a too unspecific way to put together meanings: it doesn’tdistinguish between e.g. John likes Mary and Mary likes John.

6 of 23

An old idea

Buridan

The Additive Principle, which was common in late middle ages, is morespecific, e.g. in the formulation by John Buridan:

The signification of a complex expression is the sum of thesignification of its non-logical terms.

In 1372, Pierre d’Ailly refers to the common view that it “belongs to the[very] notion of an expression that every expression has parts each one ofwhich, when separated, signifies something of what is signified by thewhole”.

Still, ‘adding’ is a too unspecific way to put together meanings: it doesn’tdistinguish between e.g. John likes Mary and Mary likes John.

6 of 23

An old idea

Making it precise

So far, the notions used in talking about (what we may now call)compositionality were

a part-whole relation, among linguistic expressions and amongconcepts/ideasan implicit notion of correspondence between the language level andthe meaning level (usually thought of as a mental level)

What is needed to make this more precise, and less metaphorical?

One way to make the structure of linguistic items precise is to usetools from algebra.

7 of 23

An old idea

Making it precise

So far, the notions used in talking about (what we may now call)compositionality were

a part-whole relation, among linguistic expressions and amongconcepts/ideasan implicit notion of correspondence between the language level andthe meaning level (usually thought of as a mental level)

What is needed to make this more precise, and less metaphorical?

One way to make the structure of linguistic items precise is to usetools from algebra.

7 of 23

An old idea

Making it precise

So far, the notions used in talking about (what we may now call)compositionality were

a part-whole relation, among linguistic expressions and amongconcepts/ideasan implicit notion of correspondence between the language level andthe meaning level (usually thought of as a mental level)

What is needed to make this more precise, and less metaphorical?

One way to make the structure of linguistic items precise is to usetools from algebra.

7 of 23

An old idea

Leibniz

Leibniz was perhaps the first to suggest an algebraic treatment of humanconcepts (when constructing his universal language).

He thought of the task as combinatorial: all concepts are combinations of asmall number of simple ideas.

The simple ideas can be coded as prime numbers and concepts as products:with 2 coding animal and 3 rational, man would be coded by 6: man =animal × rational (On Universal Synthesis and Analysis).

This is a compositional code, or translation (not of sentences so far, but ofconcept words).

It is also inversely compositional.

8 of 23

An old idea

Leibniz

Leibniz was perhaps the first to suggest an algebraic treatment of humanconcepts (when constructing his universal language).

He thought of the task as combinatorial: all concepts are combinations of asmall number of simple ideas.

The simple ideas can be coded as prime numbers and concepts as products:with 2 coding animal and 3 rational, man would be coded by 6: man =animal × rational (On Universal Synthesis and Analysis).

This is a compositional code, or translation (not of sentences so far, but ofconcept words).

It is also inversely compositional.

8 of 23

An old idea

Leibniz

Leibniz was perhaps the first to suggest an algebraic treatment of humanconcepts (when constructing his universal language).

He thought of the task as combinatorial: all concepts are combinations of asmall number of simple ideas.

The simple ideas can be coded as prime numbers and concepts as products:with 2 coding animal and 3 rational, man would be coded by 6: man =animal × rational (On Universal Synthesis and Analysis).

This is a compositional code, or translation (not of sentences so far, but ofconcept words).

It is also inversely compositional.

8 of 23

An old idea

Leibniz

Leibniz was perhaps the first to suggest an algebraic treatment of humanconcepts (when constructing his universal language).

He thought of the task as combinatorial: all concepts are combinations of asmall number of simple ideas.

The simple ideas can be coded as prime numbers and concepts as products:with 2 coding animal and 3 rational, man would be coded by 6: man =animal × rational (On Universal Synthesis and Analysis).

This is a compositional code, or translation (not of sentences so far, but ofconcept words).

It is also inversely compositional.

8 of 23

An old idea

Leibniz

Leibniz was perhaps the first to suggest an algebraic treatment of humanconcepts (when constructing his universal language).

He thought of the task as combinatorial: all concepts are combinations of asmall number of simple ideas.

The simple ideas can be coded as prime numbers and concepts as products:with 2 coding animal and 3 rational, man would be coded by 6: man =animal × rational (On Universal Synthesis and Analysis).

This is a compositional code, or translation (not of sentences so far, but ofconcept words).

It is also inversely compositional.8 of 23

An old idea

Functions

The second essential ingredient in a precise idea of compositionality isthe notion of a function; the assignment of values (‘meanings’) tolinguistic items.

Frege was the first to employ a mathematical notion of function whentalking about concepts and language. (Concepts for Frege arefunctions, from objects to truth values.)

But although he also can be said to be the first to express a modernidea of compositionality, he didn’t use functions for thecorrespondence between language and meanings.

His first formulation uses substitution, and then one doesn’t need tomention the correspondence.

His second and most explicit formulation relies on an informal idea ofcorrespondence, or isomorphism.

9 of 23

An old idea

Functions

The second essential ingredient in a precise idea of compositionality isthe notion of a function; the assignment of values (‘meanings’) tolinguistic items.

Frege was the first to employ a mathematical notion of function whentalking about concepts and language. (Concepts for Frege arefunctions, from objects to truth values.)

But although he also can be said to be the first to express a modernidea of compositionality, he didn’t use functions for thecorrespondence between language and meanings.

His first formulation uses substitution, and then one doesn’t need tomention the correspondence.

His second and most explicit formulation relies on an informal idea ofcorrespondence, or isomorphism.

9 of 23

An old idea

Functions

The second essential ingredient in a precise idea of compositionality isthe notion of a function; the assignment of values (‘meanings’) tolinguistic items.

Frege was the first to employ a mathematical notion of function whentalking about concepts and language. (Concepts for Frege arefunctions, from objects to truth values.)

But although he also can be said to be the first to express a modernidea of compositionality, he didn’t use functions for thecorrespondence between language and meanings.

His first formulation uses substitution, and then one doesn’t need tomention the correspondence.

His second and most explicit formulation relies on an informal idea ofcorrespondence, or isomorphism.

9 of 23

An old idea

Functions

The second essential ingredient in a precise idea of compositionality isthe notion of a function; the assignment of values (‘meanings’) tolinguistic items.

Frege was the first to employ a mathematical notion of function whentalking about concepts and language. (Concepts for Frege arefunctions, from objects to truth values.)

But although he also can be said to be the first to express a modernidea of compositionality, he didn’t use functions for thecorrespondence between language and meanings.

His first formulation uses substitution, and then one doesn’t need tomention the correspondence.

His second and most explicit formulation relies on an informal idea ofcorrespondence, or isomorphism.

9 of 23

An old idea

Functions

The second essential ingredient in a precise idea of compositionality isthe notion of a function; the assignment of values (‘meanings’) tolinguistic items.

Frege was the first to employ a mathematical notion of function whentalking about concepts and language. (Concepts for Frege arefunctions, from objects to truth values.)

But although he also can be said to be the first to express a modernidea of compositionality, he didn’t use functions for thecorrespondence between language and meanings.

His first formulation uses substitution, and then one doesn’t need tomention the correspondence.

His second and most explicit formulation relies on an informal idea ofcorrespondence, or isomorphism.

9 of 23

Modern versions

Frege 1

Let us assume for the time being that the sentence has areference. If we now replace one word of the sentence by anotherhaving the same reference, this can have no bearing upon thereference of the sentence. (Frege, ‘On sense and reference’, 1892)

This is (a special case of) the substitution version of compositionality. Notethat the values here are Bedeutungen (referents), such as truth values (forsentences) and individual objects (for individual-denoting terms).

Frege is often read as having implicitly endorsed a similar principle for Sinn(sense) in that paper.

10 of 23

Modern versions

Frege 1

Let us assume for the time being that the sentence has areference. If we now replace one word of the sentence by anotherhaving the same reference, this can have no bearing upon thereference of the sentence. (Frege, ‘On sense and reference’, 1892)

This is (a special case of) the substitution version of compositionality. Notethat the values here are Bedeutungen (referents), such as truth values (forsentences) and individual objects (for individual-denoting terms).

Frege is often read as having implicitly endorsed a similar principle for Sinn(sense) in that paper.

10 of 23

Modern versions

Frege 1

Let us assume for the time being that the sentence has areference. If we now replace one word of the sentence by anotherhaving the same reference, this can have no bearing upon thereference of the sentence. (Frege, ‘On sense and reference’, 1892)

This is (a special case of) the substitution version of compositionality. Notethat the values here are Bedeutungen (referents), such as truth values (forsentences) and individual objects (for individual-denoting terms).

Frege is often read as having implicitly endorsed a similar principle for Sinn(sense) in that paper.

10 of 23

Modern versions

Frege 2

A famous quote:

It is astonishing what language can do. With a few syllables it canexpress an incalculable number of thoughts, so that even athought grasped by a terrestrial being for the very first time canbe put into a form of words which will be understood bysomebody to whom the thought is entirely new. This would beimpossible, were we not able to distinguish parts in the thoughtcorresponding to the parts of a sentence, so that the structure ofthe sentence serves as an image of the structure of the thought.(Frege, ‘Compund thoughts’, 1923, opening paragraph)

Many things are notable about this passage:

Frege talks about communication, not only understanding.

He considers both the task for the speaker to find a suitable linguisticexpression to convey her Thought, and the task for the hearer to findthe appropriate Thought that corresponds to the expression uttered.

11 of 23

Modern versions

Frege 2

A famous quote:

It is astonishing what language can do. With a few syllables it canexpress an incalculable number of thoughts, so that even athought grasped by a terrestrial being for the very first time canbe put into a form of words which will be understood bysomebody to whom the thought is entirely new. This would beimpossible, were we not able to distinguish parts in the thoughtcorresponding to the parts of a sentence, so that the structure ofthe sentence serves as an image of the structure of the thought.(Frege, ‘Compund thoughts’, 1923, opening paragraph)

Many things are notable about this passage:

Frege talks about communication, not only understanding.

He considers both the task for the speaker to find a suitable linguisticexpression to convey her Thought, and the task for the hearer to findthe appropriate Thought that corresponds to the expression uttered.

11 of 23

Modern versions

Frege 2

A famous quote:

It is astonishing what language can do. With a few syllables it canexpress an incalculable number of thoughts, so that even athought grasped by a terrestrial being for the very first time canbe put into a form of words which will be understood bysomebody to whom the thought is entirely new. This would beimpossible, were we not able to distinguish parts in the thoughtcorresponding to the parts of a sentence, so that the structure ofthe sentence serves as an image of the structure of the thought.(Frege, ‘Compund thoughts’, 1923, opening paragraph)

Many things are notable about this passage:

Frege talks about communication, not only understanding.

He considers both the task for the speaker to find a suitable linguisticexpression to convey her Thought, and the task for the hearer to findthe appropriate Thought that corresponds to the expression uttered.

11 of 23

Modern versions

Frege 2

A famous quote:

It is astonishing what language can do. With a few syllables it canexpress an incalculable number of thoughts, so that even athought grasped by a terrestrial being for the very first time canbe put into a form of words which will be understood bysomebody to whom the thought is entirely new. This would beimpossible, were we not able to distinguish parts in the thoughtcorresponding to the parts of a sentence, so that the structure ofthe sentence serves as an image of the structure of the thought.(Frege, ‘Compund thoughts’, 1923, opening paragraph)

Many things are notable about this passage:

Frege talks about communication, not only understanding.

He considers both the task for the speaker to find a suitable linguisticexpression to convey her Thought, and the task for the hearer to findthe appropriate Thought that corresponds to the expression uttered.

11 of 23

Modern versions

Frege 2, cont.

Notable things cont.

That the semantics is compositional may then be a help for the hearerto find the appropriate content.

But what should the speaker do? Maybe go through the list of allsentences, interpret each one in turn, until she finds the one whoseinterpretation corresponds to what she wants to convey.For more efficiency we seem to need inverse compositionality: theexpression of a complex content is a function of the expressions of itsparts and the way they are combined.Frege seems to postulate an isomorphism between the structure oflanguage and the structure of thought (perhaps preserving the relationbeing an immediate part of).Frege not only describes compositionality, he offers an argument fromlinguistic communication that it holds, and that it is an essentialfeature of language.

12 of 23

Modern versions

Frege 2, cont.

Notable things cont.

That the semantics is compositional may then be a help for the hearerto find the appropriate content.But what should the speaker do? Maybe go through the list of allsentences, interpret each one in turn, until she finds the one whoseinterpretation corresponds to what she wants to convey.

For more efficiency we seem to need inverse compositionality: theexpression of a complex content is a function of the expressions of itsparts and the way they are combined.Frege seems to postulate an isomorphism between the structure oflanguage and the structure of thought (perhaps preserving the relationbeing an immediate part of).Frege not only describes compositionality, he offers an argument fromlinguistic communication that it holds, and that it is an essentialfeature of language.

12 of 23

Modern versions

Frege 2, cont.

Notable things cont.

That the semantics is compositional may then be a help for the hearerto find the appropriate content.But what should the speaker do? Maybe go through the list of allsentences, interpret each one in turn, until she finds the one whoseinterpretation corresponds to what she wants to convey.For more efficiency we seem to need inverse compositionality: theexpression of a complex content is a function of the expressions of itsparts and the way they are combined.

Frege seems to postulate an isomorphism between the structure oflanguage and the structure of thought (perhaps preserving the relationbeing an immediate part of).Frege not only describes compositionality, he offers an argument fromlinguistic communication that it holds, and that it is an essentialfeature of language.

12 of 23

Modern versions

Frege 2, cont.

Notable things cont.

That the semantics is compositional may then be a help for the hearerto find the appropriate content.But what should the speaker do? Maybe go through the list of allsentences, interpret each one in turn, until she finds the one whoseinterpretation corresponds to what she wants to convey.For more efficiency we seem to need inverse compositionality: theexpression of a complex content is a function of the expressions of itsparts and the way they are combined.Frege seems to postulate an isomorphism between the structure oflanguage and the structure of thought (perhaps preserving the relationbeing an immediate part of).

Frege not only describes compositionality, he offers an argument fromlinguistic communication that it holds, and that it is an essentialfeature of language.

12 of 23

Modern versions

Frege 2, cont.

Notable things cont.

That the semantics is compositional may then be a help for the hearerto find the appropriate content.But what should the speaker do? Maybe go through the list of allsentences, interpret each one in turn, until she finds the one whoseinterpretation corresponds to what she wants to convey.For more efficiency we seem to need inverse compositionality: theexpression of a complex content is a function of the expressions of itsparts and the way they are combined.Frege seems to postulate an isomorphism between the structure oflanguage and the structure of thought (perhaps preserving the relationbeing an immediate part of).Frege not only describes compositionality, he offers an argument fromlinguistic communication that it holds, and that it is an essentialfeature of language.

12 of 23

Modern versions

Carnap

Both the substitution version and the function version were explicitly statedby Rudolf Carnap (Meaning and Necessity, 1956) and labeled ‘Frege’sPrinciples of Interchangeability’:

Let . . .Aj . . . be a complex name containing an occurrence of thename Aj and . . .Ak . . . be a corresponding expression with thename Ak instead of Aj .

First principle. If Aj and Ak have the same nominatum, then. . .Aj . . . and . . .Ak . . . have the same nominatum. In otherwords, the nominatum of the whole expression is a function of thenominata of the names occurring in it.

Second principle. If Aj and Ak have the same sense, then . . .Aj . . .and . . .Ak . . . have the same sense. In other words, the sense ofthe whole expression is a function of the senses of the namesoccurring in it.

13 of 23

Modern versions

Carnap

Both the substitution version and the function version were explicitly statedby Rudolf Carnap (Meaning and Necessity, 1956) and labeled ‘Frege’sPrinciples of Interchangeability’:

Let . . .Aj . . . be a complex name containing an occurrence of thename Aj and . . .Ak . . . be a corresponding expression with thename Ak instead of Aj .

First principle. If Aj and Ak have the same nominatum, then. . .Aj . . . and . . .Ak . . . have the same nominatum. In otherwords, the nominatum of the whole expression is a function of thenominata of the names occurring in it.

Second principle. If Aj and Ak have the same sense, then . . .Aj . . .and . . .Ak . . . have the same sense. In other words, the sense ofthe whole expression is a function of the senses of the namesoccurring in it.

13 of 23

Modern versions

Putnam

In 1960 Hilary Putnam gave a talk in Oxford, later published as ‘Do trueassertions correspond to reality?’ (Mind, Language and Reality,Philosophical Papers, vol. 2, 1975), where he says

A mapping satisfying these (and suitable other conditions of thesame kind) will be called a compositional mapping. Compositionalmappings have the feature that their (value) for a complexsentence is a simple function of their value for related simplesentences.

This seems to be the first use of the word “compositional” in the sensediscussed here (though restricted to parts of sentences which are themselvessentences).

14 of 23

Modern versions

Putnam

In 1960 Hilary Putnam gave a talk in Oxford, later published as ‘Do trueassertions correspond to reality?’ (Mind, Language and Reality,Philosophical Papers, vol. 2, 1975), where he says

A mapping satisfying these (and suitable other conditions of thesame kind) will be called a compositional mapping. Compositionalmappings have the feature that their (value) for a complexsentence is a simple function of their value for related simplesentences.

This seems to be the first use of the word “compositional” in the sensediscussed here (though restricted to parts of sentences which are themselvessentences).

14 of 23

Modern versions

Katz and Fodor

Perhaps the following is the first appearance of the word “compositional” inprint:

Jerrold Katz and Jerry Fodor about adding (generative) se-mantics to Chomskyan syntax (in ‘The structure of a semantic theory’, 1964):

Since the set of sentences is infinite and each sentence is adifferent concatenation of morphemes, the fact that a speaker canunderstand any sentence must mean that the way he understandssentences which he has never previously encountered iscompositional: on the basis of his knowledge of the grammaticalproperties and the meanings of the morphemes of the language,the rules which the speaker knows enable him to determine themeaning of a novel sentence in terms of the manner in which theparts of the sentence are composed to form the whole.Correspondingly, we can expect that a system of rules whichsolves the projection problem must reflect the compositionalcharacter of the speaker’s knowledge.

15 of 23

Modern versions

Katz and Fodor

Perhaps the following is the first appearance of the word “compositional” inprint:

Jerrold Katz and Jerry Fodor about adding (generative) se-mantics to Chomskyan syntax (in ‘The structure of a semantic theory’, 1964):

Since the set of sentences is infinite and each sentence is adifferent concatenation of morphemes, the fact that a speaker canunderstand any sentence must mean that the way he understandssentences which he has never previously encountered iscompositional: on the basis of his knowledge of the grammaticalproperties and the meanings of the morphemes of the language,the rules which the speaker knows enable him to determine themeaning of a novel sentence in terms of the manner in which theparts of the sentence are composed to form the whole.Correspondingly, we can expect that a system of rules whichsolves the projection problem must reflect the compositionalcharacter of the speaker’s knowledge.

15 of 23

Modern versions

Algebraic methods in logic

Algebraic notions are implicit in the development of formal logical languagesfrom Frege and onwards.

This aspect was in particular developed by Polish logicians, notably Tarskiand Lindenbaum.

A systematic program for applying methods from algebra and topology inlogic was presented by Helena Rasiowa and Roman Sikorski: TheMathematics of Metamathematics, 1963.

16 of 23

Modern versions

Algebraic methods in logic

Algebraic notions are implicit in the development of formal logical languagesfrom Frege and onwards.

This aspect was in particular developed by Polish logicians, notably Tarskiand Lindenbaum.

A systematic program for applying methods from algebra and topology inlogic was presented by Helena Rasiowa and Roman Sikorski: TheMathematics of Metamathematics, 1963.

16 of 23

Modern versions

Algebraic methods in logic

Algebraic notions are implicit in the development of formal logical languagesfrom Frege and onwards.

This aspect was in particular developed by Polish logicians, notably Tarskiand Lindenbaum.

A systematic program for applying methods from algebra and topology inlogic was presented by Helena Rasiowa and Roman Sikorski: TheMathematics of Metamathematics, 1963.

16 of 23

Modern versions

Other languages

The algebraic approach was extended to

semantics for programming languages (Dana Scott and ChristopherStrachey, often called denotational semantics);

semantics for natural languages (from the late 60’s and onwards: DavidLewis, Richard Montague, Barbara Partee, . . . ), where compositionalitywas a fundamental assumption.

But aren’t natural languages much too unspecific and vague to apply formalmethods from logic and algebra? (Tarski thought so.)

17 of 23

Modern versions

Other languages

The algebraic approach was extended to

semantics for programming languages (Dana Scott and ChristopherStrachey, often called denotational semantics);

semantics for natural languages (from the late 60’s and onwards: DavidLewis, Richard Montague, Barbara Partee, . . . ), where compositionalitywas a fundamental assumption.

But aren’t natural languages much too unspecific and vague to apply formalmethods from logic and algebra? (Tarski thought so.)

17 of 23

Modern versions

Other languages

The algebraic approach was extended to

semantics for programming languages (Dana Scott and ChristopherStrachey, often called denotational semantics);

semantics for natural languages (from the late 60’s and onwards: DavidLewis, Richard Montague, Barbara Partee, . . . ), where compositionalitywas a fundamental assumption.

But aren’t natural languages much too unspecific and vague to apply formalmethods from logic and algebra? (Tarski thought so.)

17 of 23

Modern versions

Other languages

The algebraic approach was extended to

semantics for programming languages (Dana Scott and ChristopherStrachey, often called denotational semantics);

semantics for natural languages (from the late 60’s and onwards: DavidLewis, Richard Montague, Barbara Partee, . . . ), where compositionalitywas a fundamental assumption.

But aren’t natural languages much too unspecific and vague to apply formalmethods from logic and algebra? (Tarski thought so.)

17 of 23

Modern versions

Montague

There is in my opinion no important theoretical differencebetween natural languages and the artificial languages oflogicians; indeed, I consider it possible to comprehend the syntaxand semantics of both kinds of languages within a single naturaland mathematically precise theory.

(Montague, ‘Universal Grammar’, 1970)

18 of 23

Modern versions

Montague, cont.

In ‘Universal grammar’ Montague’s approach is explicitly algebraic.

There is a (many-sorted) syntactic algebra which generates (disambiguated)syntactic expressions of the language: (roughly)

A = 〈A, fγ ,Xδ〉γ∈Γ,δ∈∆

Here the Xδ ⊆ A are of the various (disjoint) categories or sorts, and A isthe set generated by the operations fγ from the Xδ.

A is free, so each e ∈ A is generated in a unique way.

Let B = 〈B, gγ〉γ∈Γ a semantic algebra of ‘meanings’, similar to 〈A, fγ〉γ∈Γ.

An interpretation is a homomorphism h from 〈A, fγ , 〉γ∈Γ to B (more exactly,the unique homomorphism generated from some function h0 from ∪δ∈∆Xδto B).

Thus, for example (if fγ is binary),

h(fγ(e1, e2)) = gγ(h(e1), h(e2))

So Montague builds in compositionality in his notion of a semanticinterpretation.

19 of 23

Modern versions

Montague, cont.

In ‘Universal grammar’ Montague’s approach is explicitly algebraic.

There is a (many-sorted) syntactic algebra which generates (disambiguated)syntactic expressions of the language: (roughly)

A = 〈A, fγ ,Xδ〉γ∈Γ,δ∈∆

Here the Xδ ⊆ A are of the various (disjoint) categories or sorts, and A isthe set generated by the operations fγ from the Xδ.

A is free, so each e ∈ A is generated in a unique way.

Let B = 〈B, gγ〉γ∈Γ a semantic algebra of ‘meanings’, similar to 〈A, fγ〉γ∈Γ.

An interpretation is a homomorphism h from 〈A, fγ , 〉γ∈Γ to B (more exactly,the unique homomorphism generated from some function h0 from ∪δ∈∆Xδto B).

Thus, for example (if fγ is binary),

h(fγ(e1, e2)) = gγ(h(e1), h(e2))

So Montague builds in compositionality in his notion of a semanticinterpretation.

19 of 23

Modern versions

Montague, cont.

In ‘Universal grammar’ Montague’s approach is explicitly algebraic.

There is a (many-sorted) syntactic algebra which generates (disambiguated)syntactic expressions of the language: (roughly)

A = 〈A, fγ ,Xδ〉γ∈Γ,δ∈∆

Here the Xδ ⊆ A are of the various (disjoint) categories or sorts, and A isthe set generated by the operations fγ from the Xδ.

A is free, so each e ∈ A is generated in a unique way.

Let B = 〈B, gγ〉γ∈Γ a semantic algebra of ‘meanings’, similar to 〈A, fγ〉γ∈Γ.

An interpretation is a homomorphism h from 〈A, fγ , 〉γ∈Γ to B (more exactly,the unique homomorphism generated from some function h0 from ∪δ∈∆Xδto B).

Thus, for example (if fγ is binary),

h(fγ(e1, e2)) = gγ(h(e1), h(e2))

So Montague builds in compositionality in his notion of a semanticinterpretation.

19 of 23

Modern versions

Montague, cont.

In ‘Universal grammar’ Montague’s approach is explicitly algebraic.

There is a (many-sorted) syntactic algebra which generates (disambiguated)syntactic expressions of the language: (roughly)

A = 〈A, fγ ,Xδ〉γ∈Γ,δ∈∆

Here the Xδ ⊆ A are of the various (disjoint) categories or sorts, and A isthe set generated by the operations fγ from the Xδ.

A is free, so each e ∈ A is generated in a unique way.

Let B = 〈B, gγ〉γ∈Γ a semantic algebra of ‘meanings’, similar to 〈A, fγ〉γ∈Γ.

An interpretation is a homomorphism h from 〈A, fγ , 〉γ∈Γ to B (more exactly,the unique homomorphism generated from some function h0 from ∪δ∈∆Xδto B).

Thus, for example (if fγ is binary),

h(fγ(e1, e2)) = gγ(h(e1), h(e2))

So Montague builds in compositionality in his notion of a semanticinterpretation.

19 of 23

Modern versions

Montague, cont.

In ‘Universal grammar’ Montague’s approach is explicitly algebraic.

There is a (many-sorted) syntactic algebra which generates (disambiguated)syntactic expressions of the language: (roughly)

A = 〈A, fγ ,Xδ〉γ∈Γ,δ∈∆

Here the Xδ ⊆ A are of the various (disjoint) categories or sorts, and A isthe set generated by the operations fγ from the Xδ.

A is free, so each e ∈ A is generated in a unique way.

Let B = 〈B, gγ〉γ∈Γ a semantic algebra of ‘meanings’, similar to 〈A, fγ〉γ∈Γ.

An interpretation is a homomorphism h from 〈A, fγ , 〉γ∈Γ to B (more exactly,the unique homomorphism generated from some function h0 from ∪δ∈∆Xδto B).

Thus, for example (if fγ is binary),

h(fγ(e1, e2)) = gγ(h(e1), h(e2))

So Montague builds in compositionality in his notion of a semanticinterpretation.

19 of 23

Modern versions

Montague, cont.

In ‘Universal grammar’ Montague’s approach is explicitly algebraic.

There is a (many-sorted) syntactic algebra which generates (disambiguated)syntactic expressions of the language: (roughly)

A = 〈A, fγ ,Xδ〉γ∈Γ,δ∈∆

Here the Xδ ⊆ A are of the various (disjoint) categories or sorts, and A isthe set generated by the operations fγ from the Xδ.

A is free, so each e ∈ A is generated in a unique way.

Let B = 〈B, gγ〉γ∈Γ a semantic algebra of ‘meanings’, similar to 〈A, fγ〉γ∈Γ.

An interpretation is a homomorphism h from 〈A, fγ , 〉γ∈Γ to B (more exactly,the unique homomorphism generated from some function h0 from ∪δ∈∆Xδto B).

Thus, for example (if fγ is binary),

h(fγ(e1, e2)) = gγ(h(e1), h(e2))

So Montague builds in compositionality in his notion of a semanticinterpretation.

19 of 23

Modern versions

Montague, cont.

In ‘Universal grammar’ Montague’s approach is explicitly algebraic.

There is a (many-sorted) syntactic algebra which generates (disambiguated)syntactic expressions of the language: (roughly)

A = 〈A, fγ ,Xδ〉γ∈Γ,δ∈∆

Here the Xδ ⊆ A are of the various (disjoint) categories or sorts, and A isthe set generated by the operations fγ from the Xδ.

A is free, so each e ∈ A is generated in a unique way.

Let B = 〈B, gγ〉γ∈Γ a semantic algebra of ‘meanings’, similar to 〈A, fγ〉γ∈Γ.

An interpretation is a homomorphism h from 〈A, fγ , 〉γ∈Γ to B (more exactly,the unique homomorphism generated from some function h0 from ∪δ∈∆Xδto B).

Thus, for example (if fγ is binary),

h(fγ(e1, e2)) = gγ(h(e1), h(e2))

So Montague builds in compositionality in his notion of a semanticinterpretation.

19 of 23

Modern versions

Montague, cont.

In ‘Universal grammar’ Montague’s approach is explicitly algebraic.

There is a (many-sorted) syntactic algebra which generates (disambiguated)syntactic expressions of the language: (roughly)

A = 〈A, fγ ,Xδ〉γ∈Γ,δ∈∆

Here the Xδ ⊆ A are of the various (disjoint) categories or sorts, and A isthe set generated by the operations fγ from the Xδ.

A is free, so each e ∈ A is generated in a unique way.

Let B = 〈B, gγ〉γ∈Γ a semantic algebra of ‘meanings’, similar to 〈A, fγ〉γ∈Γ.

An interpretation is a homomorphism h from 〈A, fγ , 〉γ∈Γ to B (more exactly,the unique homomorphism generated from some function h0 from ∪δ∈∆Xδto B).

Thus, for example (if fγ is binary),

h(fγ(e1, e2)) = gγ(h(e1), h(e2))

So Montague builds in compositionality in his notion of a semanticinterpretation.

19 of 23

Modern versions

Partee

The compositionality principle, in its most general form, can beexpressed as follows:

The meaning of an expression is a function of the meanings of itsparts and the way they are syntactically combined.

The principle is usually attributed to Frege, and is often called‘Frege’s principle’ . . . In its most general form, the principle isnearly uncontroversial; some version of it would appear to be anessential part of any account of how meanings are assigned tonovel sentences. (Partee, ‘Compositionality’, 1984)

20 of 23

Modern versions

Fodor

So by now you may think it is fairly clear what the compositionality principlesays, and that it is fundamental in semantics.

Not quite.

Several theorists think it is false, or unimportant, or empty. We will get backto those claims.

As to whether it is clear what the principle says . . .

Here is Jerry Fodor (2001):

So not-negotiable is compositionality that I’m not even going totell you what it is. Suffice it that the things that the expression(mutatis mutandis the concept) ‘brown cow’ applies to are exactlythe things to which the expressions ‘brown’ and ‘cow’ apply.

(Is this a good example of compositionality?)

Fodor again (ibid.):

Nobody knows exactly what compositionality demands, buteverybody knows why its demands have to be satisfied.

21 of 23

Modern versions

Fodor

So by now you may think it is fairly clear what the compositionality principlesays, and that it is fundamental in semantics.

Not quite.

Several theorists think it is false, or unimportant, or empty. We will get backto those claims.

As to whether it is clear what the principle says . . .

Here is Jerry Fodor (2001):

So not-negotiable is compositionality that I’m not even going totell you what it is. Suffice it that the things that the expression(mutatis mutandis the concept) ‘brown cow’ applies to are exactlythe things to which the expressions ‘brown’ and ‘cow’ apply.

(Is this a good example of compositionality?)

Fodor again (ibid.):

Nobody knows exactly what compositionality demands, buteverybody knows why its demands have to be satisfied.

21 of 23

Modern versions

Fodor

So by now you may think it is fairly clear what the compositionality principlesays, and that it is fundamental in semantics.

Not quite.

Several theorists think it is false, or unimportant, or empty. We will get backto those claims.

As to whether it is clear what the principle says . . .

Here is Jerry Fodor (2001):

So not-negotiable is compositionality that I’m not even going totell you what it is. Suffice it that the things that the expression(mutatis mutandis the concept) ‘brown cow’ applies to are exactlythe things to which the expressions ‘brown’ and ‘cow’ apply.

(Is this a good example of compositionality?)

Fodor again (ibid.):

Nobody knows exactly what compositionality demands, buteverybody knows why its demands have to be satisfied.

21 of 23

Modern versions

Fodor

So by now you may think it is fairly clear what the compositionality principlesays, and that it is fundamental in semantics.

Not quite.

Several theorists think it is false, or unimportant, or empty. We will get backto those claims.

As to whether it is clear what the principle says . . .

Here is Jerry Fodor (2001):

So not-negotiable is compositionality that I’m not even going totell you what it is. Suffice it that the things that the expression(mutatis mutandis the concept) ‘brown cow’ applies to are exactlythe things to which the expressions ‘brown’ and ‘cow’ apply.

(Is this a good example of compositionality?)

Fodor again (ibid.):

Nobody knows exactly what compositionality demands, buteverybody knows why its demands have to be satisfied.

21 of 23

Modern versions

Fodor

So by now you may think it is fairly clear what the compositionality principlesays, and that it is fundamental in semantics.

Not quite.

Several theorists think it is false, or unimportant, or empty. We will get backto those claims.

As to whether it is clear what the principle says . . .

Here is Jerry Fodor (2001):

So not-negotiable is compositionality that I’m not even going totell you what it is. Suffice it that the things that the expression(mutatis mutandis the concept) ‘brown cow’ applies to are exactlythe things to which the expressions ‘brown’ and ‘cow’ apply.

(Is this a good example of compositionality?)

Fodor again (ibid.):

Nobody knows exactly what compositionality demands, buteverybody knows why its demands have to be satisfied.

21 of 23

Modern versions

Fodor

So by now you may think it is fairly clear what the compositionality principlesays, and that it is fundamental in semantics.

Not quite.

Several theorists think it is false, or unimportant, or empty. We will get backto those claims.

As to whether it is clear what the principle says . . .

Here is Jerry Fodor (2001):

So not-negotiable is compositionality that I’m not even going totell you what it is. Suffice it that the things that the expression(mutatis mutandis the concept) ‘brown cow’ applies to are exactlythe things to which the expressions ‘brown’ and ‘cow’ apply.

(Is this a good example of compositionality?)

Fodor again (ibid.):

Nobody knows exactly what compositionality demands, buteverybody knows why its demands have to be satisfied.

21 of 23

Modern versions

Fodor

So by now you may think it is fairly clear what the compositionality principlesays, and that it is fundamental in semantics.

Not quite.

Several theorists think it is false, or unimportant, or empty. We will get backto those claims.

As to whether it is clear what the principle says . . .

Here is Jerry Fodor (2001):

So not-negotiable is compositionality that I’m not even going totell you what it is. Suffice it that the things that the expression(mutatis mutandis the concept) ‘brown cow’ applies to are exactlythe things to which the expressions ‘brown’ and ‘cow’ apply.

(Is this a good example of compositionality?)

Fodor again (ibid.):

Nobody knows exactly what compositionality demands, buteverybody knows why its demands have to be satisfied.21 of 23

Modern versions

Dowty

David Dowty says (‘Compositionality as an empirical problem’, 2007):

I believe that there is not and will not be — any time soon, if ever— a unique precise and ‘correct’ definition of compositionalitythat all linguists and/or philosophers can agree upon, such thatwe can ask whether natural language has that property or not.

Hmm. . . , maybe.

Dowty again (ibid.):

I propose that we let the term natural language compositionalityrefer to whatever strategies and principles natural language(s)actually do employ to construct the meaning of sentences (andother complex expressions), on the basis of whatever syntacticproperties, and contextual information they do in fact turn out todepend on.

???

22 of 23

Modern versions

Dowty

David Dowty says (‘Compositionality as an empirical problem’, 2007):

I believe that there is not and will not be — any time soon, if ever— a unique precise and ‘correct’ definition of compositionalitythat all linguists and/or philosophers can agree upon, such thatwe can ask whether natural language has that property or not.

Hmm. . . , maybe.

Dowty again (ibid.):

I propose that we let the term natural language compositionalityrefer to whatever strategies and principles natural language(s)actually do employ to construct the meaning of sentences (andother complex expressions), on the basis of whatever syntacticproperties, and contextual information they do in fact turn out todepend on.

???

22 of 23

Modern versions

Dowty

David Dowty says (‘Compositionality as an empirical problem’, 2007):

I believe that there is not and will not be — any time soon, if ever— a unique precise and ‘correct’ definition of compositionalitythat all linguists and/or philosophers can agree upon, such thatwe can ask whether natural language has that property or not.

Hmm. . . , maybe.

Dowty again (ibid.):

I propose that we let the term natural language compositionalityrefer to whatever strategies and principles natural language(s)actually do employ to construct the meaning of sentences (andother complex expressions), on the basis of whatever syntacticproperties, and contextual information they do in fact turn out todepend on.

???

22 of 23

Modern versions

Dowty

David Dowty says (‘Compositionality as an empirical problem’, 2007):

I believe that there is not and will not be — any time soon, if ever— a unique precise and ‘correct’ definition of compositionalitythat all linguists and/or philosophers can agree upon, such thatwe can ask whether natural language has that property or not.

Hmm. . . , maybe.

Dowty again (ibid.):

I propose that we let the term natural language compositionalityrefer to whatever strategies and principles natural language(s)actually do employ to construct the meaning of sentences (andother complex expressions), on the basis of whatever syntacticproperties, and contextual information they do in fact turn out todepend on.

???

22 of 23

Modern versions

Let’s not give up. . .

There is of course no point in quibbling about labels.

But Dowty’s position makes compositionality empty: (a) no natural languagecould fail to be compositional; (b) we get no idea about what property it is.

In a way, Montague’s approach trivializes matters too, since it makescompositionality part of his (precise) notion of a semantic interpretation.

Fodor’s stance is also a way of giving up the idea that there are interestingissues to discuss here.

BUT:

The idea of compositionality is not that vague or imprecise (eventhough there may be a couple of precise versions of it).

Even if interesting existing languages should turn out to be allcompositional, we need a notion that can be false of a possiblelanguage. (In Popper’s terms, the compositionality principle should befalsifiable.)

So, let’s spell out such a notion.

23 of 23

Modern versions

Let’s not give up. . .

There is of course no point in quibbling about labels.

But Dowty’s position makes compositionality empty: (a) no natural languagecould fail to be compositional; (b) we get no idea about what property it is.

In a way, Montague’s approach trivializes matters too, since it makescompositionality part of his (precise) notion of a semantic interpretation.

Fodor’s stance is also a way of giving up the idea that there are interestingissues to discuss here.

BUT:

The idea of compositionality is not that vague or imprecise (eventhough there may be a couple of precise versions of it).

Even if interesting existing languages should turn out to be allcompositional, we need a notion that can be false of a possiblelanguage. (In Popper’s terms, the compositionality principle should befalsifiable.)

So, let’s spell out such a notion.

23 of 23

Modern versions

Let’s not give up. . .

There is of course no point in quibbling about labels.

But Dowty’s position makes compositionality empty: (a) no natural languagecould fail to be compositional; (b) we get no idea about what property it is.

In a way, Montague’s approach trivializes matters too, since it makescompositionality part of his (precise) notion of a semantic interpretation.

Fodor’s stance is also a way of giving up the idea that there are interestingissues to discuss here.

BUT:

The idea of compositionality is not that vague or imprecise (eventhough there may be a couple of precise versions of it).

Even if interesting existing languages should turn out to be allcompositional, we need a notion that can be false of a possiblelanguage. (In Popper’s terms, the compositionality principle should befalsifiable.)

So, let’s spell out such a notion.

23 of 23

Modern versions

Let’s not give up. . .

There is of course no point in quibbling about labels.

But Dowty’s position makes compositionality empty: (a) no natural languagecould fail to be compositional; (b) we get no idea about what property it is.

In a way, Montague’s approach trivializes matters too, since it makescompositionality part of his (precise) notion of a semantic interpretation.

Fodor’s stance is also a way of giving up the idea that there are interestingissues to discuss here.

BUT:

The idea of compositionality is not that vague or imprecise (eventhough there may be a couple of precise versions of it).

Even if interesting existing languages should turn out to be allcompositional, we need a notion that can be false of a possiblelanguage. (In Popper’s terms, the compositionality principle should befalsifiable.)

So, let’s spell out such a notion.

23 of 23

Modern versions

Let’s not give up. . .

There is of course no point in quibbling about labels.

But Dowty’s position makes compositionality empty: (a) no natural languagecould fail to be compositional; (b) we get no idea about what property it is.

In a way, Montague’s approach trivializes matters too, since it makescompositionality part of his (precise) notion of a semantic interpretation.

Fodor’s stance is also a way of giving up the idea that there are interestingissues to discuss here.

BUT:

The idea of compositionality is not that vague or imprecise (eventhough there may be a couple of precise versions of it).

Even if interesting existing languages should turn out to be allcompositional, we need a notion that can be false of a possiblelanguage. (In Popper’s terms, the compositionality principle should befalsifiable.)

So, let’s spell out such a notion.

23 of 23

Modern versions

Let’s not give up. . .

There is of course no point in quibbling about labels.

But Dowty’s position makes compositionality empty: (a) no natural languagecould fail to be compositional; (b) we get no idea about what property it is.

In a way, Montague’s approach trivializes matters too, since it makescompositionality part of his (precise) notion of a semantic interpretation.

Fodor’s stance is also a way of giving up the idea that there are interestingissues to discuss here.

BUT:

The idea of compositionality is not that vague or imprecise (eventhough there may be a couple of precise versions of it).

Even if interesting existing languages should turn out to be allcompositional, we need a notion that can be false of a possiblelanguage. (In Popper’s terms, the compositionality principle should befalsifiable.)

So, let’s spell out such a notion.

23 of 23

Modern versions

Let’s not give up. . .

There is of course no point in quibbling about labels.

But Dowty’s position makes compositionality empty: (a) no natural languagecould fail to be compositional; (b) we get no idea about what property it is.

In a way, Montague’s approach trivializes matters too, since it makescompositionality part of his (precise) notion of a semantic interpretation.

Fodor’s stance is also a way of giving up the idea that there are interestingissues to discuss here.

BUT:

The idea of compositionality is not that vague or imprecise (eventhough there may be a couple of precise versions of it).

Even if interesting existing languages should turn out to be allcompositional, we need a notion that can be false of a possiblelanguage. (In Popper’s terms, the compositionality principle should befalsifiable.)

So, let’s spell out such a notion.

23 of 23