Compositionality Tutorial 4 Pros and Cons

Compositionality Tutorial 4 Pros and Cons

Dag Westerst˚ahl Stockholm University and University of Gothenburg

Tsinghua University, Beijing June 17, 2011

Tsinghua University, Beijing June 17, 2011 1 Dag Westerst˚ahl () Compositionality 4 / 20 Outline

1 Denials Denial 1: “X is not compositional” Denial 2: ambiguity Denial 3: idioms Conclusions

2 Emptiness

3 Praise

Tsinghua University, Beijing June 17, 2011 2 Dag Westerst˚ahl () Compositionality 4 / 20 Yes: FO is not compositional if semantic values are truth values and assignments are seen as parameters. (Because assignments are shifted in the clauses for the quantifiers.) But FO is compositional if semantic values are sets of assignments. X = Independence-Friendly logic (Hintikka, who explicitly makes the non-compositionality claim). Yes: It can be shown that no semantics for IF-logic which takes sets of assignments as values can be compositional (Cameron and Hodges). No: We already saw (Hodges’ extension theorem) that compositional semantics with certain nice properties exist. Moreover, an interesting compositional semantics which takes semantic values to be sets of sets of assignments (so-called trumps or teams) has been given by Hodges (see V¨a¨an¨anen, Dependence Logic, 2007).

Denials Denial 1: “X is not compositional” “X is not compositional”

X = first-order logic:

1 of 20 But FO is compositional if semantic values are sets of assignments. X = Independence-Friendly logic (Hintikka, who explicitly makes the non-compositionality claim). Yes: It can be shown that no semantics for IF-logic which takes sets of assignments as values can be compositional (Cameron and Hodges). No: We already saw (Hodges’ extension theorem) that compositional semantics with certain nice properties exist. Moreover, an interesting compositional semantics which takes semantic values to be sets of sets of assignments (so-called trumps or teams) has been given by Hodges (see V¨a¨an¨anen, Dependence Logic, 2007).

Denials Denial 1: “X is not compositional” “X is not compositional”

X = first-order logic: Yes: FO is not compositional if semantic values are truth values and assignments are seen as parameters. (Because assignments are shifted in the clauses for the quantifiers.)

1 of 20 X = Independence-Friendly logic (Hintikka, who explicitly makes the non-compositionality claim). Yes: It can be shown that no semantics for IF-logic which takes sets of assignments as values can be compositional (Cameron and Hodges). No: We already saw (Hodges’ extension theorem) that compositional semantics with certain nice properties exist. Moreover, an interesting compositional semantics which takes semantic values to be sets of sets of assignments (so-called trumps or teams) has been given by Hodges (see V¨a¨an¨anen, Dependence Logic, 2007).

Denials Denial 1: “X is not compositional” “X is not compositional”

X = first-order logic: Yes: FO is not compositional if semantic values are truth values and assignments are seen as parameters. (Because assignments are shifted in the clauses for the quantifiers.) But FO is compositional if semantic values are sets of assignments.

1 of 20 Yes: It can be shown that no semantics for IF-logic which takes sets of assignments as values can be compositional (Cameron and Hodges). No: We already saw (Hodges’ extension theorem) that compositional semantics with certain nice properties exist. Moreover, an interesting compositional semantics which takes semantic values to be sets of sets of assignments (so-called trumps or teams) has been given by Hodges (see V¨a¨an¨anen, Dependence Logic, 2007).

Denials Denial 1: “X is not compositional” “X is not compositional”

X = first-order logic: Yes: FO is not compositional if semantic values are truth values and assignments are seen as parameters. (Because assignments are shifted in the clauses for the quantifiers.) But FO is compositional if semantic values are sets of assignments. X = Independence-Friendly logic (Hintikka, who explicitly makes the non-compositionality claim).

1 of 20 No: We already saw (Hodges’ extension theorem) that compositional semantics with certain nice properties exist. Moreover, an interesting compositional semantics which takes semantic values to be sets of sets of assignments (so-called trumps or teams) has been given by Hodges (see V¨a¨an¨anen, Dependence Logic, 2007).

Denials Denial 1: “X is not compositional” “X is not compositional”

X = first-order logic: Yes: FO is not compositional if semantic values are truth values and assignments are seen as parameters. (Because assignments are shifted in the clauses for the quantifiers.) But FO is compositional if semantic values are sets of assignments. X = Independence-Friendly logic (Hintikka, who explicitly makes the non-compositionality claim). Yes: It can be shown that no semantics for IF-logic which takes sets of assignments as values can be compositional (Cameron and Hodges).

1 of 20 Moreover, an interesting compositional semantics which takes semantic values to be sets of sets of assignments (so-called trumps or teams) has been given by Hodges (see V¨a¨an¨anen, Dependence Logic, 2007).

Denials Denial 1: “X is not compositional” “X is not compositional”

X = first-order logic: Yes: FO is not compositional if semantic values are truth values and assignments are seen as parameters. (Because assignments are shifted in the clauses for the quantifiers.) But FO is compositional if semantic values are sets of assignments. X = Independence-Friendly logic (Hintikka, who explicitly makes the non-compositionality claim). Yes: It can be shown that no semantics for IF-logic which takes sets of assignments as values can be compositional (Cameron and Hodges). No: We already saw (Hodges’ extension theorem) that compositional semantics with certain nice properties exist.

1 of 20 Denials Denial 1: “X is not compositional” “X is not compositional”

X = first-order logic: Yes: FO is not compositional if semantic values are truth values and assignments are seen as parameters. (Because assignments are shifted in the clauses for the quantifiers.) But FO is compositional if semantic values are sets of assignments. X = Independence-Friendly logic (Hintikka, who explicitly makes the non-compositionality claim). Yes: It can be shown that no semantics for IF-logic which takes sets of assignments as values can be compositional (Cameron and Hodges). No: We already saw (Hodges’ extension theorem) that compositional semantics with certain nice properties exist. Moreover, an interesting compositional semantics which takes semantic values to be sets of sets of assignments (so-called trumps or teams) has been given by Hodges (see V¨a¨an¨anen, Dependence Logic, 2007).

1 of 20 This is too unspecified. Compositionality is relative to a syntactic analysis (a grammar) and an assignment of semantic values. X = English belief sentences, of the form “A believes that ϕ”: This is more interesting: If extensions are semantic values, and (1) Karl believes that Mark Twain is a novelist. (2) Karl believes that Samuel Clemens is a novelist. can differ in truth value, and if truth value depends on semantic value, then the semantics for (this fragment of) English is not compositional. Also, if intensions are semantic values, and (3) Karl believes that Mark Twain is a novelist. (4) Karl believes that Mark Twain is a novelist and 172 = 289. can differ in truth value, and if truth value depends on semantic value, then the semantics for (this fragment of) English is not compositional.

Denials Denial 1: “X is not compositional” “X is not compositional”, cont.

X = English:

2 of 20 X = English belief sentences, of the form “A believes that ϕ”: This is more interesting: If extensions are semantic values, and (1) Karl believes that Mark Twain is a novelist. (2) Karl believes that Samuel Clemens is a novelist. can differ in truth value, and if truth value depends on semantic value, then the semantics for (this fragment of) English is not compositional. Also, if intensions are semantic values, and (3) Karl believes that Mark Twain is a novelist. (4) Karl believes that Mark Twain is a novelist and 172 = 289. can differ in truth value, and if truth value depends on semantic value, then the semantics for (this fragment of) English is not compositional.

Denials Denial 1: “X is not compositional” “X is not compositional”, cont.

X = English: This is too unspecified. Compositionality is relative to a syntactic analysis (a grammar) and an assignment of semantic values.

2 of 20 This is more interesting: If extensions are semantic values, and (1) Karl believes that Mark Twain is a novelist. (2) Karl believes that Samuel Clemens is a novelist. can differ in truth value, and if truth value depends on semantic value, then the semantics for (this fragment of) English is not compositional. Also, if intensions are semantic values, and (3) Karl believes that Mark Twain is a novelist. (4) Karl believes that Mark Twain is a novelist and 172 = 289. can differ in truth value, and if truth value depends on semantic value, then the semantics for (this fragment of) English is not compositional.

Denials Denial 1: “X is not compositional” “X is not compositional”, cont.

X = English: This is too unspecified. Compositionality is relative to a syntactic analysis (a grammar) and an assignment of semantic values. X = English belief sentences, of the form “A believes that ϕ”:

2 of 20 Also, if intensions are semantic values, and (3) Karl believes that Mark Twain is a novelist. (4) Karl believes that Mark Twain is a novelist and 172 = 289. can differ in truth value, and if truth value depends on semantic value, then the semantics for (this fragment of) English is not compositional.

Denials Denial 1: “X is not compositional” “X is not compositional”, cont.

X = English: This is too unspecified. Compositionality is relative to a syntactic analysis (a grammar) and an assignment of semantic values. X = English belief sentences, of the form “A believes that ϕ”: This is more interesting: If extensions are semantic values, and (1) Karl believes that Mark Twain is a novelist. (2) Karl believes that Samuel Clemens is a novelist. can differ in truth value, and if truth value depends on semantic value, then the semantics for (this fragment of) English is not compositional.

2 of 20 Denials Denial 1: “X is not compositional” “X is not compositional”, cont.

X = English: This is too unspecified. Compositionality is relative to a syntactic analysis (a grammar) and an assignment of semantic values. X = English belief sentences, of the form “A believes that ϕ”: This is more interesting: If extensions are semantic values, and (1) Karl believes that Mark Twain is a novelist. (2) Karl believes that Samuel Clemens is a novelist. can differ in truth value, and if truth value depends on semantic value, then the semantics for (this fragment of) English is not compositional. Also, if intensions are semantic values, and (3) Karl believes that Mark Twain is a novelist. (4) Karl believes that Mark Twain is a novelist and 172 = 289. can differ in truth value, and if truth value depends on semantic value, then the semantics for (this fragment of) English is not compositional.

2 of 20 But can they differ in this last case? This intuition seems unacceptable. The two nouns mean the same in the idiolect of the speaker, and therefore specify the same content. They may differ in the idiolect of the attributee (Karl), but that is irrelevant. There are several proposals (starting with Frege’s indirect sense and reference) for dealing compositionally with Twain/Clemens cases. Compositional treatments of ‘and 172 = 289’ cases can make the relevant to ϕ what is believed (Pagin).

Denials Denial 1: “X is not compositional” Belief sentences, cont.

Also, if intuitive synonymy-meanings are semantic values, and (5) Karl believes that Mark Twain is a novelist. (6) Karl believes that Mark Twain is a novel-writer. can differ in truth value, and if truth value depends on semantic value, then the semantics for (this fragment of) English is not compositional.

3 of 20 This intuition seems unacceptable. The two nouns mean the same in the idiolect of the speaker, and therefore specify the same content. They may differ in the idiolect of the attributee (Karl), but that is irrelevant. There are several proposals (starting with Frege’s indirect sense and reference) for dealing compositionally with Twain/Clemens cases. Compositional treatments of ‘and 172 = 289’ cases can make the relevant to ϕ what is believed (Pagin).

Denials Denial 1: “X is not compositional” Belief sentences, cont.

Also, if intuitive synonymy-meanings are semantic values, and (5) Karl believes that Mark Twain is a novelist. (6) Karl believes that Mark Twain is a novel-writer. can differ in truth value, and if truth value depends on semantic value, then the semantics for (this fragment of) English is not compositional. But can they differ in this last case?

3 of 20 They may differ in the idiolect of the attributee (Karl), but that is irrelevant. There are several proposals (starting with Frege’s indirect sense and reference) for dealing compositionally with Twain/Clemens cases. Compositional treatments of ‘and 172 = 289’ cases can make the relevant to ϕ what is believed (Pagin).

Denials Denial 1: “X is not compositional” Belief sentences, cont.

Also, if intuitive synonymy-meanings are semantic values, and (5) Karl believes that Mark Twain is a novelist. (6) Karl believes that Mark Twain is a novel-writer. can differ in truth value, and if truth value depends on semantic value, then the semantics for (this fragment of) English is not compositional. But can they differ in this last case? This intuition seems unacceptable. The two nouns mean the same in the idiolect of the speaker, and therefore specify the same content.

3 of 20 There are several proposals (starting with Frege’s indirect sense and reference) for dealing compositionally with Twain/Clemens cases. Compositional treatments of ‘and 172 = 289’ cases can make the relevant to ϕ what is believed (Pagin).

Denials Denial 1: “X is not compositional” Belief sentences, cont.

Also, if intuitive synonymy-meanings are semantic values, and (5) Karl believes that Mark Twain is a novelist. (6) Karl believes that Mark Twain is a novel-writer. can differ in truth value, and if truth value depends on semantic value, then the semantics for (this fragment of) English is not compositional. But can they differ in this last case? This intuition seems unacceptable. The two nouns mean the same in the idiolect of the speaker, and therefore specify the same content. They may differ in the idiolect of the attributee (Karl), but that is irrelevant.

3 of 20 Compositional treatments of ‘and 172 = 289’ cases can make the relevant to ϕ what is believed (Pagin).

Denials Denial 1: “X is not compositional” Belief sentences, cont.

Also, if intuitive synonymy-meanings are semantic values, and (5) Karl believes that Mark Twain is a novelist. (6) Karl believes that Mark Twain is a novel-writer. can differ in truth value, and if truth value depends on semantic value, then the semantics for (this fragment of) English is not compositional. But can they differ in this last case? This intuition seems unacceptable. The two nouns mean the same in the idiolect of the speaker, and therefore specify the same content. They may differ in the idiolect of the attributee (Karl), but that is irrelevant. There are several proposals (starting with Frege’s indirect sense and reference) for dealing compositionally with Twain/Clemens cases.

3 of 20 Denials Denial 1: “X is not compositional” Belief sentences, cont.

Also, if intuitive synonymy-meanings are semantic values, and (5) Karl believes that Mark Twain is a novelist. (6) Karl believes that Mark Twain is a novel-writer. can differ in truth value, and if truth value depends on semantic value, then the semantics for (this fragment of) English is not compositional. But can they differ in this last case? This intuition seems unacceptable. The two nouns mean the same in the idiolect of the speaker, and therefore specify the same content. They may differ in the idiolect of the attributee (Karl), but that is irrelevant. There are several proposals (starting with Frege’s indirect sense and reference) for dealing compositionally with Twain/Clemens cases. Compositional treatments of ‘and 172 = 289’ cases can make the relevant to ϕ what is believed (Pagin).

3 of 20 Montague simply assumes that disambiguation takes place first (without saying how). Put differently, there is a common assumption that ambiguity is either lexical or structural. The term algebra framework deals with both adequately. But if grammar and syntactic structure — as many believe — are not only tools to construct unambiguous meaning bearers, but have a psychological reality tied to how we process utterances or sentences, then one needs an independent argument for why these are always unambiguous. If they are not, we seem to have a problem for the very idea of compositionality.

Denials Denial 2: ambiguity The problem of ambiguity

So far our we have assumed that meaning bearers have at most one meaning.

4 of 20 Put differently, there is a common assumption that ambiguity is either lexical or structural. The term algebra framework deals with both adequately. But if grammar and syntactic structure — as many believe — are not only tools to construct unambiguous meaning bearers, but have a psychological reality tied to how we process utterances or sentences, then one needs an independent argument for why these are always unambiguous. If they are not, we seem to have a problem for the very idea of compositionality.

Denials Denial 2: ambiguity The problem of ambiguity

So far our we have assumed that meaning bearers have at most one meaning. Montague simply assumes that disambiguation takes place first (without saying how).

4 of 20 The term algebra framework deals with both adequately. But if grammar and syntactic structure — as many believe — are not only tools to construct unambiguous meaning bearers, but have a psychological reality tied to how we process utterances or sentences, then one needs an independent argument for why these are always unambiguous. If they are not, we seem to have a problem for the very idea of compositionality.

Denials Denial 2: ambiguity The problem of ambiguity

So far our we have assumed that meaning bearers have at most one meaning. Montague simply assumes that disambiguation takes place first (without saying how). Put differently, there is a common assumption that ambiguity is either lexical or structural.

4 of 20 But if grammar and syntactic structure — as many believe — are not only tools to construct unambiguous meaning bearers, but have a psychological reality tied to how we process utterances or sentences, then one needs an independent argument for why these are always unambiguous. If they are not, we seem to have a problem for the very idea of compositionality.

Denials Denial 2: ambiguity The problem of ambiguity

So far our we have assumed that meaning bearers have at most one meaning. Montague simply assumes that disambiguation takes place first (without saying how). Put differently, there is a common assumption that ambiguity is either lexical or structural. The term algebra framework deals with both adequately.

4 of 20 If they are not, we seem to have a problem for the very idea of compositionality.

Denials Denial 2: ambiguity The problem of ambiguity

So far our we have assumed that meaning bearers have at most one meaning. Montague simply assumes that disambiguation takes place first (without saying how). Put differently, there is a common assumption that ambiguity is either lexical or structural. The term algebra framework deals with both adequately. But if grammar and syntactic structure — as many believe — are not only tools to construct unambiguous meaning bearers, but have a psychological reality tied to how we process utterances or sentences, then one needs an independent argument for why these are always unambiguous.

4 of 20 Denials Denial 2: ambiguity The problem of ambiguity

So far our we have assumed that meaning bearers have at most one meaning. Montague simply assumes that disambiguation takes place first (without saying how). Put differently, there is a common assumption that ambiguity is either lexical or structural. The term algebra framework deals with both adequately. But if grammar and syntactic structure — as many believe — are not only tools to construct unambiguous meaning bearers, but have a psychological reality tied to how we process utterances or sentences, then one needs an independent argument for why these are always unambiguous. If they are not, we seem to have a problem for the very idea of compositionality.

4 of 20 This seems OK, since the distinct meanings have to be learned one by one anyway. As we have seen, structural ambiguity, as in (8) He saw her duck under the table (9) Old men and women were evacuated first (10) John and Mary or Sue will attend the meeting is dealt with by assigning meanings to derivations of surface strings rather than to the strings themselves. In the above cases, it seems realistic that disambiguation must take place ‘before’ meaning is assigned.

Denials Denial 2: ambiguity Option 1: more fine-grained meaning bearers

For lexical ambiguity, as in (7) Sue approached the bank with caution we simply introduce distinct atomic terms for the distinct meanings: V (bank1) = V (bank2) = bank.

5 of 20 As we have seen, structural ambiguity, as in (8) He saw her duck under the table (9) Old men and women were evacuated first (10) John and Mary or Sue will attend the meeting is dealt with by assigning meanings to derivations of surface strings rather than to the strings themselves. In the above cases, it seems realistic that disambiguation must take place ‘before’ meaning is assigned.

Denials Denial 2: ambiguity Option 1: more fine-grained meaning bearers

For lexical ambiguity, as in (7) Sue approached the bank with caution we simply introduce distinct atomic terms for the distinct meanings: V (bank1) = V (bank2) = bank. This seems OK, since the distinct meanings have to be learned one by one anyway.

5 of 20 In the above cases, it seems realistic that disambiguation must take place ‘before’ meaning is assigned.

Denials Denial 2: ambiguity Option 1: more fine-grained meaning bearers

For lexical ambiguity, as in (7) Sue approached the bank with caution we simply introduce distinct atomic terms for the distinct meanings: V (bank1) = V (bank2) = bank. This seems OK, since the distinct meanings have to be learned one by one anyway. As we have seen, structural ambiguity, as in (8) He saw her duck under the table (9) Old men and women were evacuated first (10) John and Mary or Sue will attend the meeting is dealt with by assigning meanings to derivations of surface strings rather than to the strings themselves.

5 of 20 Denials Denial 2: ambiguity Option 1: more fine-grained meaning bearers

For lexical ambiguity, as in (7) Sue approached the bank with caution we simply introduce distinct atomic terms for the distinct meanings: V (bank1) = V (bank2) = bank. This seems OK, since the distinct meanings have to be learned one by one anyway. As we have seen, structural ambiguity, as in (8) He saw her duck under the table (9) Old men and women were evacuated first (10) John and Mary or Sue will attend the meeting is dealt with by assigning meanings to derivations of surface strings rather than to the strings themselves. In the above cases, it seems realistic that disambiguation must take place ‘before’ meaning is assigned.

5 of 20 A special case of this is to use set meanings: the set meaning of t is the set of ordinary meanings of t. (Essentially this idea is at work in Cooper storage.) This guarantees that meaning association is single-valued, and thus possibly compositional. However, compositionality is by no means automatic with such approaches; it needs to be verified. A problem is that set meanings are purely theoretical constructs, and it is not clear that they play any role in linguistic communication.

Denials Denial 2: ambiguity Option 2: more coarse-grained meanings

Another way is to use underspecified meanings.

6 of 20 This guarantees that meaning association is single-valued, and thus possibly compositional. However, compositionality is by no means automatic with such approaches; it needs to be verified. A problem is that set meanings are purely theoretical constructs, and it is not clear that they play any role in linguistic communication.

Denials Denial 2: ambiguity Option 2: more coarse-grained meanings

Another way is to use underspecified meanings. A special case of this is to use set meanings: the set meaning of t is the set of ordinary meanings of t. (Essentially this idea is at work in Cooper storage.)

6 of 20 However, compositionality is by no means automatic with such approaches; it needs to be verified. A problem is that set meanings are purely theoretical constructs, and it is not clear that they play any role in linguistic communication.

Denials Denial 2: ambiguity Option 2: more coarse-grained meanings

Another way is to use underspecified meanings. A special case of this is to use set meanings: the set meaning of t is the set of ordinary meanings of t. (Essentially this idea is at work in Cooper storage.) This guarantees that meaning association is single-valued, and thus possibly compositional.

6 of 20 A problem is that set meanings are purely theoretical constructs, and it is not clear that they play any role in linguistic communication.

Denials Denial 2: ambiguity Option 2: more coarse-grained meanings

Another way is to use underspecified meanings. A special case of this is to use set meanings: the set meaning of t is the set of ordinary meanings of t. (Essentially this idea is at work in Cooper storage.) This guarantees that meaning association is single-valued, and thus possibly compositional. However, compositionality is by no means automatic with such approaches; it needs to be verified.

6 of 20 Denials Denial 2: ambiguity Option 2: more coarse-grained meanings

Another way is to use underspecified meanings. A special case of this is to use set meanings: the set meaning of t is the set of ordinary meanings of t. (Essentially this idea is at work in Cooper storage.) This guarantees that meaning association is single-valued, and thus possibly compositional. However, compositionality is by no means automatic with such approaches; it needs to be verified. A problem is that set meanings are purely theoretical constructs, and it is not clear that they play any role in linguistic communication.

6 of 20 It has been argued that although the structural ambiguities just exemplified are real, other proposed structural distinctions are entirely ad hoc. For example, although most of the linguistic community treats quantifier scope ambiguity as structural — whether dealt with `ala Montague or using LFs derived by Quantifier Raising — Pelletier argues that there is no independent evidence for such an analysis. Instead, he would say, a sentence like (11) Four critics reviewed three films has a single structural analysis — the obvious phrase structure — which doesn’t distinguish between its (at least) three readings. This ambiguity (if it exists) is neither lexical nor structural; call it essential.

Denials Denial 2: ambiguity Essential ambiguity?

Are there no other kinds of ambiguities than lexical and structural ones?

7 of 20 For example, although most of the linguistic community treats quantifier scope ambiguity as structural — whether dealt with `ala Montague or using LFs derived by Quantifier Raising — Pelletier argues that there is no independent evidence for such an analysis. Instead, he would say, a sentence like (11) Four critics reviewed three films has a single structural analysis — the obvious phrase structure — which doesn’t distinguish between its (at least) three readings. This ambiguity (if it exists) is neither lexical nor structural; call it essential.

Denials Denial 2: ambiguity Essential ambiguity?

Are there no other kinds of ambiguities than lexical and structural ones? It has been argued that although the structural ambiguities just exemplified are real, other proposed structural distinctions are entirely ad hoc.

7 of 20 Instead, he would say, a sentence like (11) Four critics reviewed three films has a single structural analysis — the obvious phrase structure — which doesn’t distinguish between its (at least) three readings. This ambiguity (if it exists) is neither lexical nor structural; call it essential.

Denials Denial 2: ambiguity Essential ambiguity?

Are there no other kinds of ambiguities than lexical and structural ones? It has been argued that although the structural ambiguities just exemplified are real, other proposed structural distinctions are entirely ad hoc. For example, although most of the linguistic community treats quantifier scope ambiguity as structural — whether dealt with `ala Montague or using LFs derived by Quantifier Raising — Pelletier argues that there is no independent evidence for such an analysis.

7 of 20 This ambiguity (if it exists) is neither lexical nor structural; call it essential.

Denials Denial 2: ambiguity Essential ambiguity?

Are there no other kinds of ambiguities than lexical and structural ones? It has been argued that although the structural ambiguities just exemplified are real, other proposed structural distinctions are entirely ad hoc. For example, although most of the linguistic community treats quantifier scope ambiguity as structural — whether dealt with `ala Montague or using LFs derived by Quantifier Raising — Pelletier argues that there is no independent evidence for such an analysis. Instead, he would say, a sentence like (11) Four critics reviewed three films has a single structural analysis — the obvious phrase structure — which doesn’t distinguish between its (at least) three readings.

7 of 20 Denials Denial 2: ambiguity Essential ambiguity?

Are there no other kinds of ambiguities than lexical and structural ones? It has been argued that although the structural ambiguities just exemplified are real, other proposed structural distinctions are entirely ad hoc. For example, although most of the linguistic community treats quantifier scope ambiguity as structural — whether dealt with `ala Montague or using LFs derived by Quantifier Raising — Pelletier argues that there is no independent evidence for such an analysis. Instead, he would say, a sentence like (11) Four critics reviewed three films has a single structural analysis — the obvious phrase structure — which doesn’t distinguish between its (at least) three readings. This ambiguity (if it exists) is neither lexical nor structural; call it essential.

7 of 20 Suppose we accept that a semantics is a relation (not necessarily a function) between GT and M. Must we then give up compositionality?

Denials Denial 2: ambiguity Essential ambiguity, cont.

Jeff Pelletier: Since English has essential ambiguities, it is not compositional.

8 of 20 Must we then give up compositionality?

Denials Denial 2: ambiguity Essential ambiguity, cont.

Jeff Pelletier: Since English has essential ambiguities, it is not compositional. Suppose we accept that a semantics is a relation (not necessarily a function) between GT and M.

8 of 20 Denials Denial 2: ambiguity Essential ambiguity, cont.

Jeff Pelletier: Since English has essential ambiguities, it is not compositional. Suppose we accept that a semantics is a relation (not necessarily a function) between GT and M. Must we then give up compositionality?

8 of 20 This means that three meaning operations, rα,1, rα,2, and rα,3, correspond to the syntactic operator α. Interpreting (12) involves choosing one of them. This seems in principle no different from choosing an interpretation for bank in (14) Sue approached the bank with caution. Semantic theory specifies the possibilities but doesn’t say how to choose. So relational compositionality should roughly say: (RComp) Each meaning of a compound expression is determined by some meanings of its (immediate) parts and the mode of composition.

Denials Denial 2: ambiguity Compositionality for relational semantics: an idea

Suppose (12) corresponds to the single term (13): (12) Four critics reviewed three films (13) α(β(four, critic), γ(reviewed, β(three, film))) where all proper subterms of (13) are unambiguous.

9 of 20 Interpreting (12) involves choosing one of them. This seems in principle no different from choosing an interpretation for bank in (14) Sue approached the bank with caution. Semantic theory specifies the possibilities but doesn’t say how to choose. So relational compositionality should roughly say: (RComp) Each meaning of a compound expression is determined by some meanings of its (immediate) parts and the mode of composition.

Denials Denial 2: ambiguity Compositionality for relational semantics: an idea

Suppose (12) corresponds to the single term (13): (12) Four critics reviewed three films (13) α(β(four, critic), γ(reviewed, β(three, film))) where all proper subterms of (13) are unambiguous.

This means that three meaning operations, rα,1, rα,2, and rα,3, correspond to the syntactic operator α.

9 of 20 This seems in principle no different from choosing an interpretation for bank in (14) Sue approached the bank with caution. Semantic theory specifies the possibilities but doesn’t say how to choose. So relational compositionality should roughly say: (RComp) Each meaning of a compound expression is determined by some meanings of its (immediate) parts and the mode of composition.

Denials Denial 2: ambiguity Compositionality for relational semantics: an idea

Suppose (12) corresponds to the single term (13): (12) Four critics reviewed three films (13) α(β(four, critic), γ(reviewed, β(three, film))) where all proper subterms of (13) are unambiguous.

This means that three meaning operations, rα,1, rα,2, and rα,3, correspond to the syntactic operator α. Interpreting (12) involves choosing one of them.

9 of 20 Semantic theory specifies the possibilities but doesn’t say how to choose. So relational compositionality should roughly say: (RComp) Each meaning of a compound expression is determined by some meanings of its (immediate) parts and the mode of composition.

Denials Denial 2: ambiguity Compositionality for relational semantics: an idea

Suppose (12) corresponds to the single term (13): (12) Four critics reviewed three films (13) α(β(four, critic), γ(reviewed, β(three, film))) where all proper subterms of (13) are unambiguous.

This means that three meaning operations, rα,1, rα,2, and rα,3, correspond to the syntactic operator α. Interpreting (12) involves choosing one of them. This seems in principle no different from choosing an interpretation for bank in (14) Sue approached the bank with caution.

9 of 20 So relational compositionality should roughly say: (RComp) Each meaning of a compound expression is determined by some meanings of its (immediate) parts and the mode of composition.

Denials Denial 2: ambiguity Compositionality for relational semantics: an idea

Suppose (12) corresponds to the single term (13): (12) Four critics reviewed three films (13) α(β(four, critic), γ(reviewed, β(three, film))) where all proper subterms of (13) are unambiguous.

This means that three meaning operations, rα,1, rα,2, and rα,3, correspond to the syntactic operator α. Interpreting (12) involves choosing one of them. This seems in principle no different from choosing an interpretation for bank in (14) Sue approached the bank with caution. Semantic theory specifies the possibilities but doesn’t say how to choose.

9 of 20 Denials Denial 2: ambiguity Compositionality for relational semantics: an idea

Suppose (12) corresponds to the single term (13): (12) Four critics reviewed three films (13) α(β(four, critic), γ(reviewed, β(three, film))) where all proper subterms of (13) are unambiguous.

This means that three meaning operations, rα,1, rα,2, and rα,3, correspond to the syntactic operator α. Interpreting (12) involves choosing one of them. This seems in principle no different from choosing an interpretation for bank in (14) Sue approached the bank with caution. Semantic theory specifies the possibilities but doesn’t say how to choose. So relational compositionality should roughly say: (RComp) Each meaning of a compound expression is determined by some meanings of its (immediate) parts and the mode of composition.

9 of 20 As long as these are specified in advance, for each α, we can make sense of the notion that the meanings of a compound expression are determined by the meanings of its parts and the mode of composition. The presence of more than one operation corresponding to α is a new source of ambiguity, in principle not distinct from lexical ambiguity. It requires some work, however, to formulate a precise version of (RComp) which (i) is not trivially satisfied; (ii) reduces to ordinary compositionality when the semantics is single-valued; and (iii) works in the presence of lexical ambiguity.

Denials Denial 2: ambiguity Compositionality for relational semantics, cont.

We still have semantic operations corresponding to syntactic rules.

10 of 20 The presence of more than one operation corresponding to α is a new source of ambiguity, in principle not distinct from lexical ambiguity. It requires some work, however, to formulate a precise version of (RComp) which (i) is not trivially satisfied; (ii) reduces to ordinary compositionality when the semantics is single-valued; and (iii) works in the presence of lexical ambiguity.

Denials Denial 2: ambiguity Compositionality for relational semantics, cont.

We still have semantic operations corresponding to syntactic rules. As long as these are specified in advance, for each α, we can make sense of the notion that the meanings of a compound expression are determined by the meanings of its parts and the mode of composition.

10 of 20 It requires some work, however, to formulate a precise version of (RComp) which (i) is not trivially satisfied; (ii) reduces to ordinary compositionality when the semantics is single-valued; and (iii) works in the presence of lexical ambiguity.

Denials Denial 2: ambiguity Compositionality for relational semantics, cont.

We still have semantic operations corresponding to syntactic rules. As long as these are specified in advance, for each α, we can make sense of the notion that the meanings of a compound expression are determined by the meanings of its parts and the mode of composition. The presence of more than one operation corresponding to α is a new source of ambiguity, in principle not distinct from lexical ambiguity.

10 of 20 Denials Denial 2: ambiguity Compositionality for relational semantics, cont.

We still have semantic operations corresponding to syntactic rules. As long as these are specified in advance, for each α, we can make sense of the notion that the meanings of a compound expression are determined by the meanings of its parts and the mode of composition. The presence of more than one operation corresponding to α is a new source of ambiguity, in principle not distinct from lexical ambiguity. It requires some work, however, to formulate a precise version of (RComp) which (i) is not trivially satisfied; (ii) reduces to ordinary compositionality when the semantics is single-valued; and (iii) works in the presence of lexical ambiguity.

10 of 20 But often what is meant is just that the usual meaning composition rules, when applied to, say, µ(kick) and µ(the bucket), don’t give the idiomatic meaning of kick the bucket. This is obvious, but has little to do with compositionality. A more precise usage is from Nunberg, Sag and Wasow (1994), where kick the bucket is called non-compositional, whereas e.g. pull strings is compositional: composed from idiomatic meanings of pull and string. This is reflected in contrasts such as: (15) John kicked the bucket two days ago. (16) ?The bucket was kicked by John two days ago (17) ?John kicked the bucket two days ago, but Lucy didn’t kick it (18) Strings were pulled to secure Henry his position (19) Mary pulled some strings on Bill’s behalf, but they didn’t get him the job

Denials Denial 3: idioms Are idioms a problem for compositionality?

It is frequently stated that they are a problem (or that they are ’non-compositional’).

11 of 20 This is obvious, but has little to do with compositionality. A more precise usage is from Nunberg, Sag and Wasow (1994), where kick the bucket is called non-compositional, whereas e.g. pull strings is compositional: composed from idiomatic meanings of pull and string. This is reflected in contrasts such as: (15) John kicked the bucket two days ago. (16) ?The bucket was kicked by John two days ago (17) ?John kicked the bucket two days ago, but Lucy didn’t kick it (18) Strings were pulled to secure Henry his position (19) Mary pulled some strings on Bill’s behalf, but they didn’t get him the job

Denials Denial 3: idioms Are idioms a problem for compositionality?

It is frequently stated that they are a problem (or that they are ’non-compositional’). But often what is meant is just that the usual meaning composition rules, when applied to, say, µ(kick) and µ(the bucket), don’t give the idiomatic meaning of kick the bucket.

11 of 20 A more precise usage is from Nunberg, Sag and Wasow (1994), where kick the bucket is called non-compositional, whereas e.g. pull strings is compositional: composed from idiomatic meanings of pull and string. This is reflected in contrasts such as: (15) John kicked the bucket two days ago. (16) ?The bucket was kicked by John two days ago (17) ?John kicked the bucket two days ago, but Lucy didn’t kick it (18) Strings were pulled to secure Henry his position (19) Mary pulled some strings on Bill’s behalf, but they didn’t get him the job

Denials Denial 3: idioms Are idioms a problem for compositionality?

It is frequently stated that they are a problem (or that they are ’non-compositional’). But often what is meant is just that the usual meaning composition rules, when applied to, say, µ(kick) and µ(the bucket), don’t give the idiomatic meaning of kick the bucket. This is obvious, but has little to do with compositionality.

11 of 20 (15) John kicked the bucket two days ago. (16) ?The bucket was kicked by John two days ago (17) ?John kicked the bucket two days ago, but Lucy didn’t kick it (18) Strings were pulled to secure Henry his position (19) Mary pulled some strings on Bill’s behalf, but they didn’t get him the job

Denials Denial 3: idioms Are idioms a problem for compositionality?

It is frequently stated that they are a problem (or that they are ’non-compositional’). But often what is meant is just that the usual meaning composition rules, when applied to, say, µ(kick) and µ(the bucket), don’t give the idiomatic meaning of kick the bucket. This is obvious, but has little to do with compositionality. A more precise usage is from Nunberg, Sag and Wasow (1994), where kick the bucket is called non-compositional, whereas e.g. pull strings is compositional: composed from idiomatic meanings of pull and string. This is reflected in contrasts such as:

11 of 20 (18) Strings were pulled to secure Henry his position (19) Mary pulled some strings on Bill’s behalf, but they didn’t get him the job

Denials Denial 3: idioms Are idioms a problem for compositionality?

It is frequently stated that they are a problem (or that they are ’non-compositional’). But often what is meant is just that the usual meaning composition rules, when applied to, say, µ(kick) and µ(the bucket), don’t give the idiomatic meaning of kick the bucket. This is obvious, but has little to do with compositionality. A more precise usage is from Nunberg, Sag and Wasow (1994), where kick the bucket is called non-compositional, whereas e.g. pull strings is compositional: composed from idiomatic meanings of pull and string. This is reflected in contrasts such as: (15) John kicked the bucket two days ago. (16) ?The bucket was kicked by John two days ago (17) ?John kicked the bucket two days ago, but Lucy didn’t kick it

11 of 20 Denials Denial 3: idioms Are idioms a problem for compositionality?

It is frequently stated that they are a problem (or that they are ’non-compositional’). But often what is meant is just that the usual meaning composition rules, when applied to, say, µ(kick) and µ(the bucket), don’t give the idiomatic meaning of kick the bucket. This is obvious, but has little to do with compositionality. A more precise usage is from Nunberg, Sag and Wasow (1994), where kick the bucket is called non-compositional, whereas e.g. pull strings is compositional: composed from idiomatic meanings of pull and string. This is reflected in contrasts such as: (15) John kicked the bucket two days ago. (16) ?The bucket was kicked by John two days ago (17) ?John kicked the bucket two days ago, but Lucy didn’t kick it (18) Strings were pulled to secure Henry his position (19) Mary pulled some strings on Bill’s behalf, but they didn’t get him the job 11 of 20 If bucket ≡µ pail, how can we prevent (the idiomatic) kick the bucket ≡µ kick the pail, while maintaining compositionality? If pull has an idiomatic meaning in pull strings, how can we prevent that meaning to be used in e.g. pull the rope?

Do idioms have structure, or are some idioms atoms? Do some idioms have syntactic but not semantic structure, whereas others have both? Do answers to these questions explain the contrasting behavior with respect to passivization and anaphora? Other typical issues:

Denials Denial 3: idioms Issues about idioms

There are all kinds of interesting issues in the syntax and semantics of idioms, many of which can be formulated within the abstract framework we are using. For example:

12 of 20 If bucket ≡µ pail, how can we prevent (the idiomatic) kick the bucket ≡µ kick the pail, while maintaining compositionality? If pull has an idiomatic meaning in pull strings, how can we prevent that meaning to be used in e.g. pull the rope?

Do some idioms have syntactic but not semantic structure, whereas others have both? Do answers to these questions explain the contrasting behavior with respect to passivization and anaphora? Other typical issues:

Denials Denial 3: idioms Issues about idioms

There are all kinds of interesting issues in the syntax and semantics of idioms, many of which can be formulated within the abstract framework we are using. For example: Do idioms have structure, or are some idioms atoms?

12 of 20 If bucket ≡µ pail, how can we prevent (the idiomatic) kick the bucket ≡µ kick the pail, while maintaining compositionality? If pull has an idiomatic meaning in pull strings, how can we prevent that meaning to be used in e.g. pull the rope?

Do answers to these questions explain the contrasting behavior with respect to passivization and anaphora? Other typical issues:

Denials Denial 3: idioms Issues about idioms

There are all kinds of interesting issues in the syntax and semantics of idioms, many of which can be formulated within the abstract framework we are using. For example: Do idioms have structure, or are some idioms atoms? Do some idioms have syntactic but not semantic structure, whereas others have both?

12 of 20 If bucket ≡µ pail, how can we prevent (the idiomatic) kick the bucket ≡µ kick the pail, while maintaining compositionality? If pull has an idiomatic meaning in pull strings, how can we prevent that meaning to be used in e.g. pull the rope?

Other typical issues:

Denials Denial 3: idioms Issues about idioms

There are all kinds of interesting issues in the syntax and semantics of idioms, many of which can be formulated within the abstract framework we are using. For example: Do idioms have structure, or are some idioms atoms? Do some idioms have syntactic but not semantic structure, whereas others have both? Do answers to these questions explain the contrasting behavior with respect to passivization and anaphora?

12 of 20 If bucket ≡µ pail, how can we prevent (the idiomatic) kick the bucket ≡µ kick the pail, while maintaining compositionality? If pull has an idiomatic meaning in pull strings, how can we prevent that meaning to be used in e.g. pull the rope?

Denials Denial 3: idioms Issues about idioms

There are all kinds of interesting issues in the syntax and semantics of idioms, many of which can be formulated within the abstract framework we are using. For example: Do idioms have structure, or are some idioms atoms? Do some idioms have syntactic but not semantic structure, whereas others have both? Do answers to these questions explain the contrasting behavior with respect to passivization and anaphora? Other typical issues:

12 of 20 If pull has an idiomatic meaning in pull strings, how can we prevent that meaning to be used in e.g. pull the rope?

Denials Denial 3: idioms Issues about idioms

There are all kinds of interesting issues in the syntax and semantics of idioms, many of which can be formulated within the abstract framework we are using. For example: Do idioms have structure, or are some idioms atoms? Do some idioms have syntactic but not semantic structure, whereas others have both? Do answers to these questions explain the contrasting behavior with respect to passivization and anaphora? Other typical issues:

If bucket ≡µ pail, how can we prevent (the idiomatic) kick the bucket ≡µ kick the pail, while maintaining compositionality?

12 of 20 Denials Denial 3: idioms Issues about idioms

There are all kinds of interesting issues in the syntax and semantics of idioms, many of which can be formulated within the abstract framework we are using. For example: Do idioms have structure, or are some idioms atoms? Do some idioms have syntactic but not semantic structure, whereas others have both? Do answers to these questions explain the contrasting behavior with respect to passivization and anaphora? Other typical issues:

If bucket ≡µ pail, how can we prevent (the idiomatic) kick the bucket ≡µ kick the pail, while maintaining compositionality? If pull has an idiomatic meaning in pull strings, how can we prevent that meaning to be used in e.g. pull the rope?

12 of 20 Suppose a complex expression in a given compositional language L acquires the status of an idiom, with a new meaning (paraphrasable in L or not) and new ways of combining with other expressions. Can the given grammar and semantics be extended in a way which preserves compositionality and generates the expected meaningful expressions with the expected meanings (and is perhaps unique up to equivalence with these properties)? Different kinds of extensions may be required for different kinds of idioms, but usually these extension problems have positive solutions (cf. W-hl 2002).

In conclusion: Compositionality is not a problem specifically for idioms.

Denials Denial 3: idioms Compositional extension problems for idioms

These issues also relate to compositionality, more precisely to compositional extension problems:

13 of 20 Can the given grammar and semantics be extended in a way which preserves compositionality and generates the expected meaningful expressions with the expected meanings (and is perhaps unique up to equivalence with these properties)? Different kinds of extensions may be required for different kinds of idioms, but usually these extension problems have positive solutions (cf. W-hl 2002).

In conclusion: Compositionality is not a problem specifically for idioms.

Denials Denial 3: idioms Compositional extension problems for idioms

These issues also relate to compositionality, more precisely to compositional extension problems: Suppose a complex expression in a given compositional language L acquires the status of an idiom, with a new meaning (paraphrasable in L or not) and new ways of combining with other expressions.

13 of 20 Different kinds of extensions may be required for different kinds of idioms, but usually these extension problems have positive solutions (cf. W-hl 2002).

In conclusion: Compositionality is not a problem specifically for idioms.

Denials Denial 3: idioms Compositional extension problems for idioms

These issues also relate to compositionality, more precisely to compositional extension problems: Suppose a complex expression in a given compositional language L acquires the status of an idiom, with a new meaning (paraphrasable in L or not) and new ways of combining with other expressions. Can the given grammar and semantics be extended in a way which preserves compositionality and generates the expected meaningful expressions with the expected meanings (and is perhaps unique up to equivalence with these properties)?

13 of 20 In conclusion: Compositionality is not a problem specifically for idioms.

Denials Denial 3: idioms Compositional extension problems for idioms

These issues also relate to compositionality, more precisely to compositional extension problems: Suppose a complex expression in a given compositional language L acquires the status of an idiom, with a new meaning (paraphrasable in L or not) and new ways of combining with other expressions. Can the given grammar and semantics be extended in a way which preserves compositionality and generates the expected meaningful expressions with the expected meanings (and is perhaps unique up to equivalence with these properties)? Different kinds of extensions may be required for different kinds of idioms, but usually these extension problems have positive solutions (cf. W-hl 2002).

13 of 20 Denials Denial 3: idioms Compositional extension problems for idioms

These issues also relate to compositionality, more precisely to compositional extension problems: Suppose a complex expression in a given compositional language L acquires the status of an idiom, with a new meaning (paraphrasable in L or not) and new ways of combining with other expressions. Can the given grammar and semantics be extended in a way which preserves compositionality and generates the expected meaningful expressions with the expected meanings (and is perhaps unique up to equivalence with these properties)? Different kinds of extensions may be required for different kinds of idioms, but usually these extension problems have positive solutions (cf. W-hl 2002).

In conclusion: Compositionality is not a problem specifically for idioms.

13 of 20 (B) Important fragments of a natural language may be compositional, even though the whole language (if this phrase makes sense) isn’t.

(C) Linguistic phenomena that seem to threaten compositionality may on a closer look be compatible with it.

Still, (A) may create an impression that, if it is up to us to specify the syntax and semantics, there is no factual question of compositionality left. In other words, the compositionality issue is empty....

Denials Conclusions Conclusion

(A) Claims that some formal or natural language is not compositional may be true relative to one (syntax, semantics) pair, and false relative to another.

14 of 20 (C) Linguistic phenomena that seem to threaten compositionality may on a closer look be compatible with it.

Still, (A) may create an impression that, if it is up to us to specify the syntax and semantics, there is no factual question of compositionality left. In other words, the compositionality issue is empty....

Denials Conclusions Conclusion

(A) Claims that some formal or natural language is not compositional may be true relative to one (syntax, semantics) pair, and false relative to another.

(B) Important fragments of a natural language may be compositional, even though the whole language (if this phrase makes sense) isn’t.

14 of 20 Still, (A) may create an impression that, if it is up to us to specify the syntax and semantics, there is no factual question of compositionality left. In other words, the compositionality issue is empty....

Denials Conclusions Conclusion

(A) Claims that some formal or natural language is not compositional may be true relative to one (syntax, semantics) pair, and false relative to another.

(B) Important fragments of a natural language may be compositional, even though the whole language (if this phrase makes sense) isn’t.

(C) Linguistic phenomena that seem to threaten compositionality may on a closer look be compatible with it.

14 of 20 Denials Conclusions Conclusion

(A) Claims that some formal or natural language is not compositional may be true relative to one (syntax, semantics) pair, and false relative to another.

(B) Important fragments of a natural language may be compositional, even though the whole language (if this phrase makes sense) isn’t.

(C) Linguistic phenomena that seem to threaten compositionality may on a closer look be compatible with it.

Still, (A) may create an impression that, if it is up to us to specify the syntax and semantics, there is no factual question of compositionality left. In other words, the compositionality issue is empty....

14 of 20 µ is then compositional, even if m is not, with function application as the semantic operation (recall Heim & Kratzer), and m can be recovered from µ. NB µ(a · a) = µ(a)(µ(a)), so the function µ(a) is applied to itself, which is not possible in standard set theory. Indeed, Zadrozny shows the existence of µ by appeal to non-wellfounded set theory. Zadrozny concludes that compositionality is an empty requirement.

Emptiness Can every semantics be made compositional?

Theorem (Zadrozny, 1994) Let A = (A, ·) be a partial algebra, where the binary operation · can be thought of as concatenation (then A is a set of strings of symbols from some alphabet), and let m : A −→ M. Then there exists a set M∗ (of functions) and a function µ : A −→ M∗ such that for all a, b ∈ A, µ(a · b) = µ(a)(µ(b)) µ(a · a) = m(a)

15 of 20 NB µ(a · a) = µ(a)(µ(a)), so the function µ(a) is applied to itself, which is not possible in standard set theory. Indeed, Zadrozny shows the existence of µ by appeal to non-wellfounded set theory. Zadrozny concludes that compositionality is an empty requirement.

Emptiness Can every semantics be made compositional?

Theorem (Zadrozny, 1994) Let A = (A, ·) be a partial algebra, where the binary operation · can be thought of as concatenation (then A is a set of strings of symbols from some alphabet), and let m : A −→ M. Then there exists a set M∗ (of functions) and a function µ : A −→ M∗ such that for all a, b ∈ A, µ(a · b) = µ(a)(µ(b)) µ(a · a) = m(a)

µ is then compositional, even if m is not, with function application as the semantic operation (recall Heim & Kratzer), and m can be recovered from µ.

15 of 20 Zadrozny concludes that compositionality is an empty requirement.

Emptiness Can every semantics be made compositional?

Theorem (Zadrozny, 1994) Let A = (A, ·) be a partial algebra, where the binary operation · can be thought of as concatenation (then A is a set of strings of symbols from some alphabet), and let m : A −→ M. Then there exists a set M∗ (of functions) and a function µ : A −→ M∗ such that for all a, b ∈ A, µ(a · b) = µ(a)(µ(b)) µ(a · a) = m(a)

µ is then compositional, even if m is not, with function application as the semantic operation (recall Heim & Kratzer), and m can be recovered from µ. NB µ(a · a) = µ(a)(µ(a)), so the function µ(a) is applied to itself, which is not possible in standard set theory. Indeed, Zadrozny shows the existence of µ by appeal to non-wellfounded set theory.

15 of 20 Emptiness Can every semantics be made compositional?

Theorem (Zadrozny, 1994) Let A = (A, ·) be a partial algebra, where the binary operation · can be thought of as concatenation (then A is a set of strings of symbols from some alphabet), and let m : A −→ M. Then there exists a set M∗ (of functions) and a function µ : A −→ M∗ such that for all a, b ∈ A, µ(a · b) = µ(a)(µ(b)) µ(a · a) = m(a)

µ is then compositional, even if m is not, with function application as the semantic operation (recall Heim & Kratzer), and m can be recovered from µ. NB µ(a · a) = µ(a)(µ(a)), so the function µ(a) is applied to itself, which is not possible in standard set theory. Indeed, Zadrozny shows the existence of µ by appeal to non-wellfounded set theory. Zadrozny concludes that compositionality is an empty requirement.

15 of 20 ∗ ∗ Proof: Let µ (t) = hµ(t), ti. µ is 1-1, hence compositional. 

Zadrozny’s µ is one-one, hence trivially compositional. No independent reason is given why we should think it is an interesting semantics. Disregarding (non-wellfounded) function application, the following simpler version of Zadrozny’s claim holds: For every semantics µ for GT there is a compositional semantics µ∗ for GT from which µ can be recovered.

But µ∗ has no independent interest. It doesn’t resolve any interesting counter-instances we may have had to the compositionality of µ. For example, applied to the issue of belief sentences “A believes that ϕ” discussed earlier, this would have A believing the ordered pair of the truth value (or the intension) of ϕ and ϕ itself. Few would think this resolves the compositionality issue.

Emptiness Zadrozny, cont.

BUT:

16 of 20 ∗ ∗ Proof: Let µ (t) = hµ(t), ti. µ is 1-1, hence compositional. 

No independent reason is given why we should think it is an interesting semantics. Disregarding (non-wellfounded) function application, the following simpler version of Zadrozny’s claim holds: For every semantics µ for GT there is a compositional semantics µ∗ for GT from which µ can be recovered.

But µ∗ has no independent interest. It doesn’t resolve any interesting counter-instances we may have had to the compositionality of µ. For example, applied to the issue of belief sentences “A believes that ϕ” discussed earlier, this would have A believing the ordered pair of the truth value (or the intension) of ϕ and ϕ itself. Few would think this resolves the compositionality issue.

Emptiness Zadrozny, cont.

BUT: Zadrozny’s µ is one-one, hence trivially compositional.

16 of 20 ∗ ∗ Proof: Let µ (t) = hµ(t), ti. µ is 1-1, hence compositional. 

Disregarding (non-wellfounded) function application, the following simpler version of Zadrozny’s claim holds: For every semantics µ for GT there is a compositional semantics µ∗ for GT from which µ can be recovered.

But µ∗ has no independent interest. It doesn’t resolve any interesting counter-instances we may have had to the compositionality of µ. For example, applied to the issue of belief sentences “A believes that ϕ” discussed earlier, this would have A believing the ordered pair of the truth value (or the intension) of ϕ and ϕ itself. Few would think this resolves the compositionality issue.

Emptiness Zadrozny, cont.

BUT: Zadrozny’s µ is one-one, hence trivially compositional. No independent reason is given why we should think it is an interesting semantics.

16 of 20 ∗ ∗ Proof: Let µ (t) = hµ(t), ti. µ is 1-1, hence compositional.  But µ∗ has no independent interest. It doesn’t resolve any interesting counter-instances we may have had to the compositionality of µ. For example, applied to the issue of belief sentences “A believes that ϕ” discussed earlier, this would have A believing the ordered pair of the truth value (or the intension) of ϕ and ϕ itself. Few would think this resolves the compositionality issue.

Emptiness Zadrozny, cont.

BUT: Zadrozny’s µ is one-one, hence trivially compositional. No independent reason is given why we should think it is an interesting semantics. Disregarding (non-wellfounded) function application, the following simpler version of Zadrozny’s claim holds: For every semantics µ for GT there is a compositional semantics µ∗ for GT from which µ can be recovered.

16 of 20 But µ∗ has no independent interest. It doesn’t resolve any interesting counter-instances we may have had to the compositionality of µ. For example, applied to the issue of belief sentences “A believes that ϕ” discussed earlier, this would have A believing the ordered pair of the truth value (or the intension) of ϕ and ϕ itself. Few would think this resolves the compositionality issue.

Emptiness Zadrozny, cont.

BUT: Zadrozny’s µ is one-one, hence trivially compositional. No independent reason is given why we should think it is an interesting semantics. Disregarding (non-wellfounded) function application, the following simpler version of Zadrozny’s claim holds: For every semantics µ for GT there is a compositional semantics µ∗ for GT from which µ can be recovered. ∗ ∗ Proof: Let µ (t) = hµ(t), ti. µ is 1-1, hence compositional. 

16 of 20 For example, applied to the issue of belief sentences “A believes that ϕ” discussed earlier, this would have A believing the ordered pair of the truth value (or the intension) of ϕ and ϕ itself. Few would think this resolves the compositionality issue.

Emptiness Zadrozny, cont.

BUT: Zadrozny’s µ is one-one, hence trivially compositional. No independent reason is given why we should think it is an interesting semantics. Disregarding (non-wellfounded) function application, the following simpler version of Zadrozny’s claim holds: For every semantics µ for GT there is a compositional semantics µ∗ for GT from which µ can be recovered. ∗ ∗ Proof: Let µ (t) = hµ(t), ti. µ is 1-1, hence compositional.  But µ∗ has no independent interest. It doesn’t resolve any interesting counter-instances we may have had to the compositionality of µ.

16 of 20 Emptiness Zadrozny, cont.

BUT: Zadrozny’s µ is one-one, hence trivially compositional. No independent reason is given why we should think it is an interesting semantics. Disregarding (non-wellfounded) function application, the following simpler version of Zadrozny’s claim holds: For every semantics µ for GT there is a compositional semantics µ∗ for GT from which µ can be recovered. ∗ ∗ Proof: Let µ (t) = hµ(t), ti. µ is 1-1, hence compositional.  But µ∗ has no independent interest. It doesn’t resolve any interesting counter-instances we may have had to the compositionality of µ. For example, applied to the issue of belief sentences “A believes that ϕ” discussed earlier, this would have A believing the ordered pair of the truth value (or the intension) of ϕ and ϕ itself. Few would think this resolves the compositionality issue.

16 of 20 No simple recipe for always finding another semantics (somehow related to µ), possibly for another grammar, resolves this problem. Alternative semantics and grammars may be suggested, but must be judged on their own merits. Compare Hodges’ extension theorem: it proves mathematically the existence of a total compositional semantics, but that semantics has to agree with the given partial semantics. Nothing similar holds in Zadrozny’s case (and others). Even so, the agreement is only up to synonymy, so the representation problem remains, and is highly non-trivial. The theorem just tells us that it is worthwhile trying to solve it. In conclusion, compositionality is not an empty requirement...

Emptiness Emptiness: conclusion

Sometimes, a natural semantics µ for some grammar E turns out to be non-compositional.

17 of 20 Alternative semantics and grammars may be suggested, but must be judged on their own merits. Compare Hodges’ extension theorem: it proves mathematically the existence of a total compositional semantics, but that semantics has to agree with the given partial semantics. Nothing similar holds in Zadrozny’s case (and others). Even so, the agreement is only up to synonymy, so the representation problem remains, and is highly non-trivial. The theorem just tells us that it is worthwhile trying to solve it. In conclusion, compositionality is not an empty requirement...

Emptiness Emptiness: conclusion

Sometimes, a natural semantics µ for some grammar E turns out to be non-compositional. No simple recipe for always finding another semantics (somehow related to µ), possibly for another grammar, resolves this problem.

17 of 20 Compare Hodges’ extension theorem: it proves mathematically the existence of a total compositional semantics, but that semantics has to agree with the given partial semantics. Nothing similar holds in Zadrozny’s case (and others). Even so, the agreement is only up to synonymy, so the representation problem remains, and is highly non-trivial. The theorem just tells us that it is worthwhile trying to solve it. In conclusion, compositionality is not an empty requirement...

Emptiness Emptiness: conclusion

Sometimes, a natural semantics µ for some grammar E turns out to be non-compositional. No simple recipe for always finding another semantics (somehow related to µ), possibly for another grammar, resolves this problem. Alternative semantics and grammars may be suggested, but must be judged on their own merits.

17 of 20 Even so, the agreement is only up to synonymy, so the representation problem remains, and is highly non-trivial. The theorem just tells us that it is worthwhile trying to solve it. In conclusion, compositionality is not an empty requirement...

Emptiness Emptiness: conclusion

Sometimes, a natural semantics µ for some grammar E turns out to be non-compositional. No simple recipe for always finding another semantics (somehow related to µ), possibly for another grammar, resolves this problem. Alternative semantics and grammars may be suggested, but must be judged on their own merits. Compare Hodges’ extension theorem: it proves mathematically the existence of a total compositional semantics, but that semantics has to agree with the given partial semantics. Nothing similar holds in Zadrozny’s case (and others).

17 of 20 In conclusion, compositionality is not an empty requirement...

Emptiness Emptiness: conclusion

Sometimes, a natural semantics µ for some grammar E turns out to be non-compositional. No simple recipe for always finding another semantics (somehow related to µ), possibly for another grammar, resolves this problem. Alternative semantics and grammars may be suggested, but must be judged on their own merits. Compare Hodges’ extension theorem: it proves mathematically the existence of a total compositional semantics, but that semantics has to agree with the given partial semantics. Nothing similar holds in Zadrozny’s case (and others). Even so, the agreement is only up to synonymy, so the representation problem remains, and is highly non-trivial. The theorem just tells us that it is worthwhile trying to solve it.

17 of 20 Emptiness Emptiness: conclusion

Sometimes, a natural semantics µ for some grammar E turns out to be non-compositional. No simple recipe for always finding another semantics (somehow related to µ), possibly for another grammar, resolves this problem. Alternative semantics and grammars may be suggested, but must be judged on their own merits. Compare Hodges’ extension theorem: it proves mathematically the existence of a total compositional semantics, but that semantics has to agree with the given partial semantics. Nothing similar holds in Zadrozny’s case (and others). Even so, the agreement is only up to synonymy, so the representation problem remains, and is highly non-trivial. The theorem just tells us that it is worthwhile trying to solve it. In conclusion, compositionality is not an empty requirement...

17 of 20 These arguments presuppose that there exists an infinite (or very large) set of sentences in a language, and moreover an infinite (or very large) set of meanings already assigned to these sentences. If you think this is self-evident, the argument has some force. But the risk is that you think it is self-evident because finite grammars generate infinitely many sentences, and meanings reflect this, and then the argument may be circular. However, the success of linguistic communication between speakers is a phenomenon that can be independently tested (cf. the famous Frege quote). This phenomenon needs explanation.

Praise Are there good arguments in favor of compositionality?

There are often repeated familiar arguments that compositionality is required to explain how we can understand new sentences how we can learn a language

18 of 20 If you think this is self-evident, the argument has some force. But the risk is that you think it is self-evident because finite grammars generate infinitely many sentences, and meanings reflect this, and then the argument may be circular. However, the success of linguistic communication between speakers is a phenomenon that can be independently tested (cf. the famous Frege quote). This phenomenon needs explanation.

Praise Are there good arguments in favor of compositionality?

There are often repeated familiar arguments that compositionality is required to explain how we can understand new sentences how we can learn a language These arguments presuppose that there exists an infinite (or very large) set of sentences in a language, and moreover an infinite (or very large) set of meanings already assigned to these sentences.

18 of 20 But the risk is that you think it is self-evident because finite grammars generate infinitely many sentences, and meanings reflect this, and then the argument may be circular. However, the success of linguistic communication between speakers is a phenomenon that can be independently tested (cf. the famous Frege quote). This phenomenon needs explanation.

Praise Are there good arguments in favor of compositionality?

There are often repeated familiar arguments that compositionality is required to explain how we can understand new sentences how we can learn a language These arguments presuppose that there exists an infinite (or very large) set of sentences in a language, and moreover an infinite (or very large) set of meanings already assigned to these sentences. If you think this is self-evident, the argument has some force.

18 of 20 However, the success of linguistic communication between speakers is a phenomenon that can be independently tested (cf. the famous Frege quote). This phenomenon needs explanation.

Praise Are there good arguments in favor of compositionality?

There are often repeated familiar arguments that compositionality is required to explain how we can understand new sentences how we can learn a language These arguments presuppose that there exists an infinite (or very large) set of sentences in a language, and moreover an infinite (or very large) set of meanings already assigned to these sentences. If you think this is self-evident, the argument has some force. But the risk is that you think it is self-evident because finite grammars generate infinitely many sentences, and meanings reflect this, and then the argument may be circular.

18 of 20 Praise Are there good arguments in favor of compositionality?

There are often repeated familiar arguments that compositionality is required to explain how we can understand new sentences how we can learn a language These arguments presuppose that there exists an infinite (or very large) set of sentences in a language, and moreover an infinite (or very large) set of meanings already assigned to these sentences. If you think this is self-evident, the argument has some force. But the risk is that you think it is self-evident because finite grammars generate infinitely many sentences, and meanings reflect this, and then the argument may be circular. However, the success of linguistic communication between speakers is a phenomenon that can be independently tested (cf. the famous Frege quote). This phenomenon needs explanation.

18 of 20 However, basic compositionality must then be strengthened with a computability requirement. But then, wouldn’t a recursive semantics do just as well as a compositional one? One way to resolve this would be to show that compositional semantics is more efficient (requires fewer computation steps) than recursive semantics. Recent work by Peter Pagin appears to confirm this. Still, one wonders if there isn’t some other argument in favor of compositionality, i.e. of the substitution property, over recursivity (recursivity does not validate the substitution property). Food for further thought...

Praise Good arguments, cont.

Compositionality thus seems like an inference to the best explanation.

19 of 20 But then, wouldn’t a recursive semantics do just as well as a compositional one? One way to resolve this would be to show that compositional semantics is more efficient (requires fewer computation steps) than recursive semantics. Recent work by Peter Pagin appears to confirm this. Still, one wonders if there isn’t some other argument in favor of compositionality, i.e. of the substitution property, over recursivity (recursivity does not validate the substitution property). Food for further thought...

Praise Good arguments, cont.

Compositionality thus seems like an inference to the best explanation. However, basic compositionality must then be strengthened with a computability requirement.

19 of 20 One way to resolve this would be to show that compositional semantics is more efficient (requires fewer computation steps) than recursive semantics. Recent work by Peter Pagin appears to confirm this. Still, one wonders if there isn’t some other argument in favor of compositionality, i.e. of the substitution property, over recursivity (recursivity does not validate the substitution property). Food for further thought...

Praise Good arguments, cont.

Compositionality thus seems like an inference to the best explanation. However, basic compositionality must then be strengthened with a computability requirement. But then, wouldn’t a recursive semantics do just as well as a compositional one?

19 of 20 Recent work by Peter Pagin appears to confirm this. Still, one wonders if there isn’t some other argument in favor of compositionality, i.e. of the substitution property, over recursivity (recursivity does not validate the substitution property). Food for further thought...

Praise Good arguments, cont.

Compositionality thus seems like an inference to the best explanation. However, basic compositionality must then be strengthened with a computability requirement. But then, wouldn’t a recursive semantics do just as well as a compositional one? One way to resolve this would be to show that compositional semantics is more efficient (requires fewer computation steps) than recursive semantics.

19 of 20 Still, one wonders if there isn’t some other argument in favor of compositionality, i.e. of the substitution property, over recursivity (recursivity does not validate the substitution property). Food for further thought...

Praise Good arguments, cont.

Compositionality thus seems like an inference to the best explanation. However, basic compositionality must then be strengthened with a computability requirement. But then, wouldn’t a recursive semantics do just as well as a compositional one? One way to resolve this would be to show that compositional semantics is more efficient (requires fewer computation steps) than recursive semantics. Recent work by Peter Pagin appears to confirm this.

19 of 20 Food for further thought...

Praise Good arguments, cont.

Compositionality thus seems like an inference to the best explanation. However, basic compositionality must then be strengthened with a computability requirement. But then, wouldn’t a recursive semantics do just as well as a compositional one? One way to resolve this would be to show that compositional semantics is more efficient (requires fewer computation steps) than recursive semantics. Recent work by Peter Pagin appears to confirm this. Still, one wonders if there isn’t some other argument in favor of compositionality, i.e. of the substitution property, over recursivity (recursivity does not validate the substitution property).

19 of 20 Praise Good arguments, cont.

Compositionality thus seems like an inference to the best explanation. However, basic compositionality must then be strengthened with a computability requirement. But then, wouldn’t a recursive semantics do just as well as a compositional one? One way to resolve this would be to show that compositional semantics is more efficient (requires fewer computation steps) than recursive semantics. Recent work by Peter Pagin appears to confirm this. Still, one wonders if there isn’t some other argument in favor of compositionality, i.e. of the substitution property, over recursivity (recursivity does not validate the substitution property). Food for further thought...

19 of 20 Praise

THE END

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