# Some observations about plurals in textual mathematics

Philippe de Groote

► To cite this version:

Philippe de Groote. Some observations about plurals in textual mathematics. MCLP 2025 - International Conference on Mathematical and Computational Linguistics for Proofs, EuroProofNet, European Research Network on Formal Proofs, Sep 2025, Orsay, France. hal-05481104

HAL Id: hal-05481104

https://inria.hal.science/hal-05481104v1

Submitted on 28 Jan 2026

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Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
Some observations about plurals in textual mathematics

Philippe de Groote

Université de Lorraine, CNRS, Inria, LORIA, France

Mathematical and Computational Linguistics for Proofs
September 15-18, 2025

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Based on:

* unpublished joint work with Santiago Arambillete;

* discussions with Aarne Ranta, Hugo Herbelin, Paul-André Melliès, and Yoad Winter.

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* Textual Mathematics

* Compositional Semantics

* Plurals

* Collective vs Distributive Predicates

* Symmetric Predicates

* Conclusions

<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /></figure>
* Textual Mathematics

* Compositional Semantics

* Plurals

* Collective vs Distributive Predicates

* Symmetric Predicates

* Conclusions

<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>
* Textual Mathematics

* Compositional Semantics

* Plurals

* Collective vs Distributive Predicates

* Symmetric Predicates

* Conclusions

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /></figure>
* Textual Mathematics

* Compositional Semantics

* Plurals

* Collective vs Distributive Predicates

* Symmetric Predicates

* Conclusions

<figure><img src="image_13.png" /></figure>
* Textual Mathematics

* Compositional Semantics

* Plurals

* Collective vs Distributive Predicates

* Symmetric Predicates

* Conclusions

<figure><img src="image_14.png" /></figure>
* Textual Mathematics

* Compositional Semantics

* Plurals

* Collective vs Distributive Predicates

* Symmetric Predicates

* Conclusions

<figure><img src="image_15.png" /></figure>
* Textual Mathematics

* Compositional Semantics

* Plurals

* Collective vs Distributive Predicates

* Symmetric Predicates

* Conclusions

<figure><img src="image_16.png" /></figure>
LMCS 7805

Mohan Ganesalingam

# The Language of Mathematics

A Linguistic and Philosophical Investigation

<figure><img src="image_17.png" /></figure>

<figure><img src="image_18.png" /></figure>

Springer

<figure><img src="image_19.png" /></figure>

### Textual mathematics:

* the language used by mathematicians in textbooks and articles;

* consists of a mixture of natural language and mathematical formulas;

* it has its own idiosyncrasies that are worth studying.

<figure><img src="image_20.png" /></figure>
LMCS 7805

Mohan Ganesalingam

# The Language of Mathematics

A Linguistic and Philosophical Investigation

<figure><img src="image_21.png" /></figure>

<figure><img src="image_22.png" /></figure>

Springer

<figure><img src="image_23.png" /></figure>

### Textual mathematics:

* the language used by mathematicians in textbooks and articles;

* consists of a mixture of natural language and mathematical formulas;

* it has its own idiosyncrasies that are worth studying.

<figure><img src="image_24.png" /></figure>
LMCS 7905

Mohan Ganesalingam

# The Language of Mathematics

A Linguistic and Philosophical Investigation

<figure><img src="image_25.png" /></figure>

<figure><img src="image_26.png" /></figure>

Springer

<figure><img src="image_27.png" /></figure>

### Textual mathematics:

* the language used by mathematicians in textbooks and articles;

* consists of a mixture of natural language and mathematical formulas;

* it has its own idiosyncrasies that are worth studying.

<figure><img src="image_28.png" /></figure>
LMCS 7905

Mohan Ganesalingam

The Language
of Mathematics

A Linguistic and Philosophical Investigation

<figure><img src="image_29.png" /></figure>

<figure><img src="image_30.png" /></figure>

Springer

<figure><img src="image_31.png" /></figure>

Textual mathematics:

* the language used by mathematicians in textbooks and articles;

* consists of a mixture of natural language and mathematical formulas;

* it has its own idiosyncrasies that are worth studying.

<figure><img src="image_32.png" /></figure>
Linguistically, the study of mathematical language rather than everyday language is rewarding because it offers examples that have complicated grammatical structure but are free from ambiguities. We always know exactly what a sentence means, and there is a determinate structure to be revealed. The informal language of mathematics thus provides a kind of grammatical laboratory.

Ranta (1994)

<figure><img src="image_33.png" /></figure>
Soit $f$ une forme hermitienne sur un espace vectoriel $L$ sur $K$. On dit que deux vecteurs $x, y \in L$ sont **orthogonaux** par rapport à $f$ si

$$f(x, y) = 0.$$

(...)

Soit maintenant $M$ un sous-espace vectoriel de $L$; on appelle **orthogonal de** $M$ par rapport à $f$ l'ensemble, noté généralement

$$M^{\perp},$$

des $x \in L$ qui sont orthogonaux à tout $y \in M$.

Godement, *Cours d'Algèbre*

<figure><img src="image_34.png" /></figure>
Let $f$ be a hermitian form on a vector space $L$ over $K$. Two vectors $x, y \in L$ are said to be **orthogonal** with respect to $f$ if

$$f(x, y) = 0.$$

(...)

Now let $M$ be a vector subspace of $L$. The **orthogonal complement** of $M$ with respect to $f$ is defined to be the set, usually denoted by

$$M^{\perp},$$

of all vectors $x \in L$ which are orthogonal to every $y \in M$.

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## Abstract Syntactic Structures

*A farmer feeds a gray donkey.*

QR (SOME FARMER) (λx. QR (SOME (GRAY DONKEY))) (λy. FEED y x))

FARMER : N

DONKEY : N

GRAY : N → N

FEED : NP → NP → S

SOME : N → QNP

QR : QNP → (NP → S) → S

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## Abstract Syntactic Structures

*A farmer feeds a gray donkey.*

QR (SOME FARMER) (λx. QR (SOME (GRAY DONKEY))) (λy. FEED y x))

FARMER : N

DONKEY : N

GRAY : N → N

FEED : NP → NP → S

SOME : N → QNP

QR : QNP → (NP → S) → S

<figure><img src="image_43.png" /></figure>

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## Abstract Syntactic Structures

*A farmer feeds a gray donkey.*

QR (SOME FARMER) (λx. QR (SOME (GRAY DONKEY))) (λy. FEED y x))

FARMER : N

DONKEY : N

GRAY : N → N

FEED : NP → NP → S

SOME : N → QNP

QR : QNP → (NP → S) → S

<figure><img src="image_50.png" /></figure>

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<figure><img src="image_54.png" /></figure>

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<figure><img src="image_56.png" /></figure>
## Abstract Syntactic Structures

*A farmer feeds a gray donkey.*

QR (SOME FARMER) (λx. QR (SOME (GRAY DONKEY))) (λy. FEED y x))

FARMER : N

DONKEY : N

GRAY : N → N

FEED : NP → NP → S

SOME : N → QNP

QR : QNP → (NP → S) → S

<figure><img src="image_57.png" /></figure>

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Semantic Interpretation

$[N] = e \to t$

$[NP] = e$

$[QNP] = (e \to t) \to t$

$[S] = t$

$[[FARMER]] = \lambda x. \mathit{farmer} x$

$[[DONKEY]] = \lambda x. \mathit{donkey} x$

$[[GRAY]] = \lambda p x. (p x) \wedge (\mathit{gray} x)$

$[[FEED]] = \lambda x y. \mathit{feed} y x$

$[[SOME]] = \lambda pq, \exists x. (p x) \wedge (q x)$

$[[QR]] = \lambda f x. f x$

where
farmer, donkey, gray : e \to t
feed : e \to e \to t
Semantic Interpretation

[N] = e → t

[NP] = e

[QNP] = (e → t) → t

[S] = t

[FARMER] = λx. farmer x

[DONKEY] = λx. donkey x

[GRAY] = λpx. (px) ∧ (gray x)

[FEED] = λxy. feed y x

[SOME] = λpq. ∃x. (px) ∧ (qx)

[QR] = λfx. fx

where farmer, donkey, gray : e → t
feed : e → e → t
Semantic Interpretation

[N] = e → t

[NP] = e

[QNP] = (e → t) → t

[S] = t

[FARMER] = λx. **farmer** x

[DONKEY] = λx. **donkey** x

[GRAY] = λpx. (px) ∧ (**gray** x)

[FEED] = λxy. **feed** y x

[SOME] = λpq. ∃x. (px) ∧ (qx)

[QR] = λfx. fx

where **farmer**, **donkey**, **gray**: e → t
**feed**: e → e → t
Semantic Interpretation

$$
\begin{align*}
& \llbracket \text{QR} (\text{SOME FARMER}) (\lambda x. \text{QR} (\text{SOME (GRAY DONKEY)}) (\lambda y. \text{FEED } y x)) \rrbracket \\
&= \llbracket \text{QR} \rrbracket \llbracket (\text{SOME FARMER}) \rrbracket (\lambda x. \llbracket \text{QR} (\text{SOME (GRAY DONKEY)}) (\lambda y. \text{FEED } y x) \rrbracket) \\
&= (\lambda f x. f x) \llbracket (\text{SOME FARMER}) \rrbracket (\lambda x. \llbracket \text{QR} (\text{SOME (GRAY DONKEY)}) (\lambda y. \text{FEED } y x) \rrbracket) \\
&\rightarrow \beta \llbracket \text{SOME FARMER} \rrbracket (\lambda x. \llbracket \text{QR} (\text{SOME (GRAY DONKEY)}) (\lambda y. \text{FEED } y x) \rrbracket) \\
&= (\lambda pq. \exists x. (p x \land q x)) (\lambda x. \text{farmer } x) \\
&\qquad (\lambda x. \llbracket \text{QR} (\text{SOME (GRAY DONKEY)}) (\lambda y. \text{FEED } y x) \rrbracket) \\
&\rightarrow \beta \exists x. (\text{farmer } x) \land \llbracket (\text{QR (SOME (GRAY DONKEY)}) (\lambda y. \text{FEED } y x)) \rrbracket \\
&= \exists x. (\text{farmer } x) \land ((\lambda f x. f x) \llbracket (\text{SOME (GRAY DONKEY)}) \rrbracket) (\lambda y. \llbracket \text{FEED} \rrbracket y x) \\
&\rightarrow \beta \exists x. (\text{farmer } x) \land \llbracket (\text{SOME (GRAY DONKEY)}) \rrbracket (\lambda y. \llbracket \text{FEED} \rrbracket y x) \\
&= \exists x. (\text{farmer } x) \land ((\lambda pq. \exists y. (p y \land q y)) \llbracket (\text{GRAY DONKEY}) \rrbracket) (\lambda y. \llbracket \text{FEED} \rrbracket y x) \\
&\rightarrow \beta \exists x. (\text{farmer } x) \land (\exists y. \llbracket (\text{GRAY DONKEY}) y \rrbracket) \land \llbracket (\text{FEED} y x) \rrbracket \\
&= \exists x. (\text{farmer } x) \land (\exists y. ((\lambda px. (p x) \land \text{gray } x)) (\lambda x. \text{donkey } x) y) \land \llbracket (\text{FEED} y x) \rrbracket \\
&\rightarrow \beta \exists x. (\text{farmer } x) \land (\exists y. (\text{donkey } y) \land (\text{gray } y) \land \llbracket (\text{FEED}) y x \rrbracket) \\
&= \exists x. (\text{farmer } x) \land (\exists y. (\text{donkey } y) \land (\text{gray } y) \land ((\lambda xy. \text{feed } x) y x)) \\
&\rightarrow \beta \exists x. (\text{farmer } x) \land (\exists y. (\text{donkey } y) \land (\text{gray } y) \land (\text{feed } x y))
\end{align*}
$$

<figure><img src="image_64.png" /></figure>
Semantic Interpretation

$$
\begin{align*}
& [QR (\text{SOME FARMER}) (\lambda x. \text{QR} (\text{SOME (GRAY DONKEY)})) (\lambda y. \text{FEED } y x)] \\
&= [QR] [(\text{SOME FARMER})] (\lambda x. [QR (\text{SOME (GRAY DONKEY)})) (\lambda y. \text{FEED } y x)] \\
&= (\lambda fx. f x) [(\text{SOME FARMER})] (\lambda x. [QR (\text{SOME (GRAY DONKEY)})) (\lambda y. \text{FEED } y x)] \\
&\rightarrow \beta [\text{SOME FARMER}] (\lambda x. [QR (\text{SOME (GRAY DONKEY)})) (\lambda y. \text{FEED } y x)] \\
&= (\lambda pq. \exists x. (p x \land q x)) (\lambda x. \text{farmer } x) \\
&\qquad (\lambda x. [QR (\text{SOME (GRAY DONKEY)})) (\lambda y. \text{FEED } y x)] \\
&\rightarrow \beta \exists x. (\text{farmer } x) \land [(QR (\text{SOME (GRAY DONKEY)})) (\lambda y. \text{FEED } y x)] \\
&= \exists x. (\text{farmer } x) \land ((\lambda fx. f x) [(\text{SOME (GRAY DONKEY)})) (\lambda y. \text{[FEED]} y x)) \\
&\rightarrow \beta \exists x. (\text{farmer } x) \land [(\text{SOME (GRAY DONKEY)})) (\lambda y. \text{[FEED]} y x)] \\
&= \exists x. (\text{farmer } x) \land ((\lambda pq. \exists y. (p y \land q y)) [(\text{GRAY DONKEY})] (\lambda y. \text{[FEED]} y x)) \\
&\rightarrow \beta \exists x. (\text{farmer } x) \land (\exists y. [(\text{GRAY DONKEY})] y) \land [(\text{FEED}) y x]) \\
&= \exists x. (\text{farmer } x) \land (\exists y. ((\lambda px. (p x) \land \text{gray } x)) (\lambda x. \text{donkey } x) y) \land [(\text{FEED}) y x]) \\
&\rightarrow \beta \exists x. (\text{farmer } x) \land (\exists y. (\text{donkey } y) \land (\text{gray } y) \land [(\text{FEED}) y x]) \\
&= \exists x. (\text{farmer } x) \land (\exists y. (\text{donkey } y) \land (\text{gray } y) \land ((\lambda xy. \text{feed } x) y x)) \\
&\rightarrow \beta \exists x. (\text{farmer } x) \land (\exists y. (\text{donkey } y) \land (\text{gray } y) \land (\text{feed } x y))
\end{align*}
$$

<figure><img src="image_65.png" /></figure>
Semantic Interpretation

[(QR (SOME FARMER) (λx. QR (SOME (GRAY DONKEY)) (λy. FEED y x)))
= [(QR] [(SOME FARMER)] (λx. [QR (SOME (GRAY DONKEY)) (λy. FEED y x)]))
= (λfx. fx) [(SOME FARMER)] (λx. [QR (SOME (GRAY DONKEY)) (λy. FEED y x)]))
→β [SOME FARMER] (λx. [QR (SOME (GRAY DONKEY)) (λy. FEED y x)])
= (λpq. ∃x. (px) ∧ (qx)) (λx. farmer x)
(λx. [QR (SOME (GRAY DONKEY)) (λy. FEED y x)])
→β ∃x. (farmer x) ∧ [(QR (SOME (GRAY DONKEY)) (λy. FEED y x)))
= ∃x. (farmer x) ∧ ((λfx. fx) [(SOME (GRAY DONKEY)) (λy. [FEED] y x)])
→β ∃x. (farmer x) ∧ [(SOME (GRAY DONKEY)] (λy. [FEED] y x))
= ∃x. (farmer x) ∧ ((λpq. ∃y. (py) ∧ (qy)) [(GRAY DONKEY)] (λy. [FEED] y x))
→β ∃x. (farmer x) ∧ (∃y. [(GRAY DONKEY] y) ∧ ([FEED] y x))
= ∃x. (farmer x) ∧ (∃y. ((λpx. (px) ∧ (gray x)) (λx. donkey x) y) ∧ ([FEED] y x))
→β ∃x. (farmer x) ∧ (∃y. (donkey y) ∧ (gray y) ∧ ([FEED] y x))
= ∃x. (farmer x) ∧ (∃y. (donkey y) ∧ (gray y) ∧ ((λxy. feed x) y x))
→β ∃x. (farmer x) ∧ (∃y. (donkey y) ∧ (gray y) ∧ (feed x y))

<figure><img src="image_66.png" /></figure>
Semantic Interpretation

[(QR (SOME FARMER) (λx. QR (SOME (GRAY DONKEY)) (λy. FEED y x)))
= [(QR] [(SOME FARMER)] (λx. [QR (SOME (GRAY DONKEY)) (λy. FEED y x)]))
= (λfx. f x) [(SOME FARMER)] (λx. [QR (SOME (GRAY DONKEY)) (λy. FEED y x)]))
→β [(SOME FARMER] (λx. [QR (SOME (GRAY DONKEY)) (λy. FEED y x)]))
= (λpq. ∃x. (p x) ∧ (q y)) (λx. farmer x)
(x. [QR (SOME (GRAY DONKEY)) (λy. FEED y x)]))
→β ∃x. (farmer x) ∧ [(QR (SOME (GRAY DONKEY)) (λy. FEED y x)))
= ∃x. (farmer x) ∧ ((λfx. f x) [(SOME (GRAY DONKEY)) (λy. [FEED] y x)])
→β ∃x. (farmer x) ∧ [(SOME (GRAY DONKEY)] (λy. [FEED] y x))
= ∃x. (farmer x) ∧ ((λpq. ∃y. (p y) ∧ (q y)) [(GRAY DONKEY)] (λy. [FEED] y x))
→β ∃x. (farmer x) ∧ (∃y. [(GRAY DONKEY] y) ∧ ([FEED] y x))
= ∃x. (farmer x) ∧ (∃y. ((λpx. (p x) ∧ (gray x)) (λx. donkey x) y) ∧ ([FEED] y x))
→β ∃x. (farmer x) ∧ (∃y. (donkey y) ∧ (gray y) ∧ ([FEED] y x))
= ∃x. (farmer x) ∧ (∃y. (donkey y) ∧ (gray y) ∧ ((λxy. feed x) y x))
→β ∃x. (farmer x) ∧ (∃y. (donkey y) ∧ (gray y) ∧ (feed x y))

<figure><img src="image_67.png" /></figure>
Semantic Interpretation

$$
\begin{align*}
& [QR (\text{SOME FARMER}) (\lambda x. \text{QR} (\text{SOME (GRAY DONKEY)})) (\lambda y. \text{FEED } y x)] \\
&= [QR] [[(\text{SOME FARMER})] (\lambda x. [QR (\text{SOME (GRAY DONKEY)})) (\lambda y. \text{FEED } y x)]] \\
&= (\lambda f x. f x) [[(\text{SOME FARMER})] (\lambda x. [QR (\text{SOME (GRAY DONKEY)})) (\lambda y. \text{FEED } y x)]] \\
&\to \beta [\text{SOME FARMER}] (\lambda x. [QR (\text{SOME (GRAY DONKEY)})) (\lambda y. \text{FEED } y x)] \\
&= (\lambda pq. \exists x. (p x \wedge q x)) (\lambda x. \mathbf{farmer} x) \\
&\qquad (\lambda x. [QR (\text{SOME (GRAY DONKEY)})) (\lambda y. \text{FEED } y x)]) \\
&\to \beta \exists x. (\mathbf{farmer} x) \wedge [[(QR (\text{SOME (GRAY DONKEY)})) (\lambda y. \text{FEED } y x)]] \\
&= \exists x. (\mathbf{farmer} x) \wedge ((\lambda f x. f x) [[(\text{SOME (GRAY DONKEY)})) (\lambda y. \text{FEED} y x)] \\
&\to \beta \exists x. (\mathbf{farmer} x) \wedge [[(\text{SOME (GRAY DONKEY)})) (\lambda y. \text{FEED} y x)] \\
&= \exists x. (\mathbf{farmer} x) \wedge ((\lambda pq. \exists y. (p y \wedge q y)) [[(\text{GRAY DONKEY})] (\lambda y. \text{FEED} y x)] \\
&\to \beta \exists x. (\mathbf{farmer} x) \wedge (\exists y. [[(\text{GRAY DONKEY})] y] \wedge [[\text{FEED}] y x]) \\
&= \exists x. (\mathbf{farmer} x) \wedge (\exists y. ((\lambda px. (p x) \wedge \text{gray } x)) (\lambda x. \mathbf{donkey} x) y) \wedge [[\text{FEED}] y x] \\
&\to \beta \exists x. (\mathbf{farmer} x) \wedge (\exists y. (\text{donkey } y) \wedge (\text{gray } y) \wedge [[\text{FEED}] y x]) \\
&= \exists x. (\mathbf{farmer} x) \wedge (\exists y. (\text{donkey } y) \wedge (\text{gray } y) \wedge ((\lambda xy. \text{feed } x) y x)) \\
&\to \beta \exists x. (\mathbf{farmer} x) \wedge (\exists y. (\text{donkey } y) \wedge (\text{gray } y) \wedge (\text{feed } x y))
\end{align*}
$$

<figure><img src="image_68.png" /></figure>
Semantic Interpretation

[(QR (SOME FARMER) (λx. QR (SOME (GRAY DONKEY)) (λy. FEED y x)))
= [(QR] [(SOME FARMER)] (λx. [QR (SOME (GRAY DONKEY)) (λy. FEED y x)]))
= (λfx. f x) [(SOME FARMER)] (λx. [QR (SOME (GRAY DONKEY)) (λy. FEED y x)]))
→β [(SOME FARMER] (λx. [QR (SOME (GRAY DONKEY)) (λy. FEED y x)]))
= (λpq. ∃x. (px) ∧ (qx)) (λx. farmer x)
(λx. [QR (SOME (GRAY DONKEY)) (λy. FEED y x)])
→β ∃x. (farmer x) ∧ [(QR (SOME (GRAY DONKEY)) (λy. FEED y x)))
= ∃x. (farmer x) ∧ ((λfx. f x) [(SOME (GRAY DONKEY)) [λy. [FEED] y x)])
→β ∃x. (farmer x) ∧ [(SOME (GRAY DONKEY)] (λy. [FEED] y x))
= ∃x. (farmer x) ∧ ((λpq. ∃y. (py) ∧ (qy)) [(GRAY DONKEY)] (λy. [FEED] y x))
→β ∃x. (farmer x) ∧ (∃y. [(GRAY DONKEY] y) ∧ ([FEED] y x))
= ∃x. (farmer x) ∧ (∃y. ((λpx. (px) ∧ (gray x)) (λx. donkey x) y) ∧ ([FEED] y x))
→β ∃x. (farmer x) ∧ (∃y. (donkey y) ∧ (gray y) ∧ ([FEED] y x))
= ∃x. (farmer x) ∧ (∃y. (donkey y) ∧ (gray y) ∧ ((λxy. feed x) y x))
→β ∃x. (farmer x) ∧ (∃y. (donkey y) ∧ (gray y) ∧ (feed x y))

<figure><img src="image_69.png" /></figure>
Semantic Interpretation

[(QR (SOME FARMER) (λx. QR (SOME (GRAY DONKEY)) (λy. FEED y x)))
= [(QR] [(SOME FARMER)] (λx. [QR (SOME (GRAY DONKEY)) (λy. FEED y x)]))
= (λfx. f x) [(SOME FARMER)] (λx. [QR (SOME (GRAY DONKEY)) (λy. FEED y x)]))
→β [(SOME FARMER] (λx. [QR (SOME (GRAY DONKEY)) (λy. FEED y x)]))
= (λpq. ∃x. (px) ∧ (qx)) (λx. **farmer x**)
(λx. [QR (SOME (GRAY DONKEY)) (λy. FEED y x)])
→β ∃x. (**farmer x**) ∧ [(QR (SOME (GRAY DONKEY)) (λy. FEED y x)))
= ∃x. (**farmer x**) ∧ ((λfx. f x) [(SOME (GRAY DONKEY))) (λy. [FEED] y x))
→β ∃x. (**farmer x**) ∧ [(SOME (GRAY DONKEY)] (λy. [FEED] y x))
= ∃x. (**farmer x**) ∧ ((λpq. ∃y. (py) ∧ (qy)) [(GRAY DONKEY)] (λy. [FEED] y x))
→β ∃x. (**farmer x**) ∧ (∃y. [(GRAY DONKEY] y) ∧ ([FEED] y x))
= ∃x. (**farmer x**) ∧ (∃y. ((λpx. (px) ∧ (gray x)) (λx. **donkey** x) y) ∧ ([FEED] y x))
→β ∃x. (**farmer x**) ∧ (∃y. (donkey y) ∧ (gray y) ∧ ([FEED] y x))
= ∃x. (**farmer x**) ∧ (∃y. (donkey y) ∧ (gray y) ∧ ((λxy. feed x) y x))
→β ∃x. (**farmer x**) ∧ (∃y. (donkey y) ∧ (gray y) ∧ (feed x y))

<figure><img src="image_70.png" /></figure>
Semantic Interpretation

$$
\begin{align*}
& [QR (\text{SOME FARMER}) (\lambda x. \text{QR} (\text{SOME (GRAY DONKEY)})) (\lambda y. \text{FEED } y x)] \\
&= [QR] [[(\text{SOME FARMER})] (\lambda x. [QR (\text{SOME (GRAY DONKEY)})) (\lambda y. \text{FEED } y x)]] \\
&= (\lambda f x. f x) [[(\text{SOME FARMER})] (\lambda x. [QR (\text{SOME (GRAY DONKEY)})) (\lambda y. \text{FEED } y x)]] \\
&\to \beta [\text{SOME FARMER}] (\lambda x. [QR (\text{SOME (GRAY DONKEY)})) (\lambda y. \text{FEED } y x)] \\
&= (\lambda pq. \exists x. (p x \wedge q x)) (\lambda x. \mathbf{farmer} x) \\
&\qquad (\lambda x. [QR (\text{SOME (GRAY DONKEY)})) (\lambda y. \text{FEED } y x)]) \\
&\to \beta \exists x. (\mathbf{farmer} x) \wedge [[(QR (\text{SOME (GRAY DONKEY)})) (\lambda y. \text{FEED } y x)] \\
&= \exists x. (\mathbf{farmer} x) \wedge ((\lambda f x. f x) [[(\text{SOME (GRAY DONKEY)})) (\lambda y. \text{FEED} ] y x)) \\
&\to \beta \exists x. (\mathbf{farmer} x) \wedge [[\text{SOME (GRAY DONKEY)}] (\lambda y. \text{FEED} ] y x) \\
&= \exists x. (\mathbf{farmer} x) \wedge ((\lambda pq. \exists y. (p y \wedge q y)) [[\text{GRAY DONKEY}]] (\lambda y. \text{FEED} ] y x) \\
&\to \beta \exists x. (\mathbf{farmer} x) \wedge (\exists y. [[\text{GRAY DONKEY}]] y) \wedge [[\text{FEED}]] y x) \\
&= \exists x. (\mathbf{farmer} x) \wedge (\exists y. ((\lambda px. (p x) \wedge \text{gray } x)) (\lambda x. \text{donkey } x) y) \wedge [[\text{FEED}]] y x) \\
&\to \beta \exists x. (\mathbf{farmer} x) \wedge (\exists y. (\text{donkey } y) \wedge (\text{gray } y) \wedge [[\text{FEED}]] y x)) \\
&= \exists x. (\mathbf{farmer} x) \wedge (\exists y. (\text{donkey } y) \wedge (\text{gray } y) \wedge ((\lambda xy. \text{feed } x) y x)) \\
&\to \beta \exists x. (\mathbf{farmer} x) \wedge (\exists y. (\text{donkey } y) \wedge (\text{gray } y) \wedge (\text{feed } x) y)
\end{align*}
$$

<figure><img src="image_71.png" /></figure>
Semantic Interpretation

$$
\begin{align*}
& [QR (\text{SOME FARMER}) (\lambda x. \text{QR} (\text{SOME (GRAY DONKEY)})) (\lambda y. \text{FEED } y x)] \\
&= [QR] [[(\text{SOME FARMER})] (\lambda x. [QR (\text{SOME (GRAY DONKEY)})) (\lambda y. \text{FEED } y x)]] \\
&= (\lambda f x. f x) [[(\text{SOME FARMER})] (\lambda x. [QR (\text{SOME (GRAY DONKEY)})) (\lambda y. \text{FEED } y x)]] \\
&\to \beta [\text{SOME FARMER}] (\lambda x. [QR (\text{SOME (GRAY DONKEY)})) (\lambda y. \text{FEED } y x)] \\
&= (\lambda pq. \exists x. (p x \wedge q x)) (\lambda x. \mathbf{farmer} x) \\
&\qquad (\lambda x. [QR (\text{SOME (GRAY DONKEY)})) (\lambda y. \text{FEED } y x)]) \\
&\to \beta \exists x. (\mathbf{farmer} x) \wedge [[(QR (\text{SOME (GRAY DONKEY)})) (\lambda y. \text{FEED } y x)]] \\
&= \exists x. (\mathbf{farmer} x) \wedge ((\lambda f x. f x) [[(\text{SOME (GRAY DONKEY)})) (\lambda y. \text{FEED} ] y x)) \\
&\to \beta \exists x. (\mathbf{farmer} x) \wedge [[\text{SOME (GRAY DONKEY)}] (\lambda y. \text{FEED} ] y x) \\
&= \exists x. (\mathbf{farmer} x) \wedge ((\lambda pq. \exists y. (p y \wedge q y)) [[\text{GRAY DONKEY}]] (\lambda y. \text{FEED} ] y x) \\
&\to \beta \exists x. (\mathbf{farmer} x) \wedge ([\exists y. [[\text{GRAY DONKEY}]] y] \wedge [[\text{FEED}]] y x) \\
&= \exists x. (\mathbf{farmer} x) \wedge ([\exists y. ((\lambda px. (p x) \wedge \text{gray } y)) (\lambda x. \text{donkey } x) y] \wedge [[\text{FEED}]] y x) \\
&\to \beta \exists x. (\mathbf{farmer} x) \wedge ([\exists y. (\text{donkey } y] \wedge \text{gray } y] \wedge [[\text{FEED}]] y x)) \\
&= \exists x. (\mathbf{farmer} x) \wedge ([\exists y. (\text{donkey } y] \wedge \text{gray } y] \wedge ((\lambda xy. \text{feed } y x) y)) \\
&\to \beta \exists x. (\mathbf{farmer} x) \wedge ([\exists y. (\text{donkey } y] \wedge \text{gray } y] \wedge (\text{feed } x y))
\end{align*}
$$

<figure><img src="image_72.png" /></figure>
Semantic Interpretation

$$
\begin{align*}
& [QR (\text{SOME FARMER}) (\lambda x. \text{QR} (\text{SOME (GRAY DONKEY)})) (\lambda y. \text{FEED } y x)] \\
&= [QR] [(\text{SOME FARMER})] (\lambda x. [QR (\text{SOME (GRAY DONKEY)})) (\lambda y. \text{FEED } y x)] \\
&= (\lambda f x. f x) [(\text{SOME FARMER})] (\lambda x. [QR (\text{SOME (GRAY DONKEY)})) (\lambda y. \text{FEED } y x)] \\
\rightarrow & \beta [\text{SOME FARMER}] (\lambda x. [QR (\text{SOME (GRAY DONKEY)})) (\lambda y. \text{FEED } y x)] \\
&= (\lambda pq. \exists x. (p x \wedge q x)) (\lambda x. \mathbf{farmer} x) \\
& \phantom{=} (\lambda x. [QR (\text{SOME (GRAY DONKEY)})) (\lambda y. \text{FEED } y x)] \\
\rightarrow & \beta \exists x. (\mathbf{farmer} x) \wedge [(QR (\text{SOME (GRAY DONKEY)})) (\lambda y. \text{FEED } y x)] \\
&= \exists x. (\mathbf{farmer} x) \wedge ((\lambda f x. f x) [(\text{SOME (GRAY DONKEY)})) (\lambda y. \text{FEED} y x)) \\
\rightarrow & \beta \exists x. (\mathbf{farmer} x) \wedge [(\text{SOME (GRAY DONKEY)})) (\lambda y. \text{FEED} y x)) \\
&= \exists x. (\mathbf{farmer} x) \wedge ((\lambda pq. \exists y. (p y \wedge q y)) [(\text{GRAY DONKEY})] (\lambda y. \text{FEED} y x)) \\
\rightarrow & \beta \exists x. (\mathbf{farmer} x) \wedge (\exists y. [(\text{GRAY DONKEY})] y \wedge ([\text{FEED}] y x)) \\
&= \exists x. (\mathbf{farmer} x) \wedge (\exists y. ((\lambda px. (p x) \wedge (\text{gray } x)) (\lambda x. \text{donkey } x) y) \wedge ([\text{FEED}] y x)) \\
\rightarrow & \beta \exists x. (\mathbf{farmer} x) \wedge (\exists y. (\text{donkey } y) \wedge (\text{gray } y) \wedge ([\text{FEED}] y x)) \\
&= \exists x. (\mathbf{farmer} x) \wedge (\exists y. (\text{donkey } y) \wedge (\text{gray } y) \wedge ((\lambda xy. \text{feed } x) y x)) \\
\rightarrow & \beta \exists x. (\mathbf{farmer} x) \wedge (\exists y. (\text{donkey } y) \wedge (\text{gray } y) \wedge (\text{feed } x y))
\end{align*}
$$

<figure><img src="image_73.png" /></figure>
Semantic Interpretation

```
[QR (SOME FARMER) (λx. QR (SOME (GRAY DONKEY)) (λy. FEED y x))]
= [QR] [[(SOME FARMER)] (λx. [QR (SOME (GRAY DONKEY)) (λy. FEED y x)])]
= (λfx. f x) [[(SOME FARMER)] (λx. [QR (SOME (GRAY DONKEY)) (λy. FEED y x)])]
→β [SOME FARMER] (λx. [QR (SOME (GRAY DONKEY)) (λy. FEED y x)])
= (λpq. ∃x. (px) ∧ (qx)) (λx. farmer x)
(λx. [QR (SOME (GRAY DONKEY)) (λy. FEED y x)])
→β ∃x. (farmer x) ∧ [[(QR (SOME (GRAY DONKEY)) (λy. FEED y x))]
= ∃x. (farmer x) ∧ ((λfx. f x) [[(SOME (GRAY DONKEY)) (λy. [FEED] y x)])
→β ∃x. (farmer x) ∧ [[SOME (GRAY DONKEY)] (λy. [FEED] y x)]
= ∃x. (farmer x) ∧ ((λpq. ∃y. (py) ∧ (qy)) [[(GRAY DONKEY)] (λy. [FEED] y x)]
→β ∃x. (farmer x) ∧ (∃y. [[(GRAY DONKEY] y) ∧ ([FEED] y x)]
= ∃x. (farmer x) ∧ (∃y. ((λpx. (px) ∧ (gray x)) (λx. donkey x) y) ∧ ([FEED] y x)]
→β ∃x. (farmer x) ∧ (∃y. (donkey y) ∧ (gray y) ∧ ([FEED] y x))
= ∃x. (farmer x) ∧ (∃y. (donkey y) ∧ (gray y) ∧ ((λxy. feed x) y x))
→β ∃x. (farmer x) ∧ (∃y. (donkey y) ∧ (gray y) ∧ (feed x y))
```

<figure><img src="image_74.png" /></figure>
Semantic Interpretation

```
[QR (SOME FARMER) (λx. QR (SOME (GRAY DONKEY)) (λy. FEED y x))]
= [QR] [[(SOME FARMER)] (λx. [QR (SOME (GRAY DONKEY)) (λy. FEED y x)])]
= (λfx. f x) [[(SOME FARMER)] (λx. [QR (SOME (GRAY DONKEY)) (λy. FEED y x)])]
→β [SOME FARMER] (λx. [QR (SOME (GRAY DONKEY)) (λy. FEED y x)])
= (λpq. ∃x. (px) ∧ (qx)) (λx. farmer x)
(λx. [QR (SOME (GRAY DONKEY)) (λy. FEED y x)])
→β ∃x. (farmer x) ∧ [[(QR (SOME (GRAY DONKEY)) (λy. FEED y x))]
= ∃x. (farmer x) ∧ ((λfx. f x) [[(SOME (GRAY DONKEY)) (λy. [FEED] y x)]
→β ∃x. (farmer x) ∧ [[SOME (GRAY DONKEY)] (λy. [FEED] y x)]
= ∃x. (farmer x) ∧ ((λpq. ∃y. (py) ∧ (qy)) [[(GRAY DONKEY)] (λy. [FEED] y x)]
→β ∃x. (farmer x) ∧ (∃y. [[(GRAY DONKEY] y) ∧ ([FEED] y x)]
= ∃x. (farmer x) ∧ (∃y. ((λpx. (px) ∧ (gray x)) (λx. donkey x) y) ∧ ([FEED] y x)]
→β ∃x. (farmer x) ∧ (∃y. (donkey y) ∧ (gray y) ∧ ([FEED] y x))
= ∃x. (farmer x) ∧ (∃y. (donkey y) ∧ (gray y) ∧ (λxy. feed x y x))
→β ∃x. (farmer x) ∧ (∃y. (donkey y) ∧ (gray y) ∧ (feed x y))
```

<figure><img src="image_75.png" /></figure>
Semantic Interpretation

$$
\begin{align*}
& [[QR (SOME FARMER)] (\lambda x. QR (SOME (GRAY DONKEY)) (\lambda y. FEED y x))] \\
&= [[QR]] [[(SOME FARMER)]] (\lambda x. [[QR (SOME (GRAY DONKEY)) (\lambda y. FEED y x)]]) \\
&= (\lambda fx. f x) [[(SOME FARMER)]] (\lambda x. [[QR (SOME (GRAY DONKEY)) (\lambda y. FEED y x)]]) \\
&\rightarrow \beta [[SOME FARMER]] (\lambda x. [[QR (SOME (GRAY DONKEY)) (\lambda y. FEED y x)]]) \\
&= (\lambda pq. \exists x. (px) \wedge (qx)) (\lambda x. \mathbf{farmer} x) \\
&\qquad (\lambda x. [[QR (SOME (GRAY DONKEY)) (\lambda y. FEED y x)]]) \\
&\rightarrow \beta \exists x. (\mathbf{farmer} x) \wedge [[(QR (SOME (GRAY DONKEY)) (\lambda y. FEED y x))]] \\
&= \exists x. (\mathbf{farmer} x) \wedge ((\lambda fx. f x) [[(SOME (GRAY DONKEY))]] (\lambda y. [[FEED]] y x)) \\
&\rightarrow \beta \exists x. (\mathbf{farmer} x) \wedge [[(SOME (GRAY DONKEY))]] (\lambda y. [[FEED]] y x) \\
&= \exists x. (\mathbf{farmer} x) \wedge ((\lambda pq. \exists y. (py) \wedge (qy)) [[(GRAY DONKEY)]] (\lambda y. [[FEED]] y x)) \\
&\rightarrow \beta \exists x. (\mathbf{farmer} x) \wedge (\exists y. [[(GRAY DONKEY)]] y) \wedge [[(FEED)] y x) \\
&= \exists x. (\mathbf{farmer} x) \wedge (\exists y. ((\lambda px. (px) \wedge \mathbf{gray} x)) (\lambda x. \mathbf{donkey} x) y) \wedge [[(FEED)] y x) \\
&\rightarrow \beta \exists x. (\mathbf{farmer} x) \wedge (\exists y. (\mathbf{donkey} y) \wedge (\mathbf{gray} y) \wedge [[(FEED)] y x]) \\
&= \exists x. (\mathbf{farmer} x) \wedge (\exists y. (\mathbf{donkey} y) \wedge (\mathbf{gray} y) \wedge ((\lambda xy. \mathbf{feed} y x) y x)) \\
&\rightarrow \beta \exists x. (\mathbf{farmer} x) \wedge (\exists y. (\mathbf{donkey} y) \wedge (\mathbf{gray} y) \wedge (\mathbf{feed} x y))
\end{align*}
$$

<figure><img src="image_76.png" /></figure>
Semantic Interpretation

```
[[QR (SOME FARMER) (λx. QR (SOME (GRAY DONKEY)) (λy. FEED y x))]]
= [[QR]] [[(SOME FARMER)]] (λx. [[QR (SOME (GRAY DONKEY)) (λy. FEED y x)]]]
= (λfx. f x) [[(SOME FARMER)]] (λx. [[QR (SOME (GRAY DONKEY)) (λy. FEED y x)]]]
→β [[SOME FARMER]] (λx. [[QR (SOME (GRAY DONKEY)) (λy. FEED y x)]])
= (λpq. ∃x. (px) ∧ (qx)) (λx. **farmer x**)
    (λx. [[QR (SOME (GRAY DONKEY)) (λy. FEED y x)]])
→β ∃x. (**farmer x**) ∧ [[((QR (SOME (GRAY DONKEY)) (λy. FEED y x)))]]
= ∃x. (**farmer x**) ∧ ((λfx. f x) [[(SOME (GRAY DONKEY))]] (λy. [[FEED]] y x))
→β ∃x. (**farmer x**) ∧ [[(SOME (GRAY DONKEY))]] (λy. [[FEED]] y x)
= ∃x. (**farmer x**) ∧ ((λpq. ∃y. (py) ∧ (qy)) [[(GRAY DONKEY)]] (λy. [[FEED]] y x))
→β ∃x. (**farmer x**) ∧ (∃y. [[(GRAY DONKEY)] y] ∧ [[[FEED]] y x])
= ∃x. (**farmer x**) ∧ (∃y. ((λpx. (px) ∧ (**gray x**) (λx. **donkey** x) y) ∧ [[[FEED]] y x])
→β ∃x. (**farmer x**) ∧ (∃y. (**donkey** y) ∧ (**gray y**) ∧ [[**FEED**] y x])
= ∃x. (**farmer x**) ∧ (∃y. (**donkey** y) ∧ (**gray y**) ∧ ((λxy. **feed** y x) y x))
→β ∃x. (**farmer x**) ∧ (∃y. (**donkey** y) ∧ (**gray y**) ∧ (**feed** x y))
```

<figure><img src="image_77.png" /></figure>
## Three approaches

* Mereology: $a \oplus b$. (Blunt, 1985)

* Plural logic: $\forall x, \forall xs$. (Nicolas, 2008)

* Second-order logic: plural as sets of entities, i.e, terms of type $e \to t$. (Link, 1983)

Plural logic can be formalized as a fragment of second order-logic.

Mereology = Boolean algebra without $0$.

Stone representation theorem.

<figure><img src="image_78.png" /></figure>
## Three approaches

* **Mereology**: $a \oplus b$. (Blunt, 1985)

* **Plural logic**: $\forall x, \forall xs$. (Nicolas, 2008)

* **Second-order logic**: plural as sets of entities, i.e, terms of type $e \to t$. (Link, 1983)

Plural logic can be formalized as a fragment of second order-logic.

Mereology = Boolean algebra without $0$.

Stone representation theorem.

<figure><img src="image_79.png" /></figure>
## Three approaches

* Mereology: $a \oplus b$. (Blunt, 1985)

* Plural logic: $\forall x, \forall xs$. (Nicolas, 2008)

* Second-order logic: plural as sets of entities, i.e, terms of type $e \to t$. (Link, 1983)

Plural logic can be formalized as a fragment of second order-logic.

Mereology = Boolean algebra without $0$.

Stone representation theorem.

<figure><img src="image_80.png" /></figure>
## Three approaches

* Mereology: $a \oplus b$. (Blunt, 1985)

* Plural logic: $\forall x, \forall xs$. (Nicolas, 2008)

* Second-order logic: plural as sets of entities, i.e, terms of type $e \rightarrow t$. (Link, 1983)

Plural logic can be formalized as a fragment of second order-logic.

Mereology = Boolean algebra without $0$.

Stone representation theorem.

<figure><img src="image_81.png" /></figure>
## Three approaches

* Mereology: $a \oplus b$. (Blunt, 1985)

* Plural logic: $\forall x, \forall xs$. (Nicolas, 2008)

* Second-order logic: plural as sets of entities, i.e, terms of type $e \rightarrow t$. (Link, 1983)

Plural logic can be formalized as a fragment of second order-logic.

Mereology = Boolean algebra without 0.

Stone representation theorem.

<figure><img src="image_82.png" /></figure>
## Three approaches

* Mereology: $a \oplus b$. (Blunt, 1985)

* Plural logic: $\forall x, \forall xs$. (Nicolas, 2008)

* Second-order logic: plural as sets of entities, i.e, terms of type $e \rightarrow t$. (Link, 1983)

Plural logic can be formalized as a fragment of second order-logic.

Mereology = Boolean algebra without $0$.

Stone representation theorem.

<figure><img src="image_83.png" /></figure>
## Three approaches

* Mereology: $a \oplus b$. (Blunt, 1985)

* Plural logic: $\forall x, \forall xs$. (Nicolas, 2008)

* Second-order logic: plural as sets of entities, i.e, terms of type $e \rightarrow t$. (Link, 1983)

Plural logic can be formalized as a fragment of second order-logic.

Mereology = Boolean algebra without $0$.

Stone representation theorem.

<figure><img src="image_84.png" /></figure>
## Syntactic categories

$$ [[N_{sg}] = e \to t $$

$$ [[NP_{sg}] = e $$

$$ [[QNP_{sg}] = (e \to t) \to t $$

$$ [[N_{pl}] = (e \to t) \to t $$

$$ [[NP_{pl}] = e \to t $$

$$ [[QNP_{pl}] = ((e \to t) \to t) \to t $$

<figure><img src="image_85.png" /></figure>
## Link's distributivity operator

$$ \mathbf{distr} \triangleq \lambda p S. \forall x. (S x) \rightarrow (p x) $$

<figure><img src="image_86.png" /></figure>
A farmer feeds some donkeys.

FARMER : N<sub>sg</sub>

DONKEY : N<sub>sg</sub>

FEED : NP<sub>sg</sub> → NP<sub>sg</sub> → S

SOME<sub>sg</sub> : N<sub>sg</sub> → QNP<sub>sg</sub>

SOME<sub>pl</sub> : N<sub>pl</sub> → QNP<sub>pl</sub>

QR<sub>sg</sub> : QNP<sub>sg</sub> → (NP<sub>sg</sub> → S) → S

QR<sub>pl</sub> : QNP<sub>pl</sub> → (NP<sub>pl</sub> → S) → S

PL : N<sub>sg</sub> → N<sub>pl</sub>

DISTR : (NP<sub>sg</sub> → S) → NP<sub>pl</sub> → S

QR<sub>sg</sub> (SOME<sub>sg</sub> FARMER) (λx. QR<sub>pl</sub> (SOME<sub>pl</sub> (PL DONKEY))) (DISTR (λy. FEED y x)))
A farmer feeds some donkeys.

FARMER : N<sub>sg</sub>

DONKEY : N<sub>sg</sub>

FEED : NP<sub>sg</sub> → NP<sub>sg</sub> → S

SOME<sub>sg</sub> : N<sub>sg</sub> → QNP<sub>sg</sub>

SOME<sub>pl</sub> : N<sub>pl</sub> → QNP<sub>pl</sub>

QR<sub>sg</sub> : QNP<sub>sg</sub> → (NP<sub>sg</sub> → S) → S

QR<sub>pl</sub> : QNP<sub>pl</sub> → (NP<sub>pl</sub> → S) → S

PL : N<sub>sg</sub> → N<sub>pl</sub>

DISTR : (NP<sub>sg</sub> → S) → NP<sub>pl</sub> → S

QR<sub>sg</sub> (SOME<sub>sg</sub> FARMER) (λx. QR<sub>pl</sub> (SOME<sub>pl</sub> (PL DONKEY))) (DISTR (λy. FEED y x)))
A farmer feeds some donkeys.

FARMER : N<sub>sg</sub>

DONKEY : N<sub>sg</sub>

FEED : NP<sub>sg</sub> → NP<sub>sg</sub> → S

SOME<sub>sg</sub> : N<sub>sg</sub> → QNP<sub>sg</sub>

SOME<sub>pl</sub> : N<sub>pl</sub> → QNP<sub>pl</sub>

QR<sub>sg</sub> : QNP<sub>sg</sub> → (NP<sub>sg</sub> → S) → S

QR<sub>pl</sub> : QNP<sub>pl</sub> → (NP<sub>pl</sub> → S) → S

PL : N<sub>sg</sub> → N<sub>pl</sub>

DISTR : (NP<sub>sg</sub> → S) → NP<sub>pl</sub> → S

QR<sub>sg</sub> (SOME<sub>sg</sub> FARMER) (λx. QR<sub>pl</sub> (SOME<sub>pl</sub> (PL DONKEY))) (DISTR (λy. FEED y x)))
A farmer feeds some donkeys.

FARMER : N<sub>sg</sub>

DONKEY : N<sub>sg</sub>

FEED : NP<sub>sg</sub> → NP<sub>sg</sub> → S

SOME<sub>sg</sub> : N<sub>sg</sub> → QNP<sub>sg</sub>

SOME<sub>pl</sub> : N<sub>pl</sub> → QNP<sub>pl</sub>

QR<sub>sg</sub> : QNP<sub>sg</sub> → (NP<sub>sg</sub> → S) → S

QR<sub>pl</sub> : QNP<sub>pl</sub> → (NP<sub>pl</sub> → S) → S

PL : N<sub>sg</sub> → N<sub>pl</sub>

DISTR : (NP<sub>sg</sub> → S) → NP<sub>pl</sub> → S

QR<sub>sg</sub> (SOME<sub>sg</sub> FARMER) (λx. QR<sub>pl</sub> (SOME<sub>pl</sub> (PL DONKEY))) (DISTR (λy. FEED y x)))
$$ [[SOME pl] = \lambda pq. \exists x. (p x) \land (q x) $$

$$ [[QR pl] = \lambda fx. f x $$

$$ [[PL] = \lambda pS. (|S| \ge 2) \land (distr p S) $$

$$ [[DISTR] = distr $$

$$ \exists x. (farmer x) \land (\exists S. (|S| \ge 2) \land (\forall y. (S y) \rightarrow (donkey y)) \land (\forall y. (S y) \rightarrow (feed x y))) $$

<figure><img src="image_87.png" /></figure>
$$ [[\text{SOME pl}]] = \lambda pq. \exists x. (p x) \wedge (q x) $$

$$ [[\text{QR pl}] ] = \lambda fx. f x $$

$$ [[\text{PL}] ] = \lambda pS. (|S| \geq 2) \wedge (\textbf{distr} p S) $$

$$ [[\text{DISTR}]] = \textbf{distr} $$

$$ \exists x. (\text{farmer } x) \wedge (\exists S. (|S| \geq 2) \wedge (\forall y. (S y) \rightarrow (\text{donkey } y)) \wedge (\forall y. (S y) \rightarrow (\text{feed } x y))) $$

<figure><img src="image_88.png" /></figure>
$$[[\text{SOME pl}] = \lambda pq. \exists x. (p x) \wedge (q x)$$

$$[[\text{QR pl}] = \lambda fx. f x$$

$$[[\text{PL}] = \lambda pS. (|S| \ge 2) \wedge (\textbf{distr} p S)$$

$$[[\text{DISTR}] = \textbf{distr}$$

$$\exists x. (\textbf{farmer} x) \wedge (\exists S. (|S| \ge 2) \wedge (\forall y. (S y) \rightarrow (\textbf{donkey} y)) \wedge (\forall y. (S y) \rightarrow (\textbf{feed} x y)))$$

<figure><img src="image_89.png" /></figure>
**THEOREM 2.** Let $M$ be a finite-dimensional vector space over a division ring, let $X$ be a finite set of generators of $M$, and let $A$ be a subset of $X$. Suppose that the elements of $A$ are linearly independent. Then there exists a basis $B$ of $M$ such that

$$A \subset B \subset X.$$

<figure><img src="image_90.png" /></figure>
## Definition and Examples

A collective predicate, as opposed to a distributive one, is a predicate that applies to a plural entity considered as a whole, rather than to each individual that comprises it.

*The soldiers surrounded the fort.*

*Each soldier surrounded the fort.*

*The soldiers were numerous.*

*Each soldier was numerous.*

<figure><img src="image_91.png" /></figure>
## Definition and Examples

A collective predicate, as opposed to a distributive one, is a predicate that applies to a plural entity considered as a whole, rather than to each individual that comprises it.

*The soldiers surrounded the fort.*

*Each soldier surrounded the fort.*

*The soldiers were numerous.*

*Each soldier was numerous.*

<figure><img src="image_92.png" /></figure>
## Definition and Examples

A collective predicate, as opposed to a distributive one, is a predicate that applies to a plural entity considered as a whole, rather than to each individual that comprises it.

*The soldiers *surrounded* the fort.*

*Each soldier *surrounded* the fort.*

*The soldiers were *numerous*.*

*Each soldier was *numerous*.*

<figure><img src="image_93.png" /></figure>
## Definition and Examples

A collective predicate, as opposed to a distributive one, is a predicate that applies to a plural entity considered as a whole, rather than to each individual that comprises it.

*The soldiers *surrounded* the fort.*

*Each soldier *surrounded* the fort.*

*The soldiers were *numerous*.*

*Each soldier was *numerous*.*

<figure><img src="image_94.png" /></figure>
## Definition and Examples

A collective predicate, as opposed to a distributive one, is a predicate that applies to a plural entity considered as a whole, rather than to each individual that comprises it.

*The soldiers *surrounded* the fort.*

*Each soldier *surrounded* the fort.*

*The soldiers were *numerous*.*

*Each soldier was *numerous*.*

<figure><img src="image_95.png" /></figure>
## Mathematical Examples

There exists a finite set of elements $a_1, \dots, a_n \in M$ which generate $M$.

Let $A$ be a set of coprime numbers.

Suppose that the elements of $A$ are linearly independent.

Let $X$ be a finite set of generators of $M$.

$$ 
\begin{align*}
\text{prime} &\triangleq \lambda a. (a \neq 1) \wedge (\forall n. ((\text{Nat } n) \wedge (\text{div } n a)) \rightarrow ((n = 1) \vee (n = a))) \\
\text{coprime} &\triangleq \lambda S. \forall n. ((\text{Nat } n) \wedge (\forall a. (S a) \rightarrow (\text{div } n a))) \rightarrow (n = 1)
\end{align*}
$$

<figure><img src="image_96.png" /></figure>

<figure><img src="image_97.png" /></figure>

<figure><img src="image_98.png" /></figure>

<figure><img src="image_99.png" /></figure>

<figure><img src="image_100.png" /></figure>

<figure><img src="image_101.png" /></figure>
## Mathematical Examples

There exists a finite set of elements $a_1, \dots, a_n \in M$ which **generate** $M$.

Let $A$ be a set of **coprime** numbers.

Suppose that the elements of $A$ are **linearly independent**.

Let $X$ be a finite set of **generators** of $M$.

$$ \text{prime} \triangleq \lambda a. (a \neq 1) \wedge (\forall n. ((\text{Nat } n) \wedge (\text{div } n a)) \rightarrow ((n = 1) \vee (n = a))) $$

$$ \text{coprime} \triangleq \lambda S. \forall n. ((\text{Nat } n) \wedge (\forall a. (S a) \rightarrow (\text{div } n a))) \rightarrow (n = 1) $$

<figure><img src="image_102.png" /></figure>
Mathematical Examples

There exists a finite set of elements $a_1, \dots, a_n \in M$ which **generate** $M$.

Let $A$ be a set of **coprime** numbers.

Suppose that the elements of $A$ are **linearly independent**.

Let $X$ be a finite set of **generators** of $M$

$$ 
\begin{align*}
\text{prime} &\triangleq \lambda a. (a \neq 1) \wedge (\forall n. ((\text{Nat } n) \wedge (\text{div } n a)) \rightarrow ((n = 1) \vee (n = a))) \\
\text{coprime} &\triangleq \lambda S. \forall n. ((\text{Nat } n) \wedge (\forall a. (S a) \rightarrow (\text{div } n a))) \rightarrow (n = 1)
\end{align*}
$$

<figure><img src="image_103.png" /></figure>
## Mathematical Examples

There exists a finite set of elements $a_1, \dots, a_n \in M$ which **generate** $M$.

Let $A$ be a set of **coprime** numbers.

Suppose that the elements of $A$ are **linearly independent**.

Let $X$ be a finite set of **generators** of $M$.

$$ 
\begin{align*}
\text{prime} &\triangleq \lambda a. (a \neq 1) \wedge (\forall n. ((\text{Nat } n) \wedge (\text{div } n a)) \rightarrow ((n = 1) \vee (n = a))) \\
\text{coprime} &\triangleq \lambda S. \forall n. ((\text{Nat } n) \wedge (\forall a. (S a) \rightarrow (\text{div } n a))) \rightarrow (n = 1)
\end{align*}
$$

<figure><img src="image_104.png" /></figure>
## Mathematical Examples

There exists a finite set of elements $a_1, \dots, a_n \in M$ which **generate** $M$.

Let $A$ be a set of **coprime** numbers.

Suppose that the elements of $A$ are **linearly independent**.

Let $X$ be a finite set of **generators** of $M$.

$$ \text{prime} \triangleq \lambda a. (a \neq 1) \wedge (\forall n. ((\text{Nat } n) \wedge (\text{div } n a)) \rightarrow ((n = 1) \vee (n = a))) $$

$$ \text{coprime} \triangleq \lambda S. \forall n. ((\text{Nat } n) \wedge (\forall a. (S a) \rightarrow (\text{div } n a))) \rightarrow (n = 1) $$

<figure><img src="image_105.png" /></figure>
## Mathematical Examples

There exists a finite set of elements $a_1, \dots, a_n \in M$ which **generate** $M$.

Let $A$ be a set of **coprime** numbers.

Suppose that the elements of $A$ are **linearly independent**.

Let $X$ be a finite set of **generators** of $M$.

$$ 
\begin{aligned}
\mathbf{prime} &\triangleq \lambda a. (a \neq 1) \wedge (\forall n. ((\mathbf{Nat} n) \wedge (\mathbf{div} n a)) \rightarrow ((n = 1) \vee (n = a))) \\
\mathbf{coprime} &\triangleq \lambda S. \forall n. ((\mathbf{Nat} n) \wedge (\forall a. (S a) \rightarrow (\mathbf{div} n a))) \rightarrow (n = 1)
\end{aligned}
$$

<figure><img src="image_106.png" /></figure>
Group Nouns

The bataillon surrounded the fort.

Each bataillon surrounded the fort.

It is enough to show that B generates M (since by construction B is free).

Reification

set : e → (e → t) → t

Nat, Rat : e → t

N, Q : e

set N (λx. Nat x)

set Q (λx. Rat x)

∀s. ∀SS'. ((set s S) ∧ (set s S')) → (S' = S')
## Group Nouns

*The bataillon surrounded the fort.*

*Each bataillon surrounded the fort.*

*It is enough to show that B generates M (since by construction B is free).*

## Reification

$$ \mathit{set} : e \to (e \to t) \to t $$

$$ \mathit{Nat}, \mathit{Rat} : e \to t $$

$$ N, Q : e $$

$$ \mathit{set} N (\lambda x. \mathit{Nat} x) $$

$$ \mathit{set} Q (\lambda x. \mathit{Rat} x) $$

$$ \forall s. \forall s s' . ((\mathit{set} s S) \land (\mathit{set} s S')) \to (S' = S') $$
Group Nouns

*The bataillon surrounded the fort.*

*Each bataillon surrounded the fort.*

*It is enough to show that B generates M (since by construction B is free).*

Reification

set : e → (e → t) → t

Nat, Rat : e → t

N, Q : e

set N (λx. Nat x)

set Q (λx. Rat x)

∀s. ∀SS'. ((sets S) ∧ (sets S')) → (S' = S')
## Group Nouns

*The bataillon surrounded the fort.*

*Each bataillon surrounded the fort.*

*It is enough to show that B generates M (since by construction B is free).*

## Reification

$$ \mathit{set} : e \to (e \to t) \to t $$

$$ \mathit{Nat}, \mathit{Rat} : e \to t $$

$$ N, Q : e $$

$$ \mathit{set} N (\lambda x. \mathit{Nat} x) $$

$$ \mathit{set} Q (\lambda x. \mathit{Rat} x) $$

$$ \forall s. \forall s' . ((\mathit{set} s S) \land (\mathit{set} s S')) \to (S' = S') $$
## Group Nouns

*The bataillon surrounded the fort.*

*Each bataillon surrounded the fort.*

*It is enough to show that B generates M (since by construction B is free).*

## Reification

$\mathbf{set} : e \rightarrow (e \rightarrow t) \rightarrow t$

Nat, Rat : e \rightarrow t

N, Q : e

set N (λx. Nat x)

set Q (λx. Rat x)

$\forall s. \forall SS'. ((\mathbf{set} s S) \land (\mathbf{set} s S')) \rightarrow (S' = S')$
## Group Nouns

*The bataillon surrounded the fort.*

*Each bataillon surrounded the fort.*

*It is enough to show that B generates M (since by construction B is free).*

## Reification

`set : e → (e → t) → t`

`Nat, Rat : e → t`

`N, Q : e`

`set N (λx. Nat x)`

`set Q (λx. Rat x)`

`∀s. ∀ss'. ((set s S) ∧ (set s S')) → (S' = S')`
Group Nouns

*The bataillon surrounded the fort.*

*Each bataillon surrounded the fort.*

*It is enough to show that B generates M (since by construction B is free).*

Reification

$$ \mathbf{set} : e \to (e \to t) \to t $$

$$ \text{Nat}, \text{Rat} : e \to t $$

$$ N, Q : e $$

$$ \mathbf{set} N (\lambda x. \text{Nat} x) $$

$$ \mathbf{set} Q (\lambda x. \text{Rat} x) $$

$$ \forall s. \forall s' . ((\mathbf{set} s S) \land (\mathbf{set} s S')) \to (S' = S') $$
## Group Nouns

*The bataillon surrounded the fort.*

*Each bataillon surrounded the fort.*

*It is enough to show that B generates M (since by construction B is free).*

## Reification

$$ \mathbf{set} : e \rightarrow (e \rightarrow t) \rightarrow t $$

$$ \mathrm{Nat}, \mathrm{Rat} : e \rightarrow t $$

$$ N, Q : e $$

$$ \mathbf{set} N (\lambda x. \mathrm{Nat} x) $$

$$ \mathbf{set} Q (\lambda x. \mathrm{Rat} x) $$

$$ \forall s. \forall SS'. ((\mathbf{set} s S) \land (\mathbf{set} s S')) \rightarrow (S = S') $$
*Every set of natural numbers is a set of rational numbers.*

NATURAL-NUMBER : $N_{sg}$

RATIONAL-NUMBER : $N_{sg}$

SET-OF : $N_{pl} \rightarrow N_{sg}$

QR<sub>sg</sub> (EVERY (SET-OF (PL NATURAL-NUMBER)))

(λx. QR<sub>sg</sub> (SOME<sub>sg</sub> (SET-OF (PL RATIONAL-NUMBER))))

(λy.x = y)

<figure><img src="image_107.png" /></figure>
*Every set of natural numbers is a set of rational numbers.*

NATURAL-NUMBER : N<sub>sg</sub>

RATIONAL-NUMBER : N<sub>sg</sub>

SET-OF : N<sub>pl</sub> → N<sub>sg</sub>

QR<sub>sg</sub> (EVERY (SET-OF (PL NATURAL-NUMBER)))

(λx. QR<sub>sg</sub> (SOME<sub>sg</sub> (SET-OF (PL RATIONAL-NUMBER))))

(λy.x = y)

<figure><img src="image_108.png" /></figure>
*Every set of natural numbers is a set of rational numbers.*

NATURAL-NUMBER : N<sub>sg</sub>

RATIONAL-NUMBER : N<sub>sg</sub>

SET-OF : N<sub>pl</sub> → N<sub>sg</sub>

QR<sub>sg</sub> (EVERY (SET-OF (PL NATURAL-NUMBER)))

(λx. QR<sub>sg</sub> (SOME<sub>sg</sub> (SET-OF (PL RATIONAL-NUMBER))))

(λy.x = y)

<figure><img src="image_109.png" /></figure>
*Every set of natural numbers is a set of rational numbers.*

NATURAL-NUMBER : N<sub>sg</sub>

RATIONAL-NUMBER : N<sub>sg</sub>

SET-OF : N<sub>pl</sub> → N<sub>sg</sub>

QR<sub>sg</sub> (EVERY (SET-OF (PL NATURAL-NUMBER)))

(λx. QR<sub>sg</sub> (SOME<sub>sg</sub> (SET-OF (PL RATIONAL-NUMBER))))

(λy. x = y))

<figure><img src="image_110.png" /></figure>
$$[[\text{NATURAL-NUMBER}] = \lambda x. \text{Nat} x]$$

$$[[\text{RATIONAL-NUMBER}] = \lambda x. \text{Rat} x]$$

$$[[\text{SET-OF}] = \lambda P x. \exists S. (P S) \wedge (\text{set} x S)]$$

$$\forall x. (\exists S. (\forall z. (S z) \rightarrow (\text{Nat} z)) \wedge (\text{set} x S))$$

$$\rightarrow (\exists y. (\exists S. (\forall z. (S z) \rightarrow (\text{Rat} z)) \wedge (\text{set} y S)) \wedge (x = y))$$

$$\forall x. (\text{Nat} x) \rightarrow (\text{Rat} x)$$

<figure><img src="image_111.png" /></figure>

<figure><img src="image_112.png" /></figure>

<figure><img src="image_113.png" /></figure>

<figure><img src="image_114.png" /></figure>

<figure><img src="image_115.png" /></figure>

<figure><img src="image_116.png" /></figure>
$$[[\text{NATURAL-NUMBER}] = \lambda x. \text{Nat} x$$

$$[[\text{RATIONAL-NUMBER}] = \lambda x. \text{Rat} x$$

$$[[\text{SET-OF}] = \lambda P x. \exists S. (P S) \wedge (\textbf{set} x S)$$

$$\forall x. (\exists S. (\forall z. (S z) \rightarrow (\text{Nat} z)) \wedge (\textbf{set} x S)) \\ \rightarrow (\exists y. (\exists S. (\forall z. (S z) \rightarrow (\text{Rat} z)) \wedge (\textbf{set} y S)) \wedge (x = y))$$

$$\forall x. (\text{Nat} x) \rightarrow (\text{Rat} x)$$

<figure><img src="image_117.png" /></figure>

<figure><img src="image_118.png" /></figure>

<figure><img src="image_119.png" /></figure>

<figure><img src="image_120.png" /></figure>

<figure><img src="image_121.png" /></figure>

<figure><img src="image_122.png" /></figure>

<figure><img src="image_123.png" /></figure>
$$[[\text{NATURAL-NUMBER}] = \lambda x. \text{Nat} x$$

$$[[\text{RATIONAL-NUMBER}] = \lambda x. \text{Rat} x$$

$$[[\text{SET-OF}] = \lambda P x. \exists S. (P S) \land (\textbf{set} x S)$$

$$\begin{align*}
\forall x. (\exists S. (\forall z. (S z) \rightarrow (\textbf{Nat} z)) \land (\textbf{set} x S)) \\
\rightarrow (\exists y. (\exists S. (\forall z. (S z) \rightarrow (\textbf{Rat} z)) \land (\textbf{set} y S)) \land (x = y))
\end{align*}$$

$$\forall x. (\text{Nat} x) \rightarrow (\text{Rat} x)$$

<figure><img src="image_124.png" /></figure>

<figure><img src="image_125.png" /></figure>

<figure><img src="image_126.png" /></figure>

<figure><img src="image_127.png" /></figure>

<figure><img src="image_128.png" /></figure>

<figure><img src="image_129.png" /></figure>

<figure><img src="image_130.png" /></figure>

<figure><img src="image_131.png" /></figure>
$$[[\text{NATURAL-NUMBER}] = \lambda x. \text{Nat} x$$

$$[[\text{RATIONAL-NUMBER}] = \lambda x. \text{Rat} x$$

$$[[\text{SET-OF}] = \lambda P x. \exists S. (P S) \wedge (\mathbf{set} x S)$$

$$\begin{align*}
\forall x. (\exists S. (\forall z. (S z) \rightarrow (\mathbf{Nat} z)) \wedge (\mathbf{set} x S)) \\
\rightarrow (\exists y. (\exists S. (\forall z. (S z) \rightarrow (\mathbf{Rat} z)) \wedge (\mathbf{set} y S)) \wedge (x = y))
\end{align*}$$

$$\forall x. (\mathbf{Nat} x) \rightarrow (\mathbf{Rat} x)$$

<figure><img src="image_132.png" /></figure>
Collective vs Distributive Predicates

*Every set of prime numbers is a set of coprime numbers.*

NUMBER : N<sub>sg</sub>      [[NUMBER]] = λx. Natx
PRIME : N<sub>sg</sub> → N<sub>sg</sub>      [[PRIME]] = λpx. (px) ∧ (prime x)
COPRIME : N<sub>pl</sub> → N<sub>pl</sub>      [[COPRIME]] = λpx. (px) ∧ (coprime x)

QR<sub>sg</sub> (EVERY (SET-OF (PL (PRIME NUMBER))))
(λx. QR<sub>sg</sub> (SOME<sub>sg</sub> (SET-OF (COPRIME (PL NUMBER)))))
(λy. x = y))

<figure><img src="image_133.png" /></figure>

<figure><img src="image_134.png" /></figure>

<figure><img src="image_135.png" /></figure>

<figure><img src="image_136.png" /></figure>

<figure><img src="image_137.png" /></figure>

<figure><img src="image_138.png" /></figure>

<figure><img src="image_139.png" /></figure>
*Every set of prime numbers is a set of coprime numbers.*

NUMBER : N<sub>sg</sub>

PRIME : N<sub>sg</sub> → N<sub>sg</sub>

COPRIME : N<sub>pl</sub> → N<sub>pl</sub>

[NUMBER] = λx. Natx

[PRIME] = λpx. (px) ∧ (prime x)

[COPRIME] = λpx. (px) ∧ (coprime x)

QR<sub>sg</sub> (EVERY (SET-OF (PL (PRIME NUMBER))))

(λx. QR<sub>sg</sub> (SOME<sub>sg</sub> (SET-OF (COPRIME (PL NUMBER)))))

(λy. x = y)

<figure><img src="image_140.png" /></figure>

<figure><img src="image_141.png" /></figure>

<figure><img src="image_142.png" /></figure>

<figure><img src="image_143.png" /></figure>

<figure><img src="image_144.png" /></figure>

<figure><img src="image_145.png" /></figure>

<figure><img src="image_146.png" /></figure>
*Every set of prime numbers is a set of coprime numbers.*

NUMBER : N<sub>sg</sub>

PRIME : N<sub>sg</sub> → N<sub>sg</sub>

COPRIME : N<sub>pl</sub> → N<sub>pl</sub>

[NUMBER] = λx. Nat x

[PRIME] = λpx. (px) ∧ (prime x)

[COPRIME] = λpx. (px) ∧ (coprime x)

QR<sub>sg</sub> (EVERY (SET-OF (PL (PRIME NUMBER))))

(λx. QR<sub>sg</sub> (SOME<sub>sg</sub> (SET-OF (COPRIME (PL NUMBER)))))

(λy. x = y)

<figure><img src="image_147.png" /></figure>

<figure><img src="image_148.png" /></figure>

<figure><img src="image_149.png" /></figure>

<figure><img src="image_150.png" /></figure>

<figure><img src="image_151.png" /></figure>

<figure><img src="image_152.png" /></figure>

<figure><img src="image_153.png" /></figure>
*Every set of prime numbers is a set of coprime numbers.*

$$ 
\begin{array}{l@{\quad}l}
\text{NUMBER} : N_{\text{sg}} & [\text{NUMBER}] = \lambda x. \text{Nat} x \\
\text{PRIME} : N_{\text{sg}} \rightarrow N_{\text{sg}} & [\text{PRIME}] = \lambda p x. (p x) \wedge (\textbf{prime} x) \\
\text{COPRIME} : N_{\text{pl}} \rightarrow N_{\text{pl}} & [\text{COPRIME}] = \lambda p x. (p x) \wedge (\textbf{coprime} x)
\end{array}
$$

$$ 
\begin{array}{l}
\text{QR}_{\text{sg}} \text{ (EVERY (SET-OF (PL (PRIME NUMBER))))} \\
\quad (\lambda x. \text{QR}_{\text{sg}} \text{ (SOME}_{\text{sg}}\text{ (SET-OF (COPRIME (PL NUMBER))))} \\
\quad (\lambda y. x = y))
\end{array}
$$

<figure><img src="image_154.png" /></figure>
# Symmetric Predicates

Phrases that denote binary symmetric predicates may often be used as collective predicates:

* James agrees with Carol.
  Carol agrees with James.
  James and Carol agree.

* Boston is quite different from New York.
  New York is quite different from Boston.
  Boston and New York are quite different.

* Sue and Dan divorced.
  ? Sue divorced Dan.
  ? Dan divorced Sue.

<figure><img src="image_155.png" /></figure>
Phrases that denote binary symmetric predicates may often be used as collective predicates:

* James agrees with Carol.
  Carol agrees with James.
* James and Carol agree.

* Boston is quite different from New York.
  New York is quite different from Boston.
  Boston and New York are quite different.

* Sue and Dan divorced.
  ? Sue divorced Dan.
  ? Dan divorced Sue.

<figure><img src="image_156.png" /></figure>
Phrases that denote binary symmetric predicates may often be used as collective predicates:

* James agrees with Carol.
  Carol agrees with James.
* James and Carol agree.

* Boston is quite different from New York.
  New York is quite different from Boston.
* Boston and New York are quite different.

* Sue and Dan divorced.
  ? Sue divorced Dan.
  ? Dan divorced Sue.

<figure><img src="image_157.png" /></figure>
Phrases that denote binary symmetric predicates may often be used as collective predicates:

* James agrees with Carol.
  Carol agrees with James.
* James and Carol agree.

* Boston is quite different from New York.
  New York is quite different from Boston.
* Boston and New York are quite different.

* Sue and Dan divorced.

⁴Sue divorced Dan.

⁵Dan divorced Sue.

<figure><img src="image_158.png" /></figure>
## General Scheme

* A is orthogonal to B.

B is orthogonal to A.

A and B are orthogonal.

(1) [NP₁] is [ADJ] [PREP] [NP₂].

(2) [NP₁] and [NP₂] are [ADJ].

<figure><img src="image_159.png" /></figure>

<figure><img src="image_160.png" /></figure>

<figure><img src="image_161.png" /></figure>
## General Scheme

* A is orthogonal to B.

B is orthogonal to A.

A and B are orthogonal.

(1) [NP₁] is [ADJ] [PREP] [NP₂].

(2) [NP₁] and [NP₂] are [ADJ].

<figure><img src="image_162.png" /></figure>
The vector $\mathbf{y}$ is orthogonal to the vector $\mathbf{x}$.

Every row of $\mathbf{H}_X$ is orthogonal to every row of $\mathbf{H}_Z$.

The section $s$ is orthogonal to the first $m$ eigenfunctions of the operator.

The vector $\mathbf{y}$ and the vector $\mathbf{x}$ are orthogonal.

⁷Every row of $\mathbf{H}_X$ and every row of $\mathbf{H}_Z$ are orthogonal.

⁷The section $s$ and the first $m$ eigenfunctions of the operator are orthogonal.

<figure><img src="image_163.png" /></figure>
The vector $\mathbf{y}$ is orthogonal to the vector $\mathbf{x}$.

Every row of $\mathbf{H}_X$ is orthogonal to every row of $\mathbf{H}_Z$.

The section $s$ is orthogonal to the first $m$ eigenfunctions of the operator.

The vector $\mathbf{y}$ and the vector $\mathbf{x}$ are orthogonal.

⁷Every row of $\mathbf{H}_X$ and every row of $\mathbf{H}_Z$ are orthogonal.

⁷The section $s$ and the first $m$ eigenfunctions of the operator are orthogonal.

<figure><img src="image_164.png" /></figure>
The vector $\mathbf{y}$ is orthogonal to the vector $\mathbf{x}$.

Every row of $\mathbf{H}_X$ is orthogonal to every row of $\mathbf{H}_Z$.

The section $s$ is orthogonal to the first $m$ eigenfunctions of the operator.

The vector $\mathbf{y}$ and the vector $\mathbf{x}$ are orthogonal.

⁷Every row of $\mathbf{H}_X$ and every row of $\mathbf{H}_Z$ are orthogonal.

⁷The section $s$ and the first $m$ eigenfunctions of the operator are orthogonal.

<figure><img src="image_165.png" /></figure>
## A plea for the dual grammatical number

$$ \text{AND} : \text{NP}_{\text{sg}} \rightarrow \text{NP}_{\text{sg}} \rightarrow \text{NP}_{\text{du}} $$

$$ [[\text{NP}_{\text{du}}]] = (e \rightarrow e \rightarrow t) \rightarrow t $$

$$ [[\text{AND}]] = \lambda x y f. f x y $$

<figure><img src="image_166.png" /></figure>
## A plea for the dual grammatical number

$$ \text{AND} : \text{NP}_{\text{sg}} \rightarrow \text{NP}_{\text{sg}} \rightarrow \text{NP}_{\text{du}} $$

$$ [[\text{NP}_{\text{du}}]] = (e \rightarrow e \rightarrow t) \rightarrow t $$

$$ [[\text{AND}]] = \lambda x y f. f x y $$

<figure><img src="image_167.png" /></figure>
## A plea for the dual grammatical number

$$ \text{AND} : \text{NP}_{\text{sg}} \rightarrow \text{NP}_{\text{sg}} \rightarrow \text{NP}_{\text{du}} $$

$$ [[\text{NP}_{\text{du}}]] = (\mathbf{e} \rightarrow \mathbf{e} \rightarrow \mathbf{t}) \rightarrow \mathbf{t} $$

$$ [[\text{AND}]] = \lambda xyf. f x y $$

<figure><img src="image_168.png" /></figure>
Much more to say:

Adjectives denoting (symmetric) binary relations used with noun phrases that
denote collections of three or more elements.

Binary distributivity operator.

Strict binary distributivity operator.

Overt markers of (strict) binary distributivity: *pairwize*, *mutually*, *by pairs*...

Reciprocals: *each other*, *one another*...

...

<figure><img src="image_169.png" /></figure>
Much more to say:

Adjectives denoting (symmetric) binary relations used with noun phrases that
denote collections of three or more elements.

Binary distributivity operator.

Strict binary distributivity operator.

Overt markers of (strict) binary distributivity: *pairwize*, *mutually*, *by pairs*...

Reciprocals: *each other*, *one another*...

...

<figure><img src="image_170.png" /></figure>
* Most of the semantic phenomena that appear in natural language also arise in textual mathematics.

* However, the use of natural language in mathematics tends to be more regular.

* The language of mathematics provides indeed a quite interesting *grammatical laboratory*.

* Studying the linguistic structure of textual mathematics is an interesting source of inspiration for how to formalize mathematics.

<figure><img src="image_171.png" /></figure>
* Most of the semantic phenomena that appear in natural language also arise in textual mathematics.

* However, the use of natural language in mathematics tends to be more regular.

* The language of mathematics provides indeed a quite interesting *grammatical laboratory*.

* Studying the linguistic structure of textual mathematics is an interesting source of inspiration for how to formalize mathematics.

<figure><img src="image_172.png" /></figure>
* Most of the semantic phenomena that appear in natural language also arise in textual mathematics.

* However, the use of natural language in mathematics tends to be more regular.

* The language of mathematics provides indeed a quite interesting *grammatical laboratory*.

* Studying the linguistic structure of textual mathematics is an interesting source of inspiration for how to formalize mathematics.

<figure><img src="image_173.png" /></figure>
* Most of the semantic phenomena that appear in natural language also arise in textual mathematics.

* However, the use of natural language in mathematics tends to be more regular.

* The language of mathematics provides indeed a quite interesting *grammatical laboratory*.

* Studying the linguistic structure of textual mathematics is an interesting source of inspiration for how to formalize mathematicst.

<figure><img src="image_174.png" /></figure>
* Most of the semantic phenomena that appear in natural language also arise in textual mathematics.

* However, the use of natural language in mathematics tends to be more regular.

* The language of mathematics provides indeed a quite interesting *grammatical laboratory*.

* Studying the linguistic structure of textual mathematics is an interesting source of inspiration for how to formalize mathematicst.

<figure><img src="image_175.png" /></figure>