# On the use of binary relations as collective predicates in natural mathematics

Santiago Arambillete, Philippe de Groote

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Santiago Arambillete, Philippe de Groote. On the use of binary relations as collective predicates in natural mathematics. LENLS21 - 21st International Workshop of Logic and Engineering of Natural Language Semantics, Nagoya University, Nov 2025, Nagoya, Japan. hal-05482065

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# On the use of binary relations as collective predicates in natural mathematics*

— extended abstract —

Santiago Arambillete<fnref n="1" /> and Philippe de Groote<fnref n="2" />

**1 Introduction** This paper contributes to the linguistic and semantic study of the language of *natural mathematics*, i.e., the language that mathematicians use to express pieces of mathematics in textbooks and articles [Zinn, 2004, Ganesalingam, 2013]. This language, which consists of a mixture of natural language and mathematical formulas, has its own idiosyncrasies, and is worth studying in particular with the aim of interacting with proof assistants, such as Isabelle, Lean or Rocq, at a level closer to natural language than the formal language of mathematical logic.

More specifically, we study the conditions under which adjectives that denote binary relations can be used as collective predicates. This leads us to propose a fine-grained semantic interpretation of grammatical numbers and to introduce distributivity operators that enable a compositional semantic treatment of plurals in natural mathematics.

Our study relies on numerous examples taken from a large corpus made of mathematics papers submitted to the arXiv and some standard algebra and number theory textbooks.

**2 Collectivity versus distributivity** 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 (see Champollion [2020], for a comprehensive review). As an illustration, consider the following sophism:

(1) Every set of prime numbers is a *fortiori* a set of coprime numbers. Therefore, every prime number is a coprime number.

While this reasoning would be perfectly acceptable if *prime* and *coprime* were respectively replaced by *rational* and *real*, it just does not make any sense in the present case because *coprime* is a collective predicate.

Interpreting plural entities as sets of individuals (see Winter and Scha [2015]), *coprime* must be semantically interpreted as a predicate on sets, that is, in Montague semantics, as a term of type ($e \to t) \to t$, which is indeed the case according to definition (2a). By contrast, (2b) defines *prime* as a predicate over entities, that is, as a term of type $e \to t$.

<footnote n="1">¹ Université Paris Cité, CNRS, Inria, IRIF</footnote>

<footnote n="2">² Université de Lorraine, CNRS, Inria, LORIA</footnote>

<footnote>* The research leading to these results has received funding from the European Research Council (ERC) under the European Union's Ninth Framework Programme Horizon Europe (ERC Synergy Project Malinca, Grant Agreement n. 101167526).</footnote>
(2) a. coprime $\triangleq \lambda S. (\forall a. (Sa) \rightarrow (Na)) \wedge (\forall n. ((Nn) \wedge (\forall a. (Sa) \rightarrow (div na))) \rightarrow (n = 1))$

b. prime $\triangleq \lambda a. (Na) \wedge (a \neq 1) \wedge (\forall n. ((Nn) \wedge (div na)) \rightarrow ((n = 1) \vee (n = a)))$

To provide a compositional semantics for the common noun phrases *prime numbers* and *coprime numbers*, we need to distinguish between the category of singular noun phrases, $N_{sg}$, and the one of plural noun phrases, $N_{pl}$. Then the abstract syntactic structures of the two phrases essentially differ in the scope taken by the plural marker PL.

(3) a. $[[PL]_{N_{sg}} \rightarrow N_{pl}] [[PRIME]_{N_{sg}} \rightarrow N_{sg}] [NUMBER]_{N_{sg}}]_{N_{pl}}$

b. $[[COPRIME]_{N_{pl}} \rightarrow N_{pl}] [[PL]_{N_{sg}} \rightarrow N_{pl}] [NUMBER]_{N_{sg}}]_{N_{pl}} N_{pl}$

From a semantic perspective, the plural marker PL is an operator that transforms a predicate on entities into a predicate on sets of entities. Accordingly, we interpret it as the distributivity operator of Link [1983], the definition of which is as follows:<fnref n="3" />

(4) distr $\triangleq \lambda PS. \forall x. (Sx) \rightarrow (Px)$

The preceding discussion illustrates some of the issues related to the treatment of plurals, and emphasizes the importance of distinguishing between distributive and collective predicates in order to avoid paradoxes. In the remainder of this paper, we focus on a specific class of collective predicates used in natural mathematics, namely, those collective predicates which are derived from adjectives denoting binary relations.

**3 Symmetric predicates** It has been observed that phrases that denote binary symmetric predicates, typically verbs or adjectives, may often be used as collective predicates (see, among others, Lakoff and Peters [1969] and Winter [2018]). Consider, for instance, the verb *agree* and the adjective *different*. In sentences (5a) and (5b), they are used in their binary forms, with both verb phrases *agrees* and *is quite different* taking two syntactic arguments.

(5) a. James agrees with Carol.

b. Boston is quite different from New York.

Alternatively, they can be used in a collective form, like in sentences (6a) and (6b).

(6) a. James and Carol Agree.

b. Boston and New York are quite different.

<footnote n="3">³ For the sake of ease, we have simplified the semantic treatment of plurals. A plural entity, by definition, is a combination of at least two singular entities. Consequently, a more accurate semantic definition of the plural marker is as follows: $[[PL]] = \lambda pS. (|S| \ge 2) \wedge (distr p S)$. Interestingly enough, the formula $\forall S. (distr prime S) \rightarrow (coprime S)$, which can be considered as a possible interpretation of the first sentence of example (1), is true for every set containing at least two elements. However, it is not true for the empty set and the singletons (with the exception of the singleton {1}).</footnote>
Focusing on the case of adjectives, we can try stating the following general principle:

(7) If [ADJ] stands for an adjective that denotes a binary symmetric relation, then the sentences obtained by instantiating schemes (a) and (b), here below, are semantically equivalent.

a. $[NP_1]$ is [ADJ] [PREP] [NP_2].

b. $[NP_1]$ and $[NP_2]$ are [ADJ].

In scheme (7a), $[NP_1]$ can be a singular noun phrase denoting an entity (examples 8a and 8c) or a quantified singular noun phrase (example 8b). As for $[NP_2]$, it can be a simple singular noun phrase (example 8a), a quantified singular noun phrase (example 8b), or a plural noun phrase (example 8c).

(8) a. The vector $y$ is orthogonal to the vector $x$.

b. Every row of $H_X$ is orthogonal to every row of $H_Z$.

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

Applying principle (7) to (8), We obtain the following sentences.

(9) a. The vector $y$ and the vector $x$ are orthogonal.

b. ?Every row of $H_X$ and every row of $H_Z$ are orthogonal.

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

In the case of sentence (8a), the process is successful. Sentence (9a) is felicitous and is synonymous with (8a). By contrast, sentences (9b) and (9c) are questionable. According to some of our informants, they are ill-formed, bad style, or possibly ambiguous. In any case, they are not synonymous with the original sentences. We therefore draw the conclusion that principle (7) only applies to pairs of simple singular noun phrases, i.e., noun phrases that denote single entities. This leads us to distinguish between simple and quantified singular noun phrases, but also between general plural noun phrases and plural noun phrases denoting pairs of single entities. To this end, we introduce a special syntactic category, $NP_{du}$, to refer to pairs of entities. This category, in fact, corresponds to the dual grammatical number, which exists in several languages. We end up with four categories of noun phrase: simple singular noun phrases, $NP_{sg}$, quantified singular noun phrases, $QNP_{sg}$, dual noun phrases, $NP_{du}$, and plural noun phrases, $NP_{pl}$, with the following interpretations:

$$ (10) \qquad \begin{aligned} [[NP_{sg}]] &= e \\ [[NP_{du}]] &= (e \to e \to t) \to t \end{aligned} \qquad \begin{aligned} [[QNP_{sg}]] &= (e \to t) \to t \\ [[NP_{pl}]] &= ((e \to t) \to t) \to t \end{aligned} $$

With this type system, we can provide an appropriate compositional treatment to sentences such as those in examples (8) and (9a). For the sake of brevity, we omit the formal details of this treatment in this abstract.

**4 Binary distributivity** Adjectives denoting symmetric binary relations can also be used with noun phrases which denote collections of three or more elements.
(11) The three lines $L$, $L_-$, and $L_+$ are parallel.

A natural way of interpreting such sentences is to use a *binary distributivity operator*, analogous to the distributivity operator (4) that is used for unary predicates. Its definition is the following:

$$ (12) \quad \mathbf{bdistr} \triangleq \lambda RS. \forall xy. ((S x) \wedge (S y)) \rightarrow (R x y) $$

Applying this operator to an equivalence relation (i.e., a reflexive, symmetric, and transitive relation) amounts to interpreting the corresponding collective predicate as the set of subsets of equivalence classes of the relation. More generally, it amounts to interpreting the collective predicate corresponding to a symmetric binary relation as the set of cliques in the graph of the relation. In particular, $(\mathbf{bdistr} RS)$ implies that $R \times x$ holds for every $x$ in the set $S$, which is perfectly fine for a reflexive relation. However, in the case of example (13), using operator (12) fails at giving the proper interpretation because it requires each coefficient to be distinct from itself, which is absurd.

(13) The coefficients $\lambda_0, \dots, \lambda_n$ are all distinct.

What is needed in this case is a *strict binary distributivity operator* that adds to (12) the condition that $x \neq y$.

$$ (14) \quad \mathbf{sbdistr} \triangleq \lambda RS. \forall xy. ((S x) \wedge (S y) \wedge (x \neq y)) \rightarrow (R x y) $$

Strict binary distributivity is often marked overtly by adverbial phrases such as *pair-wise*, *mutually*, and *by pairs*, or also by reciprocals such as *each other* and *one another*, and the like.

(15) Since these generating vectors are pairwise orthogonal, they are independent.

In the case of example (13), we posit the existence of a covert strictness marker whose presence can be explained by the *Non-Vacuity Principle* of Kamp and Partee [1995] or the *Strongest Meaning Hypothesis* proposed by Dalrymple et al. [1998].

The semantic interpretation of *pairwise*, when applied to a binary relation, is given by the distributivity operator (14). The adverb *pairwise*, however, can also be applied to collective predicates that do not derive from binary relations. In this case one obtains a new predicate whose meaning is different from the original one. For a downward closed collective predicate such as *linearly independent*, the new meaning is weaker, such that being *linearly independent* always implies being *pairwise linearly independent*. Conversely, an upward closed predicate such as *coprime* is strengthened by the use of *pairwise*; that is, being *pairwise coprime* implies being *coprime*. In both cases, the meaning of *pairwise* is given by a variant of operator (14) in which the collective predicate is applied to an unordered pair:

$$ (16) \quad \lambda PS. \forall xy. (x \neq y \wedge (S x) \wedge (S y)) \rightarrow (P(\lambda z. (z = x) \vee (z = y))) $$
We conclude this extended abstract with a table that provides a qualitative account of the distribution we observed for the adverb “pairwise”.

<table><caption>**Table 1.** Use of *pairwise* with different kinds of predicates</caption><thead><tr><td>Type of predicate</td><td>Examples</td><td>Use of pairwise</td></tr></thead><tbody><tr><td>collective</td><td><i>linearly independent, coprime</i></td><td>changes meaning</td></tr><tr><td>equivalence relation</td><td><i>isomorphic, parallel, equal</i></td><td>rare</td></tr><tr><td>symmetric non-reflexive</td><td><i>orthogonal, distinct, disjoint</i></td><td>common</td></tr><tr><td>symmetric reflexive</td><td><i>commuting, intersecting, comparable</i></td><td>common?</td></tr><tr><td>matching</td><td><i>antipodal, complex conjugate</i></td><td>only with a different meaning</td></tr><tr><td>non-symmetric</td><td><i>consecutive</i></td><td>not used</td></tr></tbody></table>

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