Willem Conradie
email: wconradie@uj.ac.za

Sabine Frittella
email: sabine.frittella@insa-cvl.fr

Alessandra Palmigiano
email: a.palmigiano@tudelft.nl

Michele M Piazzai
email: m.piazzai@tudelft.nl

Apostolos Tzimoulis
email: a.tzimoulis-1@tudelft.nl

Nachoem M Wijnberg
email: n.m.wijnberg@gmail.com

Towards an epistemic-logical theory of categorization

Keywords: conference proceedings, • Information systems → Relational database model, • Theory of computation → Modal and temporal logics, Logic and databases, • Computing methodologies → Knowledge representation and reasoning, Nonmonotonic, default reasoning and belief revision, Reasoning about belief and knowledge, • Applied computing → Consumer products, Marketing, epistemic logic of categories, categorization, formal concepts, decisionmaking, common knowledge, formal concept analysis, prototype theory, category-spanning

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INTRODUCTION

Categories (understood as types of collective identities for broad classes of objects or of agents) are the most basic cognitive tools, and are key to the use of language, the construction of knowledge and identity, and the formation of agents' evaluations and decisions. The literature on categorization is expanding rapidly, motivated by-and in connection with-the theories and methodologies of a wide range of fields in the social sciences and AI. For instance, in linguistics, categories are central to the mechanisms of grammar generation [START_REF] Croft | Syntactic categories and grammatical relations: The cognitive organization of information[END_REF]; in AI, classification techniques are core to pattern recognition, data mining, text mining and knowledge discovery in databases; in sociology, categories are used to explain the construction of social identity [START_REF] Jenkins | Categorization: Identity, social process and epistemology[END_REF]; in management science, categories are used to predict how products and producers will be perceived and evaluated by consumers and investors [START_REF] Cattani | Categories and competition[END_REF][START_REF] Hsu | Typecasting, Legitimation, and Form Emergence: A Formal Theory[END_REF][START_REF] Kuijken | The producer-consumer classification gap and its effects on music festival success[END_REF][START_REF] Paolella | Category spanning, evaluation, and performance: Revised theory and test on the corporate law market[END_REF][START_REF] Nachoem | Classification systems and selection systems: The risks of radical innovation and category spanning[END_REF].

In [START_REF] Conradie | Categories: How I Learned to Stop Worrying and Love Two Sorts[END_REF], we proposed the framework of a positive (i.e. negationfree and implication-free) normal multi-modal logic as an epistemic logic of categories and agents' categorical perception, and discussed its algebraic and Kripke-style semantics. In the present paper, we introduce a simpler and more general framework than [START_REF] Conradie | Categories: How I Learned to Stop Worrying and Love Two Sorts[END_REF], in which the modal operators are regular but not normal, and the (rather technical) restrictions on the Kripke-style models of [START_REF] Conradie | Categories: How I Learned to Stop Worrying and Love Two Sorts[END_REF] are dropped. We use this logical framework to formalize core notions developed and applied in the fields mentioned above, with a focus on those relevant to management science, as a step towards building systematic connections between modern categorization theory and epistemic logic.

Structure of the paper. In Section 2, we briefly review the main views on the foundations of categorization theory together with the formal approaches inspired by some of these views. In Section 3, we discuss the basic framework of the epistemic logic of categories. We introduce its refined Kripke-style semantics and axiomatization, together with two language enrichments involving a common knowledge-type construction and hybrid-style nominal (and co-nominal) variables, respectively. In Section 4, we discuss a number of core categorization-theoretic notions from business science and our proposed formalizations of them. In Section 5, we discuss further directions. Soundness and completeness are treated in Section A.

CATEGORIZATION: FOUNDATIONS AND FORMAL APPROACHES

In the present section we review the main views, insights, and approaches to the foundations of categorization theory and to the formal models capturing these. Our account is necessarily incomplete. We refer the reader to [2] for an exhaustive overview.

Extant foundational approaches

The literature on the foundations of categorization theory displays a variety of definitions, theories, models, and methods, each of which capturing some key facets of categorization. The classical theory of categorization [START_REF] Edward | Categories and concepts[END_REF] goes back to Aristotle, and is based on the insight that all members of a category share some fundamental features which define their membership. Accordingly, categorization is viewed as a deductive process of reasoning with necessary and sufficient conditions, resulting in categories with sharp boundaries, which are represented equally well by any of their members. The classical view has inspired influential approaches in machine learning such as conceptual clustering [START_REF] Douglas | Knowledge acquisition via incremental conceptual clustering[END_REF]. However, this view runs into difficulties when trying to accommodate a new object or entity which would intuitively be part of a given category but does not share all the defining features of the category. Other difficulties, e.g. providing an exhaustive list of defining features, unclear cases, and the existence of members of given categories which are judged to be better representatives of the whole class than others, motivated the introduction of prototype theory [START_REF] Lakoff | Cognitive models and prototype theory[END_REF][START_REF] Rosch | Principles of categorization[END_REF]. This theory regards categorization as the inductive process of finding the best match between the features of an object and those of the closest prototype(s). Prototype theory addresses the above mentioned problems of the classical theory by relaxing the requirement that membership be decided through the satisfaction of an exhaustive list of features. It allows for unclear cases and embraces the empirically verified intuition that people regard membership in most categories as a matter of degrees, with certain members being more central (or prototypical) than others. (For instance, robins are regarded as prototypical birds, while penguins are not.) To account for how an ex-ante prototype is generated in the mind of agents, the exemplar theory [START_REF] Edward | The exemplar view[END_REF] was proposed, according to which individuals make category judgments by comparing new stimuli with instances already stored in memory (the "exemplars"). However, the existence of instances or prototypes of a given category presupposes that this category has already been defined. Hence, both the prototype and the exemplar view run into a circularity problem. Moreover, it has been argued that similarity-based theories of categorization (such as the prototype and the exemplar view) fail to address the problem of explaining 'why we have the categories we have', or, in other words, why certain categories seem to be more cogent and coherent than others. Even more fundamentally, similarity might be imposed rather than discovered (do things belong in the same category because they are similar, or are they similar because they belong in the same category?), i.e. might be the effect of conceptual coherence rather than its cause. Pivoting on the notion of coherence for category-formation, the theory-based view on categorization [29] posits that categories arise in connection with theories (broadly understood so as to include also informal explanations). For instance, ice, water and steam can be grouped together in the same category on the basis of the theory of phases in physical chemistry. The coherence of categories proceeds from the coherence of the theories on which they are based. This view of categorization allows one to group together entities which would be scored as dissimilar using different methods; for instance, it allows to group together a gold watch, the school report of one's grandfather, and the ownership of a piece of land in the category of "things one wants one's children to inherit", which is based on one's theory of what family is. However, the theory-based view does not account for the intuition that categories themselves are the building blocks of theory-formation, which again results in a circularity problem. Summing up, the extant views on categorization (the classical [START_REF] Edward | Categories and concepts[END_REF], prototype [START_REF] Lakoff | Cognitive models and prototype theory[END_REF][START_REF] Rosch | Principles of categorization[END_REF], exemplar [START_REF] Edward | The exemplar view[END_REF], and theory-based [29]) are difficult to reconcile and merge into a satisfactory overarching theory accommodating all the insights into categorization that researchers in the different fields have been separately developing. The present paper is one of the first steps of a research program aimed at clarifying notions developed independently, and at developing a common ground which can hopefully facilitate the build-up of such a theory.

Extant formal approaches

Conceptual spaces. The formal approach to the representation of categories and concepts which is perhaps the most widely adopted in social science and management science is the one introduced by Gärdenfors, which is based on conceptual spaces [START_REF] Gärdenfors | Conceptual spaces: The geometry of thought[END_REF]. These are multi-dimensional geometric structures, the components of which (the quality dimensions) are intended to represent basic features -e.g. colour, pitch, temperature, weight, time, price -by which objects (represented as points in the product space of these dimensions) can be meaningfully compared. Each dimension is endowed with its appropriate geometric (e.g. metric, topological) structure. Concept-formation in conceptual spaces is modelled according to a similarity-based view of concepts. Specifically, if each dimension of a conceptual space has a metric, these metrics translate in a notion of distance between the objects represented in the space, which models their similarity, so that the closer their distance, the more similar they are. Concepts (i.e. formal categories) are represented as convex sets of the conceptual space 1 . The geometric center of any such concept is a natural interpretation of the prototype of that concept. Formal Concept Analysis. A very different approach, Formal Concept Analysis (FCA) [START_REF] Ganter | Formal concept analysis: mathematical foundations[END_REF], is a method of data analysis based on Birkhoff's representation theory of complete lattices [START_REF] Davey | Lattices and Order[END_REF]. In FCA, databases are represented as formal contexts, i.e. structures (A, X, I) such that A and X are sets, and I ⊆ A × X is a binary relation. Intuitively, A is understood as a collection of objects, X as a collection of features, and for any object a and feature x, the tuple (a, x) belongs to I exactly when object a has feature x. Every formal context (A, X, I) can be associated with the collection of its formal concepts, i.e. the tuples (B, Y) such that B ⊆ A, Y ⊆ X, and B × Y is a maximal rectangle included in I. The set B is the extent of the formal concept (B, Y), and Y is its intent. Because of maximality, the extent of a formal concept uniquely identifies and is identified by its intent. Formal concepts can be partially ordered; namely, (B, Y) is a subconcept of (C, Z) exactly when B ⊆ C, or equivalently, when Z ⊆ Y. Ordered in this way, the concepts of a formal context form a complete lattice (i.e., the least upper bound and the greatest lower bound of every collection of formal concepts exist), and by Birkhoff's theorem, every complete lattice is isomorphic to some concept lattice. The link established by FCA between complete lattices and the formalization of concepts (or categories) captures an aspect of categories which is very much highlighted in the categorization theory literature. Namely, categories never occur in isolation; rather, they arise in the context of categorization systems (e.g. taxonomies), which are typically organized in hierarchies of super-(i.e. less specified) and sub-(i.e. more specified) categories. While most approaches identify concepts with their extent, in FCA, intent and extent of a concept are treated on a par, i.e., the intent of a concept is just as essential as its extent. While FCA has tried to connect itself with various cognitive and philosophical theories of concept-formation, it is most akin to the classical view.

Formal concepts as modal models. In [START_REF] Conradie | Categories: How I Learned to Stop Worrying and Love Two Sorts[END_REF], we first established a connection between FCA and modal logic, based on the idea that (enriched) formal contexts can be taken as models of an epistemic modal logic of categories/concepts. Formulas of this logic are constructed out of a set of atomic variables using the standard positive propositional connectives ∧, ∨, , ⊥, and modal operators i associated with each agent i ∈ Ag. The formulas so generated do not denote states of affairs (to which a truth-value can be assigned), but categories or concepts. In this modal language, as usual, it is easy to distinguish the 'objective' or factual information (stored in the database), encoded in the formulas of the modal-free fragment of the language, and the agents' subjective interpretation of the 'objective' information, encoded in formulas in which modal operators occur. In this language, we can talk about e.g. the category that according to Alice is the category that according to Bob is the category of Western movies. This makes it possible to define fixed points of these regressions, similarly to the way in which common knowledge is defined in classical epistemic logic [START_REF] Fagin | Reasoning about knowledge[END_REF]. Intuitively, these fixed points represent the stabilization of a process of social interaction; for instance, the consensus reached by a group of agents regarding a given category.

Models for this logic are formal contexts (A, X, I) enriched with an extra relation R i ⊆ A × X for each agent (intuitively, for every object a ∈ A and every feature x ∈ X, we read aR i x as 'object a has feature x according to agent i'. Hence, while the relation I represents reality as is recorded in the database represented by the formal context (A, X, I), each relation R i represents as usual the subjective view of the corresponding agent i about objects and their features, and is used to interpret i -formulas.

This logic arises and has been studied in the context of unified correspondence theory [START_REF] Conradie | Unified Correspondence[END_REF], and allows one to relate, via Sahlqvisttype results, sentences in the first-order language of enriched formal contexts (expressing low-level, concrete conditions about objects and features) with inequalities ϕ ≤ ψ, where ϕ and ψ are formulas in the modal language above, expressing high-level, abstract relations about categories and how they are perceived and understood by different agents. In the next section, we expand on the relevant definitions and background facts about this logic.

EPISTEMIC LOGIC OF CATEGORIES

Basic logic and intended meaning. Let Prop be a (countable or finite) set of atomic propositions and Ag be a finite set (of agents). The basic language L of the epistemic logic of categories is

ϕ := ⊥ | | p | ϕ ∧ ϕ | ϕ ∨ ϕ | i ϕ,
where p ∈ Prop. As mentioned above, formulas in this language are terms denoting categories (or concepts). Atomic propositions provide a vocabulary of category labels, such as music genres (e.g. jazz, rock, rap), movie genres (e.g. western, drama, horror), supermarket products (e.g. milk, dairy products, fresh herbs). Compound formulas ϕ ∧ ϕ and ϕ ∨ ψ respectively denote the greatest common subcategory and the smallest common supercategory of ϕ and ψ. For a given agent i ∈ Ag, the formula i ϕ denotes the category ϕ, according to i. At this stage we are deliberately vague as to the precise meaning of 'according to'. Depending on the properties of i , the formula i ϕ might denote the category known, or perceived, or believed to be ϕ by agent i. The basic, or minimal normal L-logic is a set L of sequents ϕ ψ (which intuitively read "ϕ is a subcategory of ψ") with ϕ, ψ ∈ L, containing the following axioms:

• Sequents for propositional connectives:

p p, ⊥ p, p , p p ∨ q, q p ∨ q, p ∧ q p, p ∧ q q,
• Sequents for modal operators: i p ∧ i q i (p ∧ q) and closed under the following inference rules:

ϕ χ χ ψ ϕ ψ ϕ ψ ϕ (χ/p) ψ (χ/p) χ ϕ χ ψ χ ϕ ∧ ψ ϕ χ ψ χ ϕ ∨ ψ χ ϕ ψ
i ϕ i ψ Thus, the modal fragment of L incorporates the viewpoints of individual agents into the syllogistic reasoning supported by the propositional fragment of L. By an L-logic, we understand any extension of L with L-axioms ϕ ψ. Interpretation in enriched formal contexts. Let us discuss the structures which play the role of Kripke frames. 2 An enriched formal context is a tuple

F = (P, {R i | i ∈ Ag})
such that P = (A, X, I) is a formal context, and R i ⊆ A × X for every i ∈ Ag, satisfying certain additional properties which guarantee that their associated modal operators are well defined (cf. Definition A.6). As mentioned above, formal contexts represent databases of market products (the elements of the set A), relevant features (the elements of the set X), and an incidence relation I ⊆ A × X (so that aI x reads: "market product a has feature x"). In addition, enriched formal contexts contain information about the epistemic attitudes of individual agents, so that aR i x reads: "market product a has feature x according to agent i", for any i ∈ Ag. A valuation on F is a map V : Prop → P (A) × P (B), with the restriction that V (p) is a formal concept of P = (A, X, I), i.e., every p ∈ Prop is mapped to

V (p) = (B, Y) such that B ⊆ A, Y ⊆ X,
M, a p iff a ∈ [[p]] M M, x p iff x ∈ ([p]) M
and we read M, a p as "a is a member of p", and M, x p as "x describes p". The interpretation of atomic propositions can be extended to propositional L-formulas as follows:

M, a always M, x iff aI x for all a ∈ A M, x ⊥ always M, a ⊥ iff aI x for all x ∈ X M, a ϕ ∧ ψ iff M, a ϕ and M, a ψ M, x ϕ ∧ ψ iff for all a ∈ A, if M, a ϕ ∧ ψ, then aI x M, x ϕ ∨ ψ iff M, x ϕ and M, x ψ M, a ϕ ∨ ψ iff for all x ∈ X, if M, x ϕ ∨ ψ, then aI x
Hence, in each model, is interpreted as the category generated by the set A of all objects, i.e. the widest category and hence the one with the laxest (possibly empty) description; ⊥ is interpreted as the category generated by the set X of all features, i.e. the smallest (possibly empty) category and hence the one with the most restrictive description; ϕ ∧ ψ is interpreted as the semantic category generated by the intersection of the extensions of ϕ and ψ (hence, the description of ϕ ∧ ψ certainly includes ([ϕ]) ∪ ([ψ]) but is possibly larger). Likewise, ϕ ∨ ψ is interpreted as the semantic category generated by the intersection of the intensions of ϕ and ψ (hence, objects in

[[ϕ]] ∪ [[ψ]
] are certainly members of ϕ ∨ ψ but there might be others). As to the interpretation of modal formulas: 2 Details can be found in Section A.

M, a

i ϕ iff for all x ∈ X, if M, x ϕ, then aR i x M, x i ϕ iff for all a ∈ A, if M, a ϕ, then aI x.

Thus, in each model, i ϕ is interpreted as the category whose members are those objects to which agent i attributes every feature in the description of ϕ. Finally, as to the interpretation of sequents:

M | = ϕ ψ iff for all a ∈ A, if M, a ϕ, then M, a ψ.
Adding 'common knowledge'. In [START_REF] Conradie | Categories: How I Learned to Stop Worrying and Love Two Sorts[END_REF], we observed that the environment described above is naturally suited to capture not only the factual information and the epistemic attitudes of individual agents, but also the outcome of social interaction. To this effect, we introduce an expansion L C of L with a common knowledge-type operator C. Given Prop and Ag as above, the language L C of the epistemic logic of categories with 'common knowledge' is:

ϕ := ⊥ | | p | ϕ ∧ ϕ | ϕ ∨ ϕ | i ϕ | C (ϕ) .
C-formulas are interpreted in models as follows:

M, a C (ϕ) iff for all x ∈ X, if M, x ϕ, then aR C x M, x C (ϕ) iff for all a ∈ A, if M, a C (ϕ), then aI x, where R C ⊆ A × X is defined as R C = s∈S R s , and R s ⊆ A × X is the relation associated with the modal operator s := i 1 • • • in for any element s = i 1 • • • i n in the set S of finite sequences of elements of Ag (cf. Section A.2).
The basic logic of categories with 'common knowledge' is a set L C of sequents ϕ ψ, with ϕ, ψ ∈ L C , which contains the axioms and is closed under the rules of L, and in addition contains the following axioms:

C (p) ∧ C (q) C (p ∧ q) C (p) { i p ∧ i C (p) | i ∈ Ag}
and is closed under the following inference rules:

ϕ ψ C (ϕ) C (ψ) χ i∈Ag i ϕ {χ i χ | i ∈ Ag} χ C (ϕ)
Hybrid expansions of the basic language. In several settings, it is useful to be able to talk about given objects (market-products) or given features. To this purpose, the languages L or L C can be further enriched with dedicated sets of variables in the style of hybrid logic. Let Prop be a (countable or finite) set of atomic propositions and Ag be a finite set (of agents). Given Prop and Ag as above, and (countable or finite) sets Nom and Cnom (of nominals and conominals respectively), the language L H of the hybrid logic of categories is:

ϕ := ⊥ | | p | a | x | ϕ ∧ ϕ | ϕ ∨ ϕ | i ,
where i ∈ Ag, p ∈ Prop, a ∈ Nom and x ∈ Cnom. A hybrid valuation on an enriched formal concept F maps atomic propositions to formal concepts, nominal variables to the formal concepts generated by single elements of the object domain A, and conominal variables to formal concepts generated by single elements of the feature domain X. If V (a) is the semantic category generated by a ∈ A, and V (x) is the semantic category generated by x ∈ X, then nominal and co-nominal variables are interpreted as follows:

M, y a iff aIy, M, b a iff for all y ∈ X, if aIy then bIy M, b x iff bI x M, y x iff for all b ∈ A, if bI x then bIy.

CORE CONCEPTS AND PROPOSED FORMALIZATIONS

In the present section, we use the languages L, L H and L C discussed in the previous section to capture some core notions and properties about categories, appearing and used in the literature in management science, which we discuss in the next subsection.

Core concepts

A core issue in management science is how to predict the success of a new market-product, or of a given firm over its competitors. Success clearly depends on whether the agents in the relevant audiences decide to buy the product or become clients of the firm, and a key factor in this decision is how each agent resolves a categorization problem. The ease with which products or firms are categorized affects in itself the decision-making, because the more difficult it is to categorize a product or a firm, the higher the cognitive burden and the perceived risk of the decision. This is why research has focused on the performances of category-spanning products or firms (i.e. products or firms which are members of more than one category). While being a member of more than one category can increase visibility and awareness, because audiences interested in any of these categories may pay attention to something which is also in that category, it usually lowers the success. However, the actual effects of spanning categories will depend on the properties of the categories that are spanned. The core concepts of categorization theory denote characteristics of categories or of the relation between categories that can be understood to decrease or increase the effects of spanning categories with these particular characteristics.

Typicality. The issue of whether an object a is a typical member of a given category ϕ, or to which extent a is typical of ϕ, is core to the similarity-based views of category-formation [START_REF] Lakoff | Cognitive models and prototype theory[END_REF][START_REF] Rosch | Principles of categorization[END_REF][START_REF] Edward | The exemplar view[END_REF]. As mentioned in Section 2.2, in conceptual spaces, the prototype of a formal concept is defined as the geometric center of that concept, so that the closer (i.e. more similar) any other object is to the prototype, the stronger its typicality. While this formalization is visually very appealing, it does not shed much light on the role of the agents in establishing the typicality of an object relative to a category.

Distance. The distance between two categories can be defined in different ways. One approach [START_REF] Kovács | Conceptual spaces and the consequences of category spanning[END_REF] is to express it as a negative exponential function of the categories' similarity, where the categories' similarity is calculated using a Jaccard index, i.e., cardinality of the intersection over cardinality of the union. Another approach [START_REF] Elizabeth | An ecology of social categories[END_REF] is to take the Hausdorff distance between the sets in feature space that correspond to the categories. The Hausdorff distance is the maximum of the two minimal point-to-set distances.

Contrast. Contrast is defined as the extent to which a category stands out from other categories in the same domain. It is a function of the mean typicality of objects in the category. In a high-contrast category, objects tend to be either very typical members of the category or not members at all [START_REF] Michael T Hannan | Partiality of memberships in categories and audiences[END_REF]. Objects in a high-contrast category tend to be more recognizable to agents and more positively valued [START_REF] Negro | Categorical contrast and audience appeal: Niche width and critical success in winemaking[END_REF]. Category spanning leads to greater penalties if the spanned categories have higher contrast [START_REF] Kovács | The consequences of category spanning depend on contrast[END_REF].

Leniency. By definition of contrast, members of a low-contrast category ϕ have on average low typicality in that category. This situation is compatible with each of the following alternatives: (a) there are many categories which (according to agents) have members in common with ϕ, (b) there are not many categories which (according to agents) have members in common with ϕ. The notion of leniency clarifies this issue. The leniency of ϕ is defined as the extent to which the members of ϕ are (recognized as) only members of ϕ (and of the other logically unavoidable categories), and not of other categories [START_REF] Pontikes | Two sides of the same coin: How ambiguous classification affects multiple audiences evaluations[END_REF].

Formalizations

The following proposals are not equivalent to the definitions discussed in the previous subsection, but try to capture their purely qualitative content.

Typicality. The interpretation of C-formulas on models indicates that, for every category ϕ, the members of C(ϕ) are those objects which are members of ϕ according to every agent, and moreover, according to every agent, are attributed membership in ϕ by every (other) agent, and so on. This provides justification for our proposal to regard the members of C(ϕ) as the (proto)typical members of ϕ. The main feature of this proposal is that it is explicitly based on the agents' viewpoints. This feature is compatible with empirical methodologies adopted to establish graded membership (cf. [START_REF] Hsu | Jacks of all trades and masters of none: Audiences' reactions to spanning genres in feature film production[END_REF]). Notice that there is a hierarchy of reasons why a given object fails to be a typical member of ϕ, the most severe being that some agents do not recognize its membership in ϕ, followed by some agents not recognizing that any other agent would recognize it as a member of ϕ, and so on. This observation provides a purely qualitative route to encode the gradedness of (the recognition of) category-membership. That is, two non-typical objects3 a and b can be compared in terms of the minimum number of 'epistemic iterations' needed for their typicality test to fail, so that b is more atypical than a if fewer rounds are needed for b than for a. This definition can be readily adapted so as to say that b is a more atypical member of ψ than a is of ϕ.

Distance. For four categories ϕ, ψ, χ, ξ, we can say that ϕ is closer to ψ than χ is to ξ by means of the sequent ϕ ∨ ψ ξ ∨ χ, the sequent ξ∧χ ϕ∧ψ, or by requiring the two sequents to hold simultaneously. The first sequent says that ϕ and ψ have more features in common than ξ and χ have; the second sequent says that ϕ and ψ have more common members than ξ and χ have. Notice that neither the first sequent implies or is implied by the second. This is why it might be useful to consider the information encoded in both sequents. When instantiated to ϕ = ξ, these conditions can be used to express that ϕ is closer to ψ than to χ.

Contrast. If ϕ C(ϕ) holds for a category ϕ, every member of ϕ is a typical member of ϕ, in the sense discussed above, and hence ϕ has maximal contrast. Using the formalizations of typicality and distance discussed above, we say that ϕ has equal or higher contrast than ψ if ϕ is closer to C(ϕ) than ψ is to C(ψ). 4Leniency. A category ϕ has no leniency if its members do not simultaneously belong to other categories. This property can be captured by the following condition: for any ψ and χ, if ψ ϕ and ψ χ, then either ϕ χ or χ ϕ. To understand this condition, let us instantiate ψ as the nominal category a (the category generated by one object). Then a ϕ expresses that the generator of a is a member of ϕ. The no-leniency of ϕ would require the generator a of a to not belong to other categories. However, the nature of the present formalization constrains a to be a member of every χ such that ϕ χ, so a must belong to these categories at least. Also, all the categories χ such that a χ ϕ cannot be excluded either, since the possibility that 'in-between' categories exist does not depend purely on a and ϕ alone, but depends on the context of other objects and features. Hence, we can understand no-leniency as the requirement that no other categories have a as a member than those of this minimal set of categories which cannot be excluded.

For two categories ϕ and ψ, we say that ϕ has greater or equal leniency than ψ if, for every nominal a, if a ψ and a χ for some χ such that χ ψ and ψ χ, then a ϕ and moreover, a ξ for some category ξ such that ξ ϕ and ϕ ξ. Variants of these conditions can be given also in terms of the features (using conominal variables), and also in terms of the modal operators.

CONCLUSIONS AND FURTHER DIRECTIONS

In this paper, we have introduced a basic epistemic logic of categories, expanded it with 'common knowledge'-type and 'hybrid logic'-type constructs, and used the resulting framework to capture core notions in categorization theory, as developed in management science. The logical formalizations proposed in Section 4.2 try to capture the purely qualitative content of the original definitions. The essential features of this logical framework make it particularly suitable to emphasize the different perspectives of individual agents, and how these perspectives interact. The propositional base of these logics is the positive (i.e. negation-free and implication-free) fragment of classical propositional logic (without distributivity laws). The Kripke-style semantics of this logic is given by structures known as formal contexts in Formal Concept Analysis [START_REF] Ganter | Formal concept analysis: mathematical foundations[END_REF], which we have enriched with binary relations to account for the (epistemic) interpretation of the modal operators. One fundamental difference between this semantics and the classical Kripke semantics for epistemic logics is that the relations directly encode the actual viewpoint of the individual agents, and not their uncertainty or ignorance (aR i x reads 'object a has feature x according to agent i). This paper is still very much a first step, but it already shows how logic can contribute to the vast interdisciplinary area of categorization theory, especially with regard to the analysis of various types of social interaction (e.g. epistemic, dynamic, strategic). Interestingly, the prospective contributions involve both technical aspects (some of which we discuss below) and conceptual aspects (since, as discussed in Section 2, there is no single foundational theory or view which exhaustively accounts for all the relevant aspects of categorization).

From normal to regular modal logic. The present paper refines previous work [START_REF] Conradie | Categories: How I Learned to Stop Worrying and Love Two Sorts[END_REF], which provides a conceptually independent explanation of the (rather technical) definition of the interpretation clauses of L-formulas on certain enriched formal contexts. These clauses were obtainable as the outcome of mechanical computations (cf. [6, Section 2.1.1], [4, Section A]) the soundness of which was guaranteed by certain duality-theoretic facts (cf. [START_REF] Dunn | Canonical extensions and relational completeness of some substructural logics[END_REF][START_REF] Gehrke | Generalized kripke frames[END_REF]). The treatment in Section 3 adapts these interpretation clauses to a more general and intuitively more natural environment, which drops the restrictions on formal contexts on which the underlying duality hinged. As a consequence, the duality is replaced by the weaker notion of adjunction. One consequence of this generalization is that the normality restriction on epistemic operators does not hold in general (i.e. the sequent i does not hold in every model). One way of obtaining normality is to restrict the class of (enriched) formal contexts to those such that for any feature x there exists an object a such that (a, x) I. This requirement on models is intuitively plausible if the features are to be relevant for the agents' decision-making. However, literature in philosophical logic has pointed out in other settings (cf. [START_REF] John | New Foundations for Lewis Modal Systems[END_REF]) that regular modal operators capture the epistemic reading more realistically than normal modal operators.

Fixed points. One of the most interesting aspects of the present proposal is that typicality has been captured with a 'common knowledge' operator. This operator is semantically equivalent to the usual greatest fixed point construction (cf. Section A). This paves the way to the use of languages expanded with fixed point operators to capture: for instance, as discussed in [START_REF] Conradie | Constructive canonicity for lattice-based fixed point logics[END_REF]Example 4], the formula νX. i (X ∧ p) denotes the category obtained as the limit of a process of "introspection" (in which the agent reflects on her perception of a given category p, and on her perception of her perception, and so on). A systematic exploration of this direction is work in progress.

Proof calculi. The present framework makes it possible to blend together syllogistic and epistemic reasoning. To further explore those aspects connected with reasoning and deduction in L and L C , specifically designed proof calculi will be needed. These calculi will be useful tools to explore the computational properties of these logics; moreover, the conclusions of formal inferences can provide the basis for the development of testable hypotheses. A proof-theoretic account of the basic logic L can be readily achieved by augmenting the calculus developed in [START_REF] Greco | Lattice Logic Properly Displayed[END_REF] for the propositional base with suitable rules for the modal operators, so as to fall into the general theory of [START_REF] Frittella | Multi-type Sequent Calculi[END_REF]. However, the proof theory of L C needs to be investigated. The omega rules introduced in [START_REF] Frittella | Multi-type display calculus for propositional dynamic logic[END_REF] might provide a template.

Dynamic epistemic logic of categories. An adequate formal account of the dynamic nature of categories is a core challenge facing modern categorization theory. Categories are cognitive tools that agents use as long as they are useful, which is why some categories have existed for millennia and others quickly fade away. Categories shape and are shaped by social interaction. This bidirectional causality is essential to what categories are and do, and this is why the most important and challenging further direction concerns how categories impact on social interaction and how social interaction changes agents' categorizations. One natural step in this direction is to expand the present framework with dynamic modalities, and extend the construction of dynamic updates to models based on enriched formal contexts, as done e.g. in [START_REF] Kurz | Epistemic Updates on Algebras[END_REF][START_REF] Ma | Algebraic Semantics and Model Completeness for Intuitionistic Public Announcement Logic[END_REF].

Proof. Since Y = y∈Y y ↓↑ (cf. Lemma A.2 (2)), by Lemma A.1 (4) and Lemma A.4,

R ↓ [Y] = R ↓ [ y∈Y y ↓↑ ] = y∈Y R ↓ [y ↓↑ ] = y∈Y R ↓ [y]. (1)
By the I-compatibility of R, the last term is an intersection of Galoisstable sets, which is Galois-stable (cf. Lemma A.2 (3)).

The lemma above ensures that the interpretation of L-formulas on enriched formal contexts defines a compositional semantics on formal concepts if the relations R i are I-compatible. Indeed, for every enriched formal context F = (P, {R i | i ∈ Ag}), every valuation V on F extends to an interpretation map of L-formulas defined as follows:

V(p) = ([[p]], ([p])) V( ) = (A, A ↑ ) V(⊥) = (X ↓ , X) V(ϕ ∧ ψ) = ([[ϕ]] ∩ [[ψ]], ([[ϕ]] ∩ [[ψ]]) ↑ ) V(ϕ ∨ ψ) = ((([ϕ]) ∩ ([ψ])) ↓ , ([ϕ]) ∩ ([ψ])) V( i ϕ) = (R ↓ i [([ϕ])], (R ↓ i [([ϕ])]) ↑ ) By Lemma A.5, if V(ϕ) is a formal concept, then so is V( i ϕ). Definition A.6. An enriched formal context F = (P, {R i | i ∈ Ag}) is compositional if R i is I-compatible (cf. Definition A.3) for every i ∈ Ag. A model M = (F, V) is compositional if so is F.

A.2 The interpretation of C is well defined

For any formal context P = (A, X, I) the I-product of the relations R s , R t ⊆ A × X is the relation R st ⊆ A × X defined as follows: The definition of I-product serves to characterize semantically the relation associated with the modal operators s := i 1 • • • in for any finite nonempty sequence s := i 1 • • • i n ∈ S of elements of Ag, in terms of the relations associated with each primitive modal operator. For any such s, let R s be defined recursively as follows: Proof.

a ∈ R ↓ st [x] iff a ∈ R ↓ s I ↑ R ↓ t [x ↓↑ ] . Lemma
• If s = i, then R s = R i ; • If s = it, then a ∈ R ↓ s [x] ⇐⇒ a ∈ R ↓ i I ↑ R ↓ t x ↓↑ . Lemma A.
R ↓ s [I ↑ [R ↓ t [Y]]] = R ↓ s [I ↑ [R ↓ t [ x∈Y x ↓↑ ]]] Lemma A.2 (2) = R ↓ s [I ↑ [ x∈Y R ↓ t [x ↓↑ ]]] Lemma A.1 (4) = R ↓ s [I ↑ [ x∈Y I ↓ [I ↑ [R ↓ t [x ↓↑ ]]]]] R ↓ t [x ↓↑ ] Galois-stable = R ↓ s [I ↑ [I ↓ [ x∈Y I ↑ [R ↓ t [x ↓↑ ]]]]] Lemma A.1 (4) = R ↓ s [ x∈Y I ↑ [R ↓ t [x ↓↑ ]]] Lemma A.4 = x∈Y R ↓ s [I ↑ [R ↓ t [x ↓↑ ]]] Lemma A.1 (4) = x∈Y R ↓ st [x] Definition of R st = R ↓ st [ x∈Y x] Lemma A.1 (4) = R ↓ st [Y] Y = x∈Y x
Lemma A.10. If R s , R t , R w are I-compatible, R s(tw) = R (st)w .

Proof. for any x, As for item (2), we have that

R ↓ s(tw) [x] = R ↓ s [I ↑ [R ↓ tw [x ↓↑ ]]] definition of I-product = R ↓ s [I ↑ [R ↓ t [I ↑ [R ↓ w [x ↓↑ ]]]]] Lemma A.9 = R ↓ st [I ↑ [R ↓ w [x ↓↑ ]]].
T ∈R T ↓ [Y] = T ∈R T ↓ [ y∈Y y] = T ∈R y∈Y T ↓ [y] = y∈Y T ∈R T ↓ [y] = y∈Y R ↓ [y] = ( R) ↓ [ y∈Y y] = ( R) ↓ [Y]
The lemmas above ensure that, in enriched formal contexts in which the relations R i are I-compatible, the relation R C := s∈S R s is I-compatible, and hence the interpretation of L C -formulas on the model based on these enriched formal contexts defines a compositional semantics on formal concepts. Indeed, for every such enriched formal context F = (P, {R i | i ∈ Ag}), every valuation V on F extends to an interpretation map of C-formulas as follows:

V(C(ϕ)) = (R ↓ C [([ϕ])], (R ↓ C [([ϕ])]) ↑ )

  TARK2017, July 2017, Liverpool, UK W. Conradie, S. Frittella, A. Palmigiano, M. Piazzai, A. Tzimoulis, N. Wijnberg

  and B × Y is maximal rectangle contained in I. For example, if p is the categorylabel denoting western movies, and P is a given database of movies (stored in A) and movie-features (stored in X), then V interprets the category-label p in the model M = (F, V) as the formal concept (i.e. semantic category)V (p) = (B,Y), specified by the set of movies B (i.e. the set of western movies of the database) and by the set of movie-features Y (i.e. the set of features which all western movies have). The elements of B are the members of category p in M; the elements of Y describe category p in M. The set B (resp. Y) is the extension (resp. the description) of p in M, and sometimes we will denote it [[p]] M (resp. ([p]) M ) or [[p]] (resp. ([p])) when it does not cause confusion. Alternatively, we write:

  7 immediately implies thatCorollary A.8. For every s ∈ S , the relation R s is I-compatible.Lemma A.9. If Y is Galois-stable and R s , R t are I-compatible, then R ↓ st [Y] = R ↓ s [I ↑ [R ↓ t [Y]]].

  Lemma A.9 = R ↓ (st)w [x] definition of I-product Let s = i 1 • • • i n ∈ S ,and lets := i 1 • • • in . Lemma A.11. For any model M = (F, V), M, a s ϕ iff for all x ∈ X, if M, x ϕ, then aR s x M, x s ϕ iff for all a ∈ A, if M, a ϕ, then aI x.Proof. By induction on the length of s.The base case is immediate. Let s = it. Then [[ i t ϕ]] = R ↓ i [([ t ϕ])] = R ↓ i [I ↑ [[[ t ϕ]]]] = R ↓ i [I ↑ [R ↓ t [([ϕ])]]] = R ↓ s [([ϕ])].The last equality holds by Lemma A.9. The second equivalence is trivially true. Lemma A.12. For any family R of I-compatible relations,(1)R is an I-compatible relation.(2) ( R) ↓ [Y] = T ∈R T ↓ [Y] for any Galois-stable Y. Proof. Let R = R. Then R ↓ [x] = T ∈R T ↓ [x] and R ↑ [a] = T ∈R T ↑ [a]. Then the statement follows from Lemma A.2 (3).

  A.7. If R s and R t are I-compatible, then R st is I-compatible.Proof. R ↓ st [x] being Galois-stable follows from the definition of R st , Lemma A.5, and the I-compatibility of R s and R t . To show thatR ↑ st [a] is Galois-stable, i.e. (R ↑ st [a]) ↓↑ ⊆ R ↑ st [a], by Lemma A.2 (2), it is enough to show that if y ∈ R ↑ st [a] then y ↓↑ ⊆ R ↑ st [a]. Let y ∈ R ↑ st [a], i.e. a ∈ R ↓ st [y] = R ↓ s I ↑ R ↓ t [y ↓↑ ] . If x ∈ y ↓↑ , then x ↓↑ ⊆ y ↓↑ , which implies, by the antitonicity of R ↓ s , I ↑ and R ↓ t (cf. Lemma A.1 (1)), that R ↓ s I ↑ R ↓ t [y ↓↑ ] ⊆ R ↓ s I ↑ R ↓ t [x ↓↑ ] . Hence, a ∈ R ↓ st [x], i.e. x ∈ R ↑st [a], as required.

A subset is convex if it includes the segments between any two of its points. In the Euclidian plane, squares are convex while stars are not.

represented in the language L H as nominal variables.

That is, by either requiring that ϕ ∨ C(ϕ) ψ ∨ C(ψ), or by requiring that ψ ∨ C(ψ) ϕ ∨ C(ϕ), or by requiring both sequents to hold.

so that if V(ϕ) is a formal concept, then so is V( i ϕ). Moreover, the following identity is semantically supported:

where s := i 1 • • • i n is any finite nonempty string of elements of Ag, and s := i 1 • • • in .

A.3 Soundness

Proposition A. [START_REF] Frittella | Multi-type Sequent Calculi[END_REF]. For any compositional model M and any i ∈ Ag,

Proof. By Lemma A.1 [START_REF] Cattani | Categories and competition[END_REF], if

, which proves item [START_REF] Cattani | Categories and competition[END_REF]. As to item (2),

Proof. By definition and Lemma

The following chain of (in)equalities completes the proof of item (1):

Lemma A.12 (2)

As to item (2), using Proposition A.13 [START_REF] Cattani | Categories and competition[END_REF] and the assumptions, one can show that M | = χ s ϕ for every s ∈ S . Hence,

], as required.

A.4 Completeness

The completeness of L can be proven via a standard canonical model construction. For any lattice L with regular operators i , let F L = (P L , {R i | i ∈ Ag}) be defined as follows: P L = (A, X, I) where A (resp. X) is the set of lattice filters (resp. ideals) of L, and aI x iff a ∩ x ∅. For every i ∈ Ag, let R i ⊆ A × X be defined by

In what follows, for any a ∈ A and x ∈ X, we let i

From this, it immediately follows that: Lemma A.15. For F L as above, and any a ∈ A, x ∈ X and i ∈ Ag, 3) and ( 4) of the lemma above immediately imply that:

Recall that S is the set of nonempty finite sequences of elements of Ag.

Lemma A.17. If x is the ideal generated by some u ∈ L, then, for every s

Proof. By induction on the length of s ∈ S . If

Since x is the ideal generated by u, we have that u is the greatest element of x; hence, the monotonicity of i implies that i u is the greatest element of i x. Since a is a filter, and hence is upward-closed, a ∩ i x ∅ is equivalent to i u ∈ a, which completes the proof of the base case. Let us assume that [START_REF] Gärdenfors | Conceptual spaces: The geometry of thought[END_REF] (3) and (4), and Lemma A.7, R s is I-compatible for every s ∈ S . Let z be the ideal generated by s u. Hence:

The identity marked with ( * ) follows from the fact that the filter generated by s u is the smallest element of

The canonical enriched formal context is defined by instantiating this construction to the Lindembaum-Tarski algebra of L. In this case, let V be the valuation such that [[p]] (resp. ([p])) is the set of the filters (resp. ideals) to which p belongs, and let M = (F L , V) be the canonical model. Then the following holds for M: Lemma A.18 (Truth lemma). For every ϕ ∈ L,

Using the lemma above, the following can be proven in the standard way. The proof of the weak completeness for L C proceeds along the lines of [START_REF] Fagin | Reasoning about knowledge[END_REF]Theorem 3.3.1]. For any L C -sequent ϕ ψ that is not derivable in L C , we will construct a model M ϕ,ψ based on the canonical enriched formal context such that M ϕ,ψ | = ϕ ψ, i.e. M ϕ,ψ , a ϕ but M ϕ,ψ , a ψ for some a ∈ A. Let

Let us define equivalence relations on A and on X as follows: a ≡ Φ b if and only if a ∩ Φ = b ∩ Φ (and likewise x ≡ Φ y if and only if x ∩ Φ = y ∩ Φ). By construction, there are only finitely many equivalence classes. We pick a canonical representative of each equivalence class in the following way: If a is a filter, we let [a] be the filter generated by (a ∩ Φ). Likewise if x is an ideal, we let [x] be the ideal generated by (x ∩ Φ). We let . Since this is true for any [a] ∈ G, we obtain that Using the above lemma, we can prove that:

Lemma A.21 (Truth lemma). For every σ ∈ Φ, subformula of ϕ ψ,

(1) M ϕ,ψ , a σ iff σ ∈ a;

(2) M ϕ,ψ , x σ iff σ ∈ x.

Weak completeness of L C follows from the lemma above in the standard way.


Sabine Frittella

Giuseppe Greco

Michele Piazzai

Fan Yang

Ma hew E ects via Team Semantics

Keywords: CCS Concepts:, General and reference → General conference proceedings, • Information systems → Relational database model, • Theory of computation → Logic and databases, Data modeling, • Mathematics of computing → Regression analysis, • Applied computing → Decision analysis, Matthew e ect, Team Semantics, Dependence Logic

Social scientists call Matthew e ect the self-reinforcing mechanisms whereby initially small advantages accrued by individuals, e.g. in reputation, capital, or access to opportunities, beget further advantage and result in growing inequality. While there is extensive literature on the Matthew e ect, the notion has not been explicitly de ned. In this paper, we take a rst step in this direction by providing a formalisation of Matthew e ects within the framework of team semantics, and by introducing a logic for analysing the properties of the dependence relations involved in Matthew e ects. We also show via an example how to use this logic to formalise a statistical analysis of a Matthew e ect.

of assignments (called teams), instead of single assignments as in the standard Tarskian semantics. Teams can be easily conceived as sets of rows in tables or data sets. The exible and multidisciplinary interpretations of teams engendered a rapid development of logics based on team semantics in recent years. Notable such logics are dependence logic [START_REF] Väänänen | Dependence Logic: A New Approach to Independence Friendly Logic[END_REF], independence logic [START_REF] Grädel | Dependence and Independence[END_REF] and inclusion logic [START_REF] Galliani | Inclusion and Exclusion in Team Semantics: On some logics of imperfect information[END_REF][START_REF] Galliani | Inclusion Logic and Fixed Point Logic[END_REF], which focus on characterising functional dependence, independence and inclusion relations among variables, respectively. In this paper we focus on the dependence relations relevant to Matthew e ects. To the best of our knowledge, these have not been formalised in the literature. We take a rst step in this direction by introducing a new logic based on team semantics, called logic of Matthew e ects (ML), and analysing basic properties of these dependence relations within the framework of ML.

Structure of the paper. In Section 2, we de ne the syntax and semantics of our logic ML. In Section 3, we formalise four distinct types of Matthew e ects and study their properties. In Section 4, we use ML to formalise an existing study on the Matthew e ect in science [START_REF] Azoulay | Matthew: E ect or fable?[END_REF].

LOGIC OF MATTHEW EFFECTS

We rst present the general framework of the logic ML, including team semantics, mathematical de nitions, and notation. We provide a brief presentation of time series regression, a common tool to analyse trends in dynamic data. For more details on this topic, the reader can refer to [START_REF] Hamilton | Time Series Analysis[END_REF]. In Section 2.2, we introduce the basic dependence relations we use as building blocks to de ne Matthew e ects. In Section 2.3, we present the syntax and semantics of ML.

General framework

We introduce the general framework to talk about data sets and regressions over data sets. In Appendix A.1, we present a toy example of a data set (Table 1) and a Matthew e ect to exemplify the team semantics. We refer to this example throughout this subsection. In this paper, we assume that the empirical context where the Matthew e ect occurs is described by a rst-order model M of some signature L that will be speci ed in the sequel. The model M is assumed to have a three-sorted domain: data sort, regression sort, and duration sort.

Three-sorted domain. Elements of the data sort are all possible values in the data sets, e.g. real numbers or names. We include the value undefined to the possible values for variables of type data sort. We use w, x, , z, . . . (possibly with subscripts) to stand for variables of this sort. Data sets can be presented as tables (e.g. Table 1), and the variables of data sort can be understood as attributes in such tables. We assume that the time attribute is present in every data set, and we reserve the letter t for the time variable. In this setting, an observation in a data set or a row of a table corresponds to an assignment s that assigns to each variable x of data sort a value a of data sort in the domain of the underlying model M. Formally, a data set D consisting of several observations is a set of assignments, which we also call team. In this paper, the terms data set and team are used interchangeably. As we explained in the introduction, the logic we introduce for Matthew e ects adopts team semantics, where formulas are evaluated on teams.

Elements of the regression sort are names of regressions, and we use r, r ′ , . . . to stand for variables of this sort. For a given model M, we call granularity the minimal time interval δ , between two observations. For instance, in Table 1 the granularity is δ = 1 ear . Di erent regressions can be performed on any data set using di erent variables. Consecutive observations for a single variable can be aggregated over time intervals that correspond to particular multiples of the granularity δ . The intervals of time (i.e., the natural numbers that correspond to the multiples of the minimal time interval) form the domain of the duration sort. We use , ′ , . . . to stand for variables of this sort.

Regression functions. Each regression generates one regression function for each dependent variable under consideration. A regression function can be represented informally as:

(t) = {i 1 , ...,i k }∈X β i 1 ...i k (x i 1 ) (t-lδ ) ⋯(x i k ) (t-lδ ) + ϵ, (1)
where {x 1 , . . . , x n } is a set of independent variables, z (t) is the value of variable z at time t, X is a non-empty downward closed subset of P({1, . . . , n}) (i.e., B ⊆ A ∈ X implies B ∈ X), the positive real number l ⋅ δ is the time interval of the regression, l is a natural number, ϵ is an error term, and each coe cient β i 1 ...i k is a real number. Most studies in the social sciences, including those on Matthew e ects, use regressions of degree two or less, for instance:

(t) = α + β 1 (x 1 ) t-lδ + β 2 (x 2 ) t-lδ + β 3 (x 3 ) t-lδ + β 4 (x 1 ) t-lδ (x 2 ) t-lδ + β 5 (x 2 ) t-lδ (x 3 ) t-lδ + ϵ. (2)
In a data set D, if s and s ′ are two observations that respectively contain the data for time t and t-lδ (i.e., s(t) = s ′ (t)+l⋅δ ), then the above regression equation ( 2) can be formally represented as:

s( ) = α + β 1 s ′ (x 1 ) + β 2 s ′ (x 2 ) + β 3 s ′ (x 3 ) + β 4 s ′ (x 1 )s ′ (x 2 ) + β 5 s ′ (x 2 )s ′ (x 3 ) + ϵ. If x 1
and x 2 are the focal variables, we will sometimes abbreviate the expression above as:

s( ) = β 1 s ′ (x 1 ) + β 2 s ′ (x 2 ) + β 4 s ′ (x 1 )s ′ (x 2 ) + q(s ′ (x 1 ), s ′ (x 2 ), s ′ (x 3 )).
Analysed data set. We call a data set D associated with regression R 1 , . . . , R k with durations of time l 1 , . . . , l k , respectively, an analysed data set, denoted (D, R 1 , l 1 , . . . , R k , l k ). Formally, we view this analysed data set as a data set extended from D by adding 2k columns with attributes r 1 , . . . , r k , 1 , . . . , k , where each column r i has a constant value R i , and each column i has a constant value l i . For simplicity, we denote this analysed data set also by D, and write D(r i ) for the unique value of the attribute r i in D, and D( i ) for the unique value of the attribute i . Technically, we may view a name R of a regression analysis as a function mapping each dependent variable under consideration to a polynomial R( ) of the form (1).

Basic dependence relations.

We de ne basic dependence relations and atomic formulas that we later use to de ne Matthew e ects.

De nition 2.1 (Basic dependence relations and their team semantics). Let x 1 , . . . , x n , be data sort variables, r be a regression sort variable, and be a duration sort variable.

• The atomic formula x 1 , . . . , x n r characterises the notion of being positively -dependent on x 1 , . . . , x n with respect to the regression r. We say that the de ned relation is true in a data set D with an underlying model M, denoted M ⊧ D x 1 , . . . , x n r , i for all s, s ′ ∈ D,

s(t) = s ′ (t) + D( ) ⋅ δ M ⇒ s( ) = β 1 s ′ (x 1 ) + ⋅ ⋅ ⋅ + β n s ′ (x n ) + q(s ′ (⃗ x), s ′ ( ⃗ w)).
where each β i is signi cantly greater than 0 (see Appendix B.2), and D(r)( ) is the polynomial represented above. • The atomic formula x 1 , . . . , x n r characterises the notion of being negatively -dependent on x 1 , . . . , x n with respect to r, and we de ne M ⊧ D x 1 , . . . , x n r the same way as above except that we now require each β i to be signi cantly smaller than 0.

• The atomic formula x 1 ⊗ . . . ⊗ x n r characterises the notion of being positively moderated -dependent on x 1 , . . . , x n with respect to r, and we de ne

M ⊧ D x 1 ⊗ . . . ⊗ x n r i for all s, s ′ ∈ D, s(t) = s ′ (t) + D( ) ⋅ δ M ⇒ s( ) = n k =1 1≤i 1 <⋅⋅⋅<i k ≤n β i 1 ...i k s ′ (x i 1 )⋯s ′ (x i k ) + q(s ′ (⃗ x), s ′ ( ⃗ w)),
where β i ...n is signi cantly greater than 0, and D(r)( ) is the polynomial represented as above. • The atomic formula x 1 ⊗ . . . ⊗ x n r characterises the notion of being negatively moderated -dependent on x 1 , . . . , x n with respect to r, and we de ne M ⊧ D x 1 ⊗ . . . ⊗ x n r the same way as above except that we now require β i ...n to be signi cantly smaller than 0.

De nition 2.2 (Atomic formulas for positive and negative dependency). Let X denote the set of strings X of dependent variables of the form

x 11 ⊗ ⋯ ⊗ x 1n 1 , . . . , x k 1 ⊗ ⋯ ⊗ x kn k .
For every string X ∈ X such that X = x 11 ⊗ ⋯ ⊗ x 1n 1 , . . . , x k1 ⊗ ⋯ ⊗ x kn k and for every duration of time ⋅ δ , we introduce the atomic formulas X r and X r , de ned as follows:

• M ⊧ D x 11 ⊗ ⋯ ⊗ x 1n 1 , . . . , x k 1 ⊗ ⋯ ⊗ x kn k r i for all s, s ′ ∈ D, s(t) = s ′ (t) + D( ) ⋅ δ M ⇒ s( ) = k i=1 n i m=1 1≤j 1 <⋅⋅⋅<j m ≤n i β i, j 1 ...j m s ′ (x i j 1 )⋯s ′ (x i j m ) + q(s ′ (⃗ x), s ′ ( ⃗ w)), (3)
where each β i,1, ...,n i is signi cantly greater than 0, and D(r)( ) is the polynomial represented represented as above.

• M ⊧ D x 11 ⊗ ⋯ ⊗ x 1n 1 , . . . , x k 1 ⊗ ⋯ ⊗ x kn k r
i the same as the above holds except that each β i,1, ...,n i is now required to be signi cantly smaller than 0.

The logic for Ma hew e ects

Here, we de ne the signature, the syntax, and the semantics of ML.

De nition 2.3 (The signature L). The signature L of ML contains the constant functions 0 and δ , the unary functions {E ect ,X ∈ Var, X ∈ X}, the unary predicate Small, and the binary predicates ≫, ≪, and ≈.

The function symbols E ect ,X are used to refer to certain combinations of the coe cients in the polynomials of the regression functions. For every variables x, , every string X of dependent variables, E ect ,X (r) is a regression sort argument, whose interpretation in the intended models M under a data set D is de ned as:

• i X = x 11 ⊗ ⋯ ⊗ x 1n 1 , . . . , x k 1 ⊗ ⋯ ⊗ x kn k , and β 1 x 11 ⋯x 1n 1 , ..., β k x k 1 ⋯x kn k are terms in the polynomial D(r)( ), then for every s ∈ D, E ect M ,X (s(r)) = β 1 + ⋅ ⋅ ⋅ + β k . Otherwise, set E ect M ,X (s(r)) = undefined.
We equivalently denote E ect ,X (r) by E ect(r, , X ). The use of these terms becomes clearer in Section 4. The constant symbols 0 and δ are to be interpreted as the natural number 0 and the granularity of the data sets δ respectively. The predicate symbols Small, ≪, ≫ and ≈ are to be interpreted as "small, " "signi cantly smaller than, " "signi cantly greater than, " and "equivalent to, " respectively. De nition 2.4 (Syntax). Let Var 0 , Var 1 and Var 2 be respectively countable sets of variables of data sort, regression sort, and duration sort. The syntax of ML is de ned as follows:

Terms of data sort α ∶∶= x 0 δ E ect ,X (r) Terms of duration sort β ∶∶= Terms of regression sort γ ∶∶= r Formulas ϕ ∶∶= X r X r Small(α) α ≪ α α ≫ α α ≈ α ϕ ∧ ϕ ∃ 1 x ϕ ∃ 1 ϕ ∃ 1 r ϕ where x ∈ Var 0 , r ∈ Var 1 and ∈ Var 2 .
De nition 2.5 (Free variables). For compound formulas, the sets of free variables of each sort are de ned as usual. For atomic formulas, the sets of free variables of each sort are de ned as follows:

• The set Fv 0 (ϕ) of free variables of data sort is de ned as

-for X = x 11 ⊗ ⋯ ⊗ x 1n 1 , . . . , x k 1 ⊗ ⋯ ⊗ x kn k , Fv 0 (X r ) = Fv 0 (X r ) = {x i, j 1 ≤ i ≤ k and 1 ≤ j ≤ n i } ∪ { , t} 1 , -Fv 0 (Small(α)) = Fv 0 (α), -Fv 0 (α ≪ β) = Fv 0 (α ≫ β) = Fv 0 (α ≈ β) = Fv 0 (α) ∪ Fv 0 (β),
and the set Fv 0 (α) is de ned inductively as Fv 0 (x) = {x} and Fv 0 (0) = Fv 0 (δ ) = Fv 0 (E ect ,X (r)) = ∅.

• The set Fv 1 (ϕ) of free variables of regression sort is de ned as

-Fv 1 (X r ) = Fv 1 (X r ) = {r}, -Fv 1 (Small(α)) = Fv 1 (α) and Fv 1 (α ≪ β) = Fv 1 (α ≫ β) = Fv 1 (α ≈ β) = Fv 1 (α) ∪ Fv 1 (β), and
-the set Fv 1 (α) is de ned inductively as Fv 1 (x) = Fv 1 (0) = Fv 1 (δ ) = ∅ and Fv 1 (E ect ,X (r)) = {r}. • The set Fv 2 (ϕ) of free variables of duration sort is de ned as -Fv 2 (X r ) = Fv 2 (X r ) = { } and -Fv 2 (ϕ) = ∅ for any other atomic formula.
Formulas with sets Var 0 , Var 1 , Var 2 of free variables of data sort, regression sort, and duration sort are evaluated on a model M with respect to teams over

V 0 ∪ V 1 ∪ V 2 , i.e., sets D of assignments s ∶ V 0 ∪ V 1 ∪ V 2 → M.
De nition 2.6 (Semantics). We de ne inductively the satisfaction relation M ⊧ D ϕ as follows:

• See Theorem 2.2 for the team semantics of the atomic formulas X r k and X r k .

• For the other atomic formula θ , M ⊧ D θ i M ⊧ s θ in the usual sense for all s ∈ D.

• M ⊧ D ϕ ∧ ψ i M ⊧ D ϕ and M ⊧ D ψ . • M ⊧ D ∃ 1 x ϕ i M ⊧ D(a x ) ϕ
for some element a ∈ M of data sort, where D(a x) = {s(a x) s ∈ D}.

• M ⊧ D ∃1 r ϕ and M ⊧ D ∃ 1 ϕ are de ned as above respecting the sorts of the variables.

For any set Γ ∪ {ϕ} of formulas, we write Γ ⊧ ϕ if for all models M and teams D, M ⊧ D γ for all γ ∈ Γ implies M ⊧ D ϕ. We write ϕ ⊧ ψ for {ϕ} ⊧ ψ .

FORMALISING MATTHEW EFFECTS

In this section, we formalise four distinct types of Matthew e ects in ML and we investigate the properties of the dependence relations and Matthew e ects.

De nition 3.1 (Matthew e ects). We de ne the following notions:

• being subject to a positive direct -Ma hew e ect with respect to r (see also Table 2

(b)): DME r ∶∶= r .
• being subject to a positive x-mediated -Ma hew e ect with respect to r (see also Table 2

(c)): MME r (x) ∶∶= x r ∧ r x .
• being subject to a positive x-complete -Ma hew e ect with respect to r (see also Table 2

(d)): CME r (x) ∶∶= MME r (x) ∧ DME r .
• x and being subjects to a positive complete -Ma hew e ect with respect to r (see also Table 2(e)):

CME r (x, ) ∶∶= MME r (x) ∧ DME r x ∧ DME r .
Properties. It is not hard to verify that the dependence relation X r satis es the following properties:

• (Re exivity) ⊧ x r 0 x • (Enhancing) x r x ⊧ ∃ 1 r x r k x • (Commutativity) X 1 , ⋯, X n r ⊧ X i 1 , ⋯, X i n r and W , x 1 ⊗ ⋯ ⊗ x n , Z r ⊧ W , x i 1 ⊗ ⋯ ⊗ x i n , Z r , where i 1 , . . . , i n is any permutation of 1, . . . , n • (Duplication) X 1 , . . . , X n r ⊧ X i , X 1 , . . . , X n r , where X i ∈ {X 1 , . . . , X n } • (Projection) X 1 , . . . , X n r ⊧ X i 1 , . . . , X i k r , where X i 1 , . . . , X i k is any subsequence of X 1 , . . . , X n • (Regrouping) (X r ), (Z r ) ⊧ X , Z r • (Transitivity) (X r ), ( r k z) ⊧ ∃ 1 r X r (
k+1) z As a consequence of the transitivity of the dependence relation, mediated and direct Matthew e ects satisfy the following properties:

• (Transitivity) MME r x( ), MME r (z) ⊧ ∃ 1 r MME r 2 x(z) • (Scaling) MME r (x) ⊧ ∃ 1 r 0 DME r 0 2 x ∧ ∃ 1 r 1 DME r 1 2
Moreover, a direct Matthew e ect of a variable x is clearly a mediated Matthew e ect where x itself is the mediator, namely, DME r ⊧ MME r ( ), and a mediated Matthew e ect is reciprocal for the two variables involved, namely MME r (x) ⊧ MME r x( ).

CASE STUDY

In Section 4.1, we informally present the results of Azoulay, Stuart, and Wang (henceforth: ASW) reported in [START_REF] Azoulay | Matthew: E ect or fable?[END_REF] about the Matthew e ect in science. In Section 4.2, we formalise their analysis.

Informal presentation

ASW propose an empirical test for the following proposition: scientists of higher status will have even higher status in the future. According to the de nitions presented in Section 2, this is equivalent to saying that a direct Matthew e ect exists with regard to a scientist's status. The empirical test revolves around the conferral to medical scientists of the prestigious title of Howard Hughes Medical Investigator (HHMI). ASW examine how the yearly number of citations for an article published by a HHMI-appointed scientist before the appointment changes as a result of the appointment. ASW assume that both the HHMI appointment and an article's yearly number of citations re ect the scientist's status.

The hypothesis that a direct Matthew e ect exists with regard to a scientist's status is accepted if the articles published by HHMI appointees receive more citations after the appointment, compared to articles of similar quality published by non-appointees. The results of the analysis show that:

• The citation boost is small and it a ects only the articles published up to 1 year before the HHMI appointment.

Older articles do not witness a change in citations received as a result of the appointment. • The citation boost is larger for articles published in journals with low impact factor, articles that use more novel keywords, and articles that cite a greater number of studies from other elds (i.e., that are recombinant). ASW argue that this is because the quality of these articles is more di cult to assess; therefore, the HHMI appointment acts as a signal of quality and more strongly a ects the yearly citations these articles receive. • The citation boost is larger for articles published by scientists who have a smaller total number of citations attached to their name or who are younger at the time of the HHMI appointment. On the basis of these results, ASW conclude that there is a Matthew e ect with regard to scientists' status, but the extent to which this is observable depends on the age of the articles published by scientists and on how easily the quality of these articles can be assessed. In addition, they conclude that the Matthew e ect more strongly a ects scientists who have lower status at the time they are appointed.

Formalisation

ASW perform a complex empirical test involving multiple variables and regression models. These are presented in detail in Appendix B. In this subsection, we rst formalise ASW's statements in isolation, then we analyse the reasoning they use to draw their conclusions.

Statements. Each statement is presented in natural language (text in italic) and then formalised using ML. ASW aim at proving that there is a direct Matthew e ect with regard to a scientist's status. This can be formalised by the formula:

∃ 1 ∃ 1 r a DME r a Status. (4)
ASW observe (regression r 1 ) that ACitAF positively depends on HHMI, but the e ect is small [2, page 21], which can be formalised by the formula:

HHMI r 1 ear ACitAF ∧ Small(r 1 , ACitAF, {HHMI}). (5)
In addition, they observe (regressions r 2 , r 3 , and r 4 ) that this positive dependency only a ects the articles published up to 1 year before the HHMI appointment:

HHMI r 2 ear ACitAF (6)
∧ E ect(r 2 , ACitAF, {HHMI}) ≫ E ect(r 1 , ACitAF, {HHMI})

∧ E ect(r 3 , ACitAF, {HHMI}) ≈ 0 ∧ E ect(r 4 , ACitAF, {HHMI}) ≈ 0.

Recall that the regressions r 2 , r 3 and r 4 are respectively based on articles published up to 1 year, 2 years, and 3 to 10 years before the HHMI appointment (see Table 4). ASW also observe (regression r 5 ) that there is a stronger increase in citations after the HHMI appointment if the article is published in a journal with low impact factor: HHMI ⊗ LIF r 5 ear ACitAF

Furthermore, they observe that there is a stronger increase in citations if the article is novel (regression r 6 ):

HHMI ⊗ Novel r 6 ear ACitAF, (10)
or if it is recombinant (regression r 7 ):

HHMI ⊗ Recombinant r 7 ear ACitAF . ( 11
)
ASW assume that the quality of articles is more uncertain if they are published in journals with low impact factor, if they are novel, or if they are recombinant:

∃ 1 ∃ 1 r b {LIF, Novel, Recombinant} r b UArtQ . (12)
The assumption [START_REF] R K Merton | The Matthew e ect in science[END_REF] and the observations ( 9), ( 10) and [START_REF] Kontinen | Axiomatizing rst-order consequences in dependence logic[END_REF] suggest that the positive dependency of ACitAF on HHMI more strongly a ects articles of uncertain quality:

∃ 1 ∃ 1 r c HHMI r c ACitAF ∧ HHMI ⊗ UArtQ r c ACitAF . (13)
Moreover, ASW observe that there is a stronger increase in the number of citations after the HHMI appointment if the article is published by a scientist who is less cited at the time of the appointment (regression r 8 ):

HHMI ⊗ Hnotwellcited r 8 ear ACitAF . ( 14
)
or who is younger at the time of the appointment (regression r 9 ):

HHMI ⊗ Hyoung r 9 ear ACitAF . ( 15
)
The assumption that the variables Hnotwellcited and Hyoung have a negative e ect on Status,

∃ 1 ∃ 1 r d {Hnotwellcited, Hyoung} r d Status , (16)
leads to the conclusion that the positive dependency of ACitAF on HHMI more strongly a ects scientists who have lower status at the time of the HHMI appointment than it a ects scientists who have higher status:

∃ 1 ∃ 1 r e HHMI r e ACitAF ∧ HHMI ⊗ Status r e ACitAF . ( 17
)
Reasoning. The observations ( 5), ( 6), ( 7), (8) ( 9), ( 10), ( 11), ( 14) and ( 15) and the assumptions ( 12) and ( 16) are considered as axioms. Here we list the deduction steps ASW use to deduce (4), ( 13) and (17). To deduce [START_REF] R K Merton | The Matthew e ect in science, II: Cumulative advantage and the symbolism of Intellectual Property[END_REF], ASW use the following reasoning: if ACitAF is subject to a positively moderated ear -dependency on HHMI ⊗ LIF (resp. HHMI ⊗ Novel or HHMI ⊗ Recombinant) witnessed by the regression r, and if UArtQ is subject to a positive LIF (resp. Novel or Recombinant) dependency, then there is an hypothetical regression r ′ that witnesses the fact that ACitAF is subject to a positively moderated ′ -dependency on HHMI ⊗ UArtQ. This reasoning is captured by the following deduction step:

h ⊗ n r c ∃ 1 ∃ 1 r n r u ∃ 1 ′ ∃ 1 r ′ h ⊗ u r ′ ′ c
.

To deduce (17), ASW use the following reasoning: if ACitAF is subject to a positively moderated ear -dependency on HHMI ⊗ Hnotwellcited (resp. HHMI ⊗ Hyoung) witnessed by the regression r 8 (resp. r 9 ), and if Hnotwellcited (resp. Hyoung) has a negative e ect on Status, then there is an hypothetical regression r ′ that witnesses the fact that ACitAF is subject to a positively moderated ′ -dependency on HHMI ⊗ Status. This reasoning is captured by the following deduction step:

h ⊗ n r c ∃ 1 ∃ 1 r n r s ∃ 1 ′ ∃ 1 r ′ h ⊗ s ′ r ′ c .
To deduce (4), one rst need to express the assumption that HHMI is positively dependent on Status:

∃ 1 ∃ 1 r f Status r f HHMI . ( 18
)
Based on this, ASW use the following reasoning: if HHMI is positively dependent on Status, if ACitAF is positively dependent on HHMI as witnessed by the regression r 1 , then Status is positively dependent on Status, which means that Status is subject to a direct Matthew e ect. This reasoning is captured by the following deduction steps:

∃ 1 ∃ 1 r s r h h r 1 ear c ∃ 1 ∃ 1 r s r c and ∃ 1 ∃ 1 r s r c ∃ 1 ∃ 1 r c r s ∃ 1 ∃ 1 r DME r s.

CONCLUSION AND FURTHER RESEARCH

Conclusion. While there is a great deal of literature in the social sciences invoking the Matthew e ect to explain important phenomena, from career dynamics to economic inequality, the concept of Matthew e ect has never been properly formalised. This makes it di cult to compare and synthesise the results of di erent studies. This paper o ers a rst formalisation of the Matthew e ect via a logic based on team semantics.

An original contribution of this paper is that our formalisation allows for a clear distinction between di erent types of Matthew e ects: direct, mediated, and complete. This shows just how complicated self-reinforcing phenomena can be, because an observed Matthew e ect can actually result from the interplay of direct and mediated Matthew e ects. In addition, our formalisation serves a number of interrelated purposes: rst, as shown in our case study, it can be used to better understand the import of empirical research and to make explicit the assumptions needed to support the researchers' conclusions; second, it can be used to compare and relate the results of di erent studies to one another, and thereby develop new theory on a rmer foundation; third, it allows empirical scientists to ask new research questions and formulate more precise hypotheses.

This study is only a rst step in exploring logical formalisms that address the intricate phenomena concerning the Matthew e ect. Further work will expand the logical analysis in the following directions:

Studying properties of ML. The logic ML, we introduced is de ned on the basis of team semantics. A team or a data set is essentially a relation of the model, which is a second-order object. As a consequence, logics based on team semantics are usually second-order in expressive power. Indeed, the two major team-based logics, dependence logic and independence logic, are expressibly equivalent to existential second-order logic [START_REF] Galliani | Inclusion and Exclusion in Team Semantics: On some logics of imperfect information[END_REF][START_REF] Väänänen | Dependence Logic: A New Approach to Independence Friendly Logic[END_REF]. The atomic dependency notions formalised in ML are more involved than the functional dependency, independence, and other dependency notions studied so far in team-based logics. Yet the language of our ML is very simple, as the only complex formulas are the conjunctions and the existentially quanti ed statements with weak existential quanti ers of team semantics. One natural conjecture would be that this logic is strictly weaker in expressive power than existential second-order or even rst-order logic.

A richer language with a good proof calculus. Although we demonstrated in the case study that the simple language ML can already express interesting facts about Matthew e ects, in future work we will introduce stronger logics by expanding the language to include disjunction, negation, implication, and strong quanti ers. We also want to introduce proof calculi for these extensions or their su ciently strong fragments.

Re ecting the complexity of empirical tests. Our formalisation suggests that empirical ndings about Matthew e ects are contingent on particular statistical analyses, performed on particular data, where particular variables are observed over particular time intervals. Choices related to research design can thus a ect the empirical evidence researchers nd about Matthew e ects. For example, choosing to observe the variables yearly rather than monthly, weekly, or daily can determine whether a direct Matthew e ect is found in the place of a mediated one. Moreover, the fact that some variables like quality and uncertainty about quality remain unobserved can conceal important dependencies; as a result, a Matthew e ect may appear to be mediated by a certain (observed) variable whereas in fact it is mediated by another (unobserved) one, which depends or is dependent on the apparent mediator. In formalising Matthew e ects, or indeed any dependency tested via statistical analysis, one must be able to express these details within the syntax of the logic.

A APPENDIX -MATTHEW EFFECTS A.1 Toy example of a Ma hew e ect

Matthew e ects are often detected by researchers while analyzing empirical data. In the statistical literature, these represent a form of autocorrelation [START_REF] Hamilton | Time Series Analysis[END_REF]. Table 1 presents a hypothetical dataset that shows prima facie evidence of a Matthew e ect. This data concerns the careers of visual artists: each row contains information about an artist during a given year. The rst column includes the artist ID; the second column includes the number of artworks sold by the artist during the observation year; the third column includes the number of times the artist or her work were reviewed by the media during the observation year; the fourth column speci es the observation year. Assume that all artists started their career in 2010 and that they were equally productive during the study period. The data suggests that Artist A started accumulating reviews from the very beginning, and sales quickly followed. A similar pattern can be observed for Artist B, though with some delay. In the case of Artist C, however, this trend never began.

We may presume that both sales and reviews are subject to a Matthew e ect because selling more artworks leads to greater odds of selling artwork in the future. Similarly, being reviewed increases the odds of future reviews. However, it is also possible-and indeed highly likely-that being reviewed increases the odds of future sales, and that selling artwork increase the odds of future reviews. The fact that these dependencies occur at the same time makes the individual e ects di cult to isolate.

(t) = α + βx (t-lδ ) + γ 1 w 1 (t-lδ ) + ⋅ ⋅ ⋅ + γ 1 w 1 (t-lδ ) + ϵ,
where β is signi cantly greater than 0. Since we also have

(t-lδ ) = α + βx (t-2lδ ) + γ 1 w 1 (t-2lδ ) + ⋅ ⋅ ⋅ + γ 1 w 1 (t-2lδ ) + ϵ,
when all the independent variables except x are held constant, we obtain

(t) -(t-lδ ) = β(x (t-lδ ) -x (t-2lδ ) ),
meaning that the value of increases as the value of x increases (see also O O : :

x (t-2l) O O (b) Direct Matthew e ect (c) Mediated Matthew e ect (t) (t-l) K S K S (t-2l) O O (t) (t-l) K S x (t-l) d d (t-2l) x (t-2l) O O : : (t-3l) O O x (t-3l) (d)
Complete Matthew e ect for (e) Complete Matthew e ect for (x, )

x • • • K S K S • K S : : • K S • K S K S d d • K S K S • O O K S K S : : • O O x • • • K S K S • K S : : • K S • K S K S d d • K S K S • O O K S K S : : K S K S • O O B APPENDIX -CASE STUDY B.

Variables

The key variables used by ASW to analyze the Matthew e ect among scientists in [START_REF] Azoulay | Matthew: E ect or fable?[END_REF] is reported in Table 3. Some of these variables measure relevant characteristics of a scientist, including Author, HHMI, Hdate, Hage, and Hcit. Other variables measure characteristics related to the article published by the scientist, including Article, ArtY, Journal, NbAut, Apos, ACitAF, ACitBF, IF, Novelty, and Recombination.

In [START_REF] Azoulay | Matthew: E ect or fable?[END_REF], the variables Hage, Hcit, IF, Novelty, and Recombination are split along the median value observed in the data, giving rise to the following boolean variables: Hyoung, which is true if the scientist is younger than the median at the time of the HHMI appointment; Hwellcited, which is true if the scientist has a greater number of total citations than the median at the time of the HHMI appointment; HIF, which is true if the journal where the article is published has higher impact factor than the median; Novel, which is true if the keywords associated with the article are more novel than the median; and Recombinant, which is true if the proportion of out-of-eld literature cited by the article is greater than the median.

In addition to these variables, which are measured and actually used in ASW's statistical analysis, the claims made in [START_REF] Azoulay | Matthew: E ect or fable?[END_REF] about the Matthew e ect involve a number of additional variables, which cannot be measured directly but are assumed to depend on some of the observed variables. These include Status, i.e., the status of the focal scientist, as well as ArtQ and UArtQ, which represent the quality of the scientist's article and the uncertainty about the quality of the scientist's article, respectively.

B.2 Regressions

A regression is performed via the application of an algorithm, i.e., an estimator, to the observed data. The algorithm yields a set of coe cients, which correspond to the β mentioned in Section 2. Each coe cient represents an e ect, i.e., a change in the value of the dependent variable that results from a one-unit increase in the value of the independent variable x. Each coe cient is associated with a level of statistical signi cance, i.e., a value between 0 and 1 that represents the probability of observing the estimated e ect in the data. The lower this value, the greater the probability of observing the e ect. A coe cient with a level of statistical signi cance below some predetermined threshold is said to be signi cantly greater than 0 if the coe cient is positive, and signi cantly smaller than 0 if the coe cient is negative. If the level of statistical signi cance is above the predetermined threshold, the coe cient is said to be non-signi cant or equivalent to zero. In [START_REF] Azoulay | Matthew: E ect or fable?[END_REF], the chosen signi cance threshold is 0.05, as is conventional in the social sciences.

Table 4 reports the full list of the regressions performed by ASW in [START_REF] Azoulay | Matthew: E ect or fable?[END_REF]. These are indexed by numbers 1-9. These regressions concern the observed variables listed in Table 3. In addition, Table 4 reports a list regressions that are not actually performed by ASW in [START_REF] Azoulay | Matthew: E ect or fable?[END_REF], but their results are nonetheless relevant to ASW's analysis. We call these hypothetical regressions. These are not actually performed because they concern the unobserved variables listed in Table 3. However, they could be performed if it were possible to observe these variables. These hypothetical regressions are indexed by letters a-e. In every regression, the dependent variable is ACitAF. Based on the full sample. Independent variables include all the observed variables listed in Table 3 r 2 Like r 1 , but the sample includes only articles published up to 1 year before the HHMI appointment r 3

Like r 1 , but the sample includes only articles published 2 years before the HHMI appointment r 4

Like r 1 , but the sample includes only articles published 3 to 10 years before the HHMI appointment r 5

Like r 2 , but estimating a di erent e ect of HHMI when HIF is true or false r 6

Like r 2 , but estimating a di erent e ect of HHMI when Novel is true or false r 7

Like r 2 , but estimating a di erent e ect of HHMI when Recombinant is true or false r 8

Like r 2 , but estimating a di erent e ect of HHMI when Hwellcited is true or false Like r 2 , but estimating a di erent e ect of HHMI when Hyoung is true or false

  r a Hypothetical regression where the direct Matthew e ect on Status is estimated r b Hypothetical regression where the e ects of HIF, Novel, and Recombinant on UArtQ are estimated r c Hypothetical regression where the e ect of UArtQ on ACitAF is estimated r d Hypothetical regression where the e ects of Hwellcited and Hyoung on Status are estimated r e Hypothetical regression where the e ect of Status on ACitAF is estimated r f Hypothetical regression where the e ect of Status on HHMI is estimated

Table 1 .

 1 A data set (or a team)

	Artist Sales Reviews Time . . .	Artist Sales Reviews Time . . .	Artist Sales Reviews Time . . .
	A	0	1	2010 . . .	B	0	0	2010 . . .	C	0	0	2010 . . .
	A	1	2	2011 . . .	B	0	0	2011 . . .	C	1	1	2011 . . .
	A	1	1	2012 . . .	B	0	1	2012 . . .	C	0	0	2012 . . .
	A	2	4	2013 . . .	B	0	2	2013 . . .	C	0	1	2013 . . .
	A	4	7	2014 . . .	B	2	5	2014 . . .	C	1	2	2014 . . .
	A	7	9	2015 . . .	B	4	8	2015 . . .	C	0	1	2015 . . .
	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮
	A.2 Di erent Ma hew e ects

Table 2

 2 contains a graphic representation of the de nitions of the positive dependency introduced in Theorem 2.1 and of the Matthew e ects de ned in Section 3. To help understand the de nitions of the di erent Matthew e ects and Table2, we present a simple example of positive dependency.Example A.1. Consider a very simple dependence relation x r with a linear regression function

Table 2

 2

(a)).

Table 2 .

 2 Types of Ma hew e ect

	(a) Dependence relation
		(t)
	x (t-l)	(t-l)

Table 3 .

 3 List of variables

		Variable	Meaning	Type	Time
					dependent
		Author	Scientist ID	N
		HHMI	Author is appointed HHMI	0 1
		Hdate	Date of the HHMI appointment	N
		Hage	Age of the author	N
			at HHMI appointment
		Hyoung	Author was young	0 1
	Observed		at HHMI appointment
		Hold	Author was old	0 1
	Author related	Hcit	at HHMI appointment Author's total citations at HHMI appointment	N
		Hwellcited	Author was well cited	0 1
			at HHMI appointment
		Hnotwellcited

Table 4 .

 4 List of regressions

	Regression Description
	r 1

Notice that we assume the variable t to be always present in these atoms, although for simplicity we do not explicitly write the variable in the formulas. Cf. the team semantics given in (3).