This dissertation addresses the question of how the information encoded by category labels is interpreted by agents in a market for the purpose of decision-making. To this end, we first examine the influence of categorization on economic and strategic outcomes with two empirical studies, and then use the insights provided by these studies to develop a formal theory of classification systems. Consistently with Formal Concept Analysis (FCA), this theory builds on the fundamental mathematical notions of lattices and order, and it is thus uniquely suited to yield an ontological perspective on category representations. As a result, we are much better equipped to understand how categories serve as the "cognitive infrastructures'" of markets and affect economic activity. Chapter 1 offers a concise overview of the extant research on categorization in cognitive psychology, economic sociology, and organization theory. We build extensively on this diverse literature during the course of our exposition. The first part of this thesis includes our empirical studies. In Chapter 2, we synthesize insights from industrial economics, strategic management, and organizational ecology to examine the effects of product proliferation strategies. Conceptualizing the market as a multidimensional (Lancastrian) space of product features, we argue that product categories guide firms' strategic decisions by partitioning the space into subsets or regions. Product proliferation occurs when a firm bids to occupy a product category at the expense of competitors by saturating the corresponding region of space. Consistently with game-theoretic models of product competition in differentiated markets, we predict proliferation to have a negative effect on the likelihood of rival product introductions in the targeted category; however, we also predict that this effect is weaker if the region of space to which the category maps is more complex (i.e., heterogeneous in terms of product features). Our analysis of firms' patterns of new product introductions in the US recording industry supports these hypotheses; in addition, it suggests that product proliferation effectively deters competitors who can alter their positioning in feature space, but those who are constrained to particular positions remain virtually unaffected. In Chapter 3, we turn to consumers' perspective and examine how the categorization of products according to different classification systems affects the attribution of value. Focusing on the distinction between categories based on prototypes and categories based on goals, we argue that these category labels of these two kinds map to structurally different regions of the feature space. Valuation requires consumers to infer the location of products from their labels, but because type- and goal-based categories have different internal structures, they enable different sorts of inferences. Building on this argument, we theorize that under particular conditions spanning type-based categories has a U-shaped effect on consumers' evaluations, whereas spanning goal-based categories has a negative effect. At the same time, we predict that spanning goal-based categories can moderate the U-shaped effect of spanning type-based categories by enabling consumers to make more precise inferences from fewer type-based labels. Our analysis of product ratings on a popular music website offers empirical support for these hypotheses. In the second part of this thesis, we develop a formal theory of categorization that accounts for the key aspects highlighted by our empirical studies. In Chapter 4, we introduce an order-theoretic account of classification systems as RS-frames. These are algebraic structures based on RS-polarities, which we enrich with additional relations to interpret modalities. Consistently with FCA, we propose to interpret an RS-polarity as a database consisting of a set of objects (such as products or organizations in a market), a set of features, and an incidence relation linking objects with their features. All the possible categories whereby the objects and the features may be grouped arise as the Galois-stable sets of this polarity, just like formal concepts in FCA. An agent's perception of the objects and their features, which can be unique, incomplete, or even mistaken, is modeled by a relation giving rise to a normal modal operator that expresses an agent's beliefs about a category's intensional and extensional meaning. The fixed points of the iterations of belief modalities are used to model categories whose meaning is shared as they arise from social interaction. In Chapter 5, we clarify how the order-theoretic perspective on concepts enabled by FCA complements the geometric perspective allowed by the theory of conceptual spaces. In addition to introducing a sound and complete epistemic-logical language, we refine the framework presented in the previous chapter both technically and conceptually: Technically, because we free its semantics from the restrictions imposed by the RS-conditions and generalize to more natural Kripke-style frames. This makes our formalism better suited to represent formal contexts (i.e., databases) as they occur in real-world domains. Conceptually, because we enhance our theory of classification systems as concept lattices and propose formalizations for some of the most important theoretical constructs in the categorization literature, including typicality, similarity, contrast, and leniency. In particular, we elaborate our interpretation of the fixed-point construction introduced before by tying it directly to the notion of typicality. Possible extensions are discussed, especially with regard to dynamic updates. Chapter 6 summarizes the main findings of this dissertation, elucidates their implications for organizational research, identifies key areas for improvement, and presents promising directions for future study. Special consideration is given to the possibility of unifying FCA and conceptual spaces using the framework of correspondence theory. We conclude with a general reflection on the role of logic in the social sciences.@en

This dissertation pertains to algebraic proof theory, a research field aimed at solving problems in structural proof theory using results and insights from algebraic logic, universal algebra, duality and representation theory for classes of algebras. The main contributions of this dissertation involve the very recent theory of multi-type calculi on the proof-theoretic side, and the well established theory of heterogeneous algebras on the algebraic side. Given a cut-admissible sequent calculus for a basic logic (e.g. the full Lambek calculus), a core question in structural proof theory concerns the identification of axioms which can be added to the given basic logic so that the resulting axiomatic extensions can be captured by calculi which are again cut-admissible. This question is very hard, since the cut elimination theorem is notoriously a very fragile result. However, algebraic proof theory has given a very satisfactory answer to this question for substructural logics, by identifying a hierarchy (Nn, Pn) of axioms in the language of the full Lambek calculus, referred to as the substructural hierarchy, and guaranteeing that, up to the level N2, these axioms can be effectively transformed into special structural rules (called analytic) which can be safely added to a cut-admissible calculus without destroying cut-admissibility. The research program of algebraic proof theory can be exported to arbitrary signatures of normal lattice expansions, to the study of the systematic connections between algebraic logic and display calculi, and even beyond display calculi, to the study of the systematic connections between the theory of heterogeneous algebras and multi-type calculi, a proof-theoretic format generalizing display calculi, which has proven capable to encompass logics which fall out of the scope of the proof-theoretic hierarchy, and uniformly endow them with calculi enjoying the same excellent properties which (single-type) proper display calculi have. The defining feature of the multi-type calculi format is that it allows entities of different types to coexist and interact on equal ground: each type has its own internal logic (i.e. language and deduction relation), and the interaction between logics of different types is facilitated by special heterogeneous connectives, primitive to the language, and treated on a par with all the others. The fundamental insight justifying such a move is the very natural consideration, stemming from the algebraic viewpoint on (unified) correspondence, that the fundamental properties underlying this theory are purely order-theoretic, and that as long as maps or logical connectives have these fundamental properties, there is very little difference whether these maps have one and the same domain and codomain, or bridge different domains and codomains. This enriched environment is specifically designed to address the problem of expressing the interactions between entities of different types by means of analytic structural rules. In the present dissertation, we extend the semantic cut elimination and finite model property from the signature of residuated lattices to arbitrary signatures of normal lattice expansions, and build or refine the multi-type algebraic proof theory of three logics, each of which arises in close connection with a well known class of algebras (semi De Morgan algebras, bilattices, and Kleene algebras) and is problematic for standard proof-theoretic methods. @en

This thesis is part of a line of research aimed at providing a strong and modular mathematical backbone to a wide and inherently diverse class of logics, introduced to capture different facets of social behaviour. The contributions of this thesis are rooted methodologically in duality, algebraic logic and structural proof theory, pertain to and advance three theories (unified correspondence, multi-type calculi, and updates on algebras) aimed at improving the semantic and proof-theoretic environment of wide classes of logics, and apply these theories to the introduction of logical frameworks specifically designed to capture concrete aspects of social behaviour, such as agents’ coordination and planning concerning the transformation and use of resources, and agents’ decision-making under uncertainty. The results of this thesis include: the characterization of the axiomatic extensions of the basic DLE-logics which admit proper display calculi; an algorithm computing the analytic structural rules capturing these axiomatic extensions; the introduction of a multi-type environment to describe and reason about agents’ abilities and capabilities to use and transform resources; the introduction of a proper display calculus for firstorder logic; the introduction of the intuitionistic counterpart of Probabilistic Dynamic Epistemic Logic, specifically designed to address situations in which truth is socially constructed. The results and methodologies developed in this thesis pave the way to the logical modelling of the inner workings of organizations and their dynamics, and of social phenomena such as reputational Matthew effects and bank runs. @en