What are mental constructs consisting of clusters or collections of related concepts that serve to categorize new information in order to make sense of the world?

Prototype theory, as developed by Rosch, has had repercussions in two main areas of linguistics: lexical semantics and syntax. Word meanings are the names of categories, and the meanings of many words display characteristic prototype effects (fuzziness of category boundaries, degrees of representativity of category members). Further areas of application have been semantic change, and the structure of polysemy networks. The prototype approach does, however, encounter problems in connection with theories of semantic compositionality. Linguistic constructs, such as syntactic and lexical categories, also display prototype effects. The application of prototype theory to the study of parts of speech and syntactic constructions has been especially fruitful.

1 Prototype Theories

Prototype theories make a simple assumption about concept representation, namely that the concept is represented as the ‘ideal’ or ‘average’ category exemplar. Thus, representations of specific category exemplars do not influence classification. Numerous forms have been suggested for the (probabilistic) response rule; here we describe one common example (for other examples, see Categorization and Similarity Models). Assume that the probability that stimulus i is classified in category J, P(RJ/Si), is given by the equation:

(1)P(RJ/Si)=bJsiPJ∑KbKsiP K

where bJ (0≤bJ≤1,ΣbJ=1) represents the bias towards making category response J and siPJ is the similarity between exemplar i and the prototype of category J. The idea is that classification depends on the similarity between exemplar i and the category J prototype relative to i's similarity to the prototypes of all other categories. Similarity is usually measured with multidimensional scaling methods (see also Multidimensional Scaling in Psychology; Signal Detection Theory: Multidimensional).

One piece of evidence favoring prototype models is that a previously unseen prototype may in some circumstances be classified with higher accuracy in the test phase than the actual training stimuli. Moreover, compared to the training stimuli, the prototype may be particularly resistant to forgetting (Homa et al. 1981). Despite this, and notwithstanding their attractive simplicity, there is an abundance of evidence against prototype models of concept learning, at least within the domain of perceptual classification. Prototype models have often been shown to provide poorer fits than other models to large sets of classification data, they make a number of predictions that have been falsified, and they fail to account for a number of well-established phenomena (see Nosofsky 1992 for a thorough review).

Theories of categorisation

One way of considering knowledge systems is as formal mechanisms for classifying and categorising objects. Graphically, a typical ontology resembles a hierarchical taxonomy—though, technically, it is a directed acyclic graph, meaning that concepts can have more than a single ‘parent’ as well as multiple ‘siblings’ and ‘children’. (Ontologies also can support other sorts of conceptual relations, but the relationship of subsumption is axiomatised into the semantics of the OWL directly, as are several other relations.) In such systems, concept application relies on objects meeting necessary and sufficient conditions for class membership. This general model accords well with the broad tradition of category application stretching back to Aristotle. However, ontologies are intended to be machine-oriented representations of conceptualisations, with only an analogical relation to mental cognitive models. What, then, can be gleaned from contemporary theories of categorisation?

Since the 1960s alternative models have been proposed for how mental concepts are organised and applied. Like ontologies, semantic networks, pioneered by Quillian (1967), model cognitive conceptual networks as directed graphs, with concepts connected by one-way associative links. Unlike ontologies these links do not imply any logical (or other) kind of relation between the concepts—only that a general association exists. Semantic networks were adapted for early knowledge representation systems, such as frame systems, which utilise the same graphic structure of conceptual nodes and links: ‘We can think of a frame as a network of nodes and relations’ (Minsky 1974). Minsky also explicitly notes the similarity between frame systems and Kuhnian paradigms—what results from the construction of a frame system as a viewpoint of a slice of the world. By extension, semantic networks can be viewed as proto-paradigms in the Kuhnian sense, though it is not clear what the limits between one network and another might be—this analogy should not, then, be over-strained.

A feature of semantic networks is the lack of underlying logical formalism. While Minskian frame systems and other analogues in the 1970s were ‘updated’ with formal semantic layers, notably through the development of description logics in the 1980s, according to Minsky the lack of formal apparatus is a ‘feature’ rather than a ‘bug’—imposition of checks on consistency, for example, impose an unrealistic constraint on attempts to represent human kinds of knowledge, precisely because humans are rarely consistent in their use of concepts (Minsky 1974). At best they are required to be consistent across a localised portion of their cognitive semantic network, relevant to a given problem at hand, and the associated concepts and reasoning required to handle it. Similarly the authors of semantic network models note the difficulty in assuming neatly structured graphs model mental conceptual organisation: ‘Dictionary definitions are not very orderly and we doubt that human memory, which is far richer, is even as orderly as a dictionary’ (Collins and Quillian 1969). Semantic networks represent an early—and enduring—model of cognition, which continues to be influential in updated models such as neural networks and parallel distributed processing (Rogers and McClelland 2004). Such networks also exhibit two features of relevance to the theory adopted here: first, the emphasis on structural, connectionist models of cognition—that concepts are not merely accumulated quantitatively as entries in a cognitive dictionary, but are also interconnected, so that the addition of new concepts makes a qualitative difference in how existing concepts are applied; and second, the implied coherence of networks, which suggests concepts are not merely arranged haphazardly but form coherent and explanatory schemes or structures.

In the mid-1970s prototype theory, another cognitive model, was proposed for describing concept use. Building on Wittgenstein’s development of ‘language games’ (Wittgenstein 1967), Rosch (1975) demonstrated through a series of empirical experiments that the process of classifying objects under conceptual labels was generally not undertaken by looking for necessary and sufficient conditions for concept-hood. Rather, concepts are applied based on similarities between a perceived object and a conceptual ‘prototype’—a typical or exemplary instance of a concept. Possession of necessary and sufficient attributes is a weaker indicator for object inclusion within a category than the proximation of the values of particularly salient attributes—markers of family resemblance—to those of the ideal category member. For example, a candidate dog might be classified so by virtue of the proximity of key perceptual attributes to those of an ideal ‘dog’ in the mind of the perceiver—fur, number of legs, size, shape of head, and so on. Applying categories on the basis of family resemblances rather than criterial attributes suggests that, at least in everyday circumstances, concept application is a vague and error-prone affair, guided by fuzzy heuristics rather than strict adherence to definitional conditions. Also, by implication, concept application is part of learning—repeated use of concepts results in prototypes which are more consistent with those used by other concept users. This would suggest a strong normative and consensual dimension to concept use. Finally, Rosch (1975) postulated that there exist ‘basic level semantic categories’, containing concepts most proximate to human experience and cognition. Superordinate categories have less contrastive features, while subordinate categories have less common features—hence basic categories tend to be those with more clearly identifiable prototypical instances, and so tend to be privileged in concept learning and use.

While semantic network and prototype models provide evocative descriptive theories that seem to capture more intuitive features of categorisation, they provide relatively little causal explanation of how particular clusters of concepts come to be organised cognitively. Several new theories were developed in the 1980s with a stronger explanatory emphasis (Komatsu 1992). Medin and Schaffer (1978), for example, propose an exemplar-based ‘context’ theory rival to prototype theory, which eschews the inherent naturalism of ‘basic level’ categorial identification for a more active role of cognition in devising ‘strategies and hypotheses’ when retrieving memorised category exemplar candidates. Concept use, then, involves agents not merely navigating a conceptual hierarchy or observing perceptual family resemblances when they apply concepts; they are also actively formulating theories derived from the present context, and drawing on associative connections between concept candidates and other associated concepts. In this model, concept use involves scientific theorising; in later variants, the model becomes ‘theory theory’ (Medin 1989). As one proponent puts it:

In particular, children develop abstract, coherent systems of entities and rules, particularly causal entities and rules. That is, they develop theories. These theories enable children to make predictions about new evidence, to interpret evidence, and to explain evidence. Children actively experiment with and explore the world, testing the predictions of the theory and gathering relevant evidence. Some counter-evidence to the theory is simply reinterpreted in terms of the theory. Eventually, however, when many predictions of the theory are falsified, the child begins to seek alternative theories. If the alternative does a better job of predicting and explaining the evidence it replaces the existing theory (Gopnik 2003, p. 240).

Empirical research on cognitive development in children (Gopnik 2003) and cross-cultural comparisons of conceptual organisation and preference (Atran et al. 1999; Medin et al. 2006; Ross and Medin 2005) has shown strong support for ‘theory theory’ accounts. Quine’s view of science as ‘self-conscious common sense’ provides a further form of philosophical endorsement to this view.

For the purposes of this study, a strength of the ‘theory theory’ account is its orientation towards conceptual holism and schematism—concepts do not merely relate to objects in the world, according to this view (although assuredly they do this too); they also stand within a dynamic, explanatory apparatus, with other concepts, relations and rules. Moreover theories are used by agents not to explain phenomena to themselves, but also to others; concept use has then a role both in one’s own sense making of the world, and also in how one describes, explains, justifies and communicates with others. In short, concepts are understood as standing not only in relation to objects in the world, as a correspondence theory would have it; they stand in relation to one another, to form at least locally coherent mental explanations; and they also bind together participating users into communities and cultures. The account presented here similarly draws on supplemental coherentist and consensual notions of truth to explain commensurability.

2.3 Conceptual Approaches

Conceptual approaches locate meaning in the mind, rather than in language or in the extralinguistic world. Much discussion has had to do with the meanings of words in relation to conceptual categories (for general discussion, see Taylor 1989, and Ungerer and Schmid 1996).

The currently dominant view of what natural conceptual categories (concepts) are like is that they cannot in general be defined by any finite set of necessary and sufficient features, but are somewhat indeterminate in nature. One important theory, prototype theory, holds that natural categories are organized around ideal examples (prototypes), and that other items belong to the category to the extent that they resemble the prototype. There is normally no definite cut-off point dividing members from non-members: category boundaries are typically fuzzy, or vague.

A major difference of opinion exists over whether words map directly onto conceptual categories, or whether there is an intermediate level of semantic structure which is purely linguistic in nature. Those, like the cognitive linguistics, who argue for direct mapping, claim that there is no theoretical work for a linguistic level of semantics to do: all semantic phenomena can be given a conceptual explanation. Those who believe in an independent linguistic level of semantics argue that the fuzziness of conceptual distinctions contrasts with the sharpness of linguistic distinctions, and indicates a separate level of structure. (Think of the uncertain dividing line between ‘alive’ and ‘dead’ in ‘real life’ in relation to the fact that The rabbit is dead logically entails The rabbit is not alive.) They also point to cases like book, which (it is claimed) has a single sense (at the linguistic level), but corresponds to two different concepts, a concrete physical object (The book fell from the shelf) and an abstract text (It is a very difficult book).

A conceptual approach is not necessarily incompatible with a componential approach. Many prototype theorists characterize categories/concepts in terms of a set of prototype features, which may or may not have the status of semantic primitives. Such features are not of the necessary-and-sufficient sort, however, but have the property that the more of them something has, the better an example of the relevant category it is.

6 Grammatical Relations and the Functional Approach to Language

The concept of grammatical relation has been known in linguistics since the time of the grammarians of antiquity. However, significant progress in understanding the nature of grammatical relations has been achieved only in the last decades thanks to the orientation towards the functional explanation of grammar (see Grammar: Functional Approaches), which has permitted establishing the polyfunctional character of the elementary categories which form the clusters known as grammatical relations. The majority of these categories were themselves discovered and investigated in the framework of the functional approach.

In this connection the question of the existence of different patterns of encoding S, A, and P arguments merits special attention. Proceeding from the now widely established assumption of the iconicity of the linguistic sign (see Linguistics: Iconicity), replacing the Saussurean assumption of the arbitrariness of the sign, one can allow for the existence of a functional motivation for the various coding patterns. These patterns unite arguments with different elementary roles in different ways. The basic prototype of all these groupings (see Linguistics: Prototype Theory) is given by the agent and patient of the transitive construction. From this viewpoint active alignment is a natural metonymic extension of the elementary roles of agent and patient, generalizing them to the hyperroles (macroroles) of actor (the participant in the event which performs, effects, instigates, or controls it) and undergoer (the participant in the event which does not perform, effect, instigate, or control it, but rather is affected by it in some way). Arguments with different elementary roles can be interpreted in terms of these hyperroles.

Accusative alignment is also motivated essentially not by grammatical relations, but by the hyperroles principal and patientive. The principal indicates the main participant, the ‘hero’ of the situation, who is primarily responsible for the fact that this situation takes place. In the transitive clause the principal is, of course, the agent-like argument, while in the intransitive clause the single nuclear argument has no competitors for the role of principal. The patientive is the most patient-like participant of a multi-participant event. This role unites, in addition to the patient, arguments of other types.

Ergative alignment articulates the S, A, P arguments by means of other hyperroles, absolutive and agentive. The absolutive is the metonymic extension of the patient to intransitive verbs, it is the immediate, nearest, most involved or affected participant in the situation. In the transitive clause the patient is most involved in the situation. In the intransitive clause there is no competition for the role of closest and most involved participant in the situation. The absolutive in the transitive clause is in opposition to the agentive, the most agent-like participant of the multi-participant event.

Tripartite alignment distinguishes agentive and patientive of the transitive clause, distinguishing them from the single nuclear argument of the intransitive clause.

All these means of generalizing the elementary roles are in principle motivated, but a language has the option of grammaticalizing just one of them as its basic pattern. In addition, a language may have different patterns in different contexts, and examples of this kind have been attested.

1.07.2.6 Cognitive Geographic Concepts and Qualitative Reasoning

Cognitive geographic concepts generally refer to the informal geographic knowledge that people acquire and accumulate during the interactions with the surrounding environment (Golledge and Spector, 1978). Such informal knowledge was termed as naïve geography by Egenhofer and Mark (1995) and can be differentiated from the formal geographic knowledge that requires intentional learning and systematical training (Golledge, 2002). The training requirements of formal geographic knowledge can be seen from the special concepts and terminologies, such as projected coordinate systems, raster, vector, and map algebra, discussed in many GIS textbooks (Bolstad, 2005; Clarke, 1997; Longley et al., 2001). Since not every GIS user has received formal training, understanding the conceptualization of general people toward geographic concepts can facilitate the design of GIS (Smith and Mark, 1998). This section focuses on the informal understanding of general people toward geographic concepts and spatial relations, given the focus of this article on geospatial semantics. However, it is worth noting that the content discussed in this section is only part of the field of cognitive geography (Montello, 2009; Montello et al., 2003a) which involves many other topics such as geovisualization (MacEachren and Kraak, 2001).

Geographic concepts and spatial relations are two types of informal geographic knowledge that people develop in everyday life. Studies on the former often examine the typical examples that people associate with the corresponding geographic concepts. For example, Smith and Mark (2001) found that nonexpert individuals usually think about entities in the physical environment (e.g., mountains and rivers) when asked to give examples of geographic features or objects, whereas they are more likely to answer with social or built features (e.g., roads and cities) when asked things that could be portrayed on a map. Such typical examples can be explained by the prototype theory from Rosch (1973) and Rosch and Lloyd (1978) in psychology, in which some members are better examples of a category. For example, robin is generally considered as a better example for the category of bird compared with penguin. There are values in understanding the typical examples of geographic concepts. For instance, it can help increase the precision of GIR by identifying the default geographic entities that are more likely to match the search terms of a user. Besides, it has been found that different communities, especially the communities with different languages, may establish their own conceptual systems (Mark and Turk, 2003). Understanding these conceptualization differences on geographic concepts can help develop GIS that can better fit local needs (Smith and Mark, 2001).

Spatial relation is another type of informal geographic knowledge that we acquire by interacting with the environment. Fig. 5 provides an example which illustrates the spatial relations that a person may develop for different places near the campus of the University of California Santa Barbara. Such spatial relations are qualitative: we may know the general locations and directions of these places but not the exact distances between them (e.g., the distance between Costco and Camino Cinemas in meters is unknown to the person). Yet, these informal spatial relations are useful and sufficient for many of our daily tasks such as wayfinding and route descriptions (Brosset et al., 2007; Klippel and Winter, 2005; Klippel et al., 2005; Montello, 1998). In addition, these relations are convenient to acquire since we do not always carry a ruler to measure the exact distances and angles between objects. These informal spatial relations also present an abstraction from some quantitative details and are not restricted to a set of specific values (e.g., the spatial relation A is to the west of B can represent an infinite number of A and B, as long as they satisfy this relative spatial constraint) (Freksa, 1991; Gelsey and McDermott, 1990).

The informal conceptualization of people on geographic concepts and spatial relations can be formally and computationally modeled to support qualitative reasoning. The term qualitative reasoning should be differentiated from the term qualitativeness which may imply descriptive rather than analytical methods (Egenhofer and Mark, 1995). Spatial calculi can be employed to encode spatial relations (Renz and Nebel, 2007), such as the mereotopology (Clarke, 1981), 9-intersection relations (Egenhofer, 1991; Egenhofer and Franzosa, 1991), double-cross calculus (Freksa, 1992), region connection calculus (Randell et al., 1992), flip-flop calculus (Ligozat, 1993), and cardinal direction calculus (Frank, 1996). In addition to spatial relations, temporal relations, such as the relative relations between events, can also be formally represented using, for example, the interval algebra proposed by Allen (1983). Algorithmically, informal knowledge can be modeled as a graph with nodes representing geographic concepts (and their typical instances) and edges representing their spatial relations. If we restrict the nodes to be only place instances, we can derive a place graph. This graph representation is fundamentally different from the Cartesian coordinates and geometric rules widely adopted in existing GIS and could become the foundation of place-based GIS (Goodchild, 2011). Platial operations, as counterparts of the spatial operations (Gao et al., 2013), could then be developed by reusing and extending the existing graph-based algorithms as well as designing new ones. Constructing such a geographic knowledge graph can be challenging, since different individuals often conceptualize places and spatial relations differently. However, such a challenge also brings the opportunity of designing more personalized GIS for supporting the tasks of individuals (Abdalla and Frank, 2011; Abdalla et al., 2013). Qualitative reasoning and place-based GIS should not be seen as replacements for quantitative reasoning and geometry-based GIS (Egenhofer and Mark, 1995). Instead, they complement existing methods and systems and should be used when the application context is appropriate.

4 Establishing a Classification

After a measure of similarity has been selected, the next step is the actual classification of the objects based on the similarities between them. Formally (e.g., Biggs 1999), classes can be thought equivalent to (a) partitioning a set into subsets, (b) classifying a set of objects, and (c) distributing a set of objects into a set of ‘boxes.’ These various perspectives differ markedly in their implications for classification. For example, in most mathematical conceptualizations, an element is classified into exactly one class. Some clustering procedures, however, allow for residual elements, which are not considered clusterable.

Depending on the approach to classification that a researcher has chosen, certain considerations and precautions are necessary. For example, similarity judgment data may not fulfill some necessary assumptions: Generally, related objects are to be located in the same class. The relationship xRx when ‘x is related to x’ has the properties of reflexivity: xRx, symmetry: xRy↠yRx, and transitivity: xRy and yRz↠xRz. These are the properties of an equivalence relation. If the empirical data are similarity judgments, they do not necessarily fulfill this relation. Some properties of this relation can be tested statistically. Another important consideration applies if the objects are classified by using rules referring to features. In such cases, these rules need to be free of contradictions (for a test, see Feger 1994).

Only a few substantive theories in the behavioral and social sciences allow one to deduce the number and kind of classes needed to describe a given range of phenomena. Therefore in many cases inductive procedures have to be used to generate classes. For this, one needs a concept of what constitutes a class. Many researchers apply inductive classification methods without ever considering explicitly the class concept that their method implies. The following part of the paper gives a brief discussion of the class concepts implied in frequently used methods for finding classes. The list is not complete, and the ‘cluster analysis proper’ dominates all other approaches, because of the frequency of its use.

Before discussing class concepts in detail, one more general distinction needs to be made. If classes are defined by properties of objects, two levels of definition can be distinguished. A general definition specifies the relationship between the properties and the classes. Specific definitions provide detailed translations of the general definition into formal operations for assigning the objects to the classes. Obviously there can be many different specific definitions. General definitions can be ordered by the kind and amount of variability they allow among objects within the class. There are two general positions with respect to within-class variability. The ‘monothetic’ position (Sutcliffe 1993) assumes that a class is defined by one or a few necessary properties. The ‘polythetic’ counter-position (Gyllenberg and Koski 1996) assumes that some properties of a specified total set, not necessarily the same for every object, are sufficient. According to this position, a property is shared by most, but not necessarily all objects of a given class. Proponents of the monothetic camp tend to stress that some properties are more important than others, and that these properties should be used to establish the classification. The opposite position assumes equal importance of all properties. As a third type of general definition, one may add definitions referring to a ‘prototype,’ that is, the most typical example of a class or a hypothetical mean object. In this last case, ‘closeness’ or similarity decides about class membership, and the prototype may be defined with or without allowing for variation in its properties.

Given properties as the base for a classification, the actual observations often are represented as a data matrix containing, for example, the objects as the columns and their properties as the rows. The cells of the matrix contain either the values ‘0’ or ‘1’ to indicate the absence or presence of properties, or they contain frequencies, durations, intensities, or symbols (in the case of qualitative polytomous items) that indicate the type or degree of the respective property in the respective object. As this enumeration shows, the procedure can accommodate data of all scale types. The goal now is to find a ‘feature by classes’ matrix, called—corresponding to its purpose—the reference or identification matrix, or simply ‘a classification.’