This paper takes as its basis, Gärdenfors' [1] description of conceptual spaces and sketches an initial geography of the conceptual space for temperature. It highlights important diferences between the space's constituents, including: how category membership difers for concepts along diferent parts of the dimension; how comparatives and compounds are formed; how context afects concept location and behaviour; and how non-literal usage allows concepts to be used outside of their conceptual space or in a diferent way within their own conceptual space. The purpose of this tour of temperature space is to aid the practical implementation of semantic representations based upon conceptual spaces.
In the conceptual spaces approach to the representation of meaning, concepts and their instances occupy
spaces composed of quality dimensions. The proximity of objects along those dimensions equates to
semantic similarity. The approach allows for graded and ambiguous class membership. Gärdenfors [
An important yet seemingly simple conceptual space is that for temperature. Perception of temperature is important for survival (and comfort) so is also a frequent topic of conversation. Although temperatures occupy just one dimension — a number line — close consideration of English words for temperature unveils a complex arrangement of that space and a range of types of concept within it.
This paper forms a guide through temperature space which highlights the issues that need to be considered when developing a practical implementation of conceptual spaces. It begins with a summary of relevant literature on the representation of adjective semantics along dimensions. The constituent concepts of temperature space are then introduced, followed by a discussion of their positions, compound forms, and behaviour in diferent contexts.
Conceptual Spaces order concepts along quality dimensions, with similar concepts close to one another.
Gärdenfors [
Paradis [
Extreme adjectives These also exist on a scale with two antonyms (such as terrible and excellent).
But the scale is not open-ended; instead the adjectives exist at the extremes of the scale. For this reason, they cannot combine with scalar modifiers (* very terrible), but can combine with totality modifiers ( completely terrible). There is disagreement over whether or not these adjectives have a comparative form.
Limit adjectives These are adjectives (such as dead) which do not exist on a scale and do not have graded membership. They do not have a comparative form (*deader) and combine only with totality modifiers ( completely dead, almost dead).
A large number of basic adjectives fit within the scalar category. According to Barsalou [
But, Hampton [
As described above, only scalar adjectives have comparative forms. According to Gärdenfors [2, p.136], the meaning of comparatives depends upon the dimensionality of the space.
Comparison of objects along a single dimension, such as height, requires consideration of their
position along the dimension, but not the position of the prototype. Dietz [
On the other hand, comparison of objects within a multi-dimensional space, such as colour, involves
comparing their proximity to the prototype. For to be redder than , must be closer than to the
prototype for red. It also seems meaningless to use the term redder if neither nor are at all red [
Temperature space has one dimension [
Temperature contains the antonyms cold and hot. But depending on context, a greater variety of adjectives are also available. The words cool and warm act as gentler compliments, and sometimes mild can also be used. Along the dimension of temperature, the antonyms cold and hot sit opposite one another, leaning towards the two extremes; the antonyms cool and warm are closer to the middle; and mild sits in the middle.
The antonym of mild, is extreme, which can occupy either end of temperature space. Alternatively, mild and extreme can sit along a mildness/extremeness dimension which maps onto temperature in the same way that the integers map onto the natural numbers via the absolute value function. Figure 1 shows such an arrangement, with temperature concepts ordered along a coldness/hotness dimension that has mild at its origin. The Temperature line is folded at the origin in order to demonstrate its mapping onto mildness/extremeness.
Concepts not shown in figure 1 include limit adjectives such as boiling and freezing, which have strict definitions when used literally, and adjectives such as lukewarm and tepid, which are restricted to certain contexts. It makes less sense to place these adjectives along a scale.
Hot and cold are unbounded scalar adjectives which behave analogously to tall and short in height space. The typicality of instances of these concepts increases as they get further along the dimension, even as they surpass the prototype value of the concept.
On the other hand, temperatures become more typical instances of mild as they get closer to its prototype. This behaviour is more typical of concepts in a multi-dimensional space, but mild is perhaps not strictly native to temperature space, but rather to mildness/extremeness space where its behaviour is more similar to that of short in height space.
Warm and cool seem to be in between these two types of behaviour. They are not strictly bounded, but at distances increasingly far from their prototype, alternative concepts become more salient.
It is not clear that hard boundaries should be drawn between each of these concepts with Voronoi
tessellation as suggested by Gärdenfors [
Because temperature concepts for the most part have the behaviour of scalar adjectives, they have comparative forms and can combine with scalar modifiers.
To determine semantically valid and pragmatically appropriate comparative concepts for the description of a given situation, both the ordering of points in temperature space and their proximity to prototypes is important.
To describe a point x as hotter or colder, less hot, or less cold, than point y, only the relative position of the two points along the temperature dimension is required in order to produce a logically true predicate.
But further consideration of their position in temperature is necessary in order to produce a pragmatically sensible sentence of English. The situation is further complicated by the presence of warm and cool, which provide a further array of comparatives to choose from.
The phrases is hotter than , is warmer than , and is less cold than all correspond with the inequality > , but to English speakers they each have a slightly diferent meaning, which relates to the original adjective’s location along the temperature dimension.
For example, in figure 1: a is strictly speaking hotter than e, but it is better to describe it as less cold than e; similarly c is strictly speaking hotter than a, but it is better to describe it as warmer than a
Adjectives’ comparative forms (hotter, colder) are equivalent to prefixing the adjective with more. If A is more X than B then A is further out along the unbounded dimension than B. If A is less X than B then A is closer to the centre of the space than B. In the case of mild, A is milder/more mild than B means A is closer to the prototype for mild. On the other hand, A is less mild than B means that A is further away than B (in either direction) from the prototype for mild. Figure 2 illustrates these diferences in behaviour. While warm and hot have positive polarities, and cool and cold have negative polarities, mild has neither.
An adjective’s polarity also governs the interpretation of grammatical modifiers such as very, quite, and extremely. When conjoined with an adjective they form a compound with a prototype that is shifted from the location of the lone adjective (see figure 3).
Words such as very and extremely, which increase the magnitude of the adjective they are attached to, follow the same rules as the comparative more: when attached to a positive adjective such as warm, they shift meaning further in the positive direction; when attached to a negative adjective such as cold, they shift meaning further in the negative direction; when attached to an adjective at the space’s origin (mild), they shift meaning closer to the origin and contract its boundaries: extremely mild has a more limited purview than mild.
Words such as quite, slightly, or a bit have the opposite efect and are therefore more akin to the comparative less. Since they shift meaning away from the extreme of the prototype, when combined with mild two prototypes are formed: one in the colder half of space, and one in the warmer half. Quite mild therefore occupies a non-convex region of temperature space. This is a further sign that mild does not really belong in temperature space, but is temporarily introduced in certain contexts.
When an unbounded concept such as hot has a magnitude-increasing modifier such as extremely applied to it, the resulting compound is also unbounded. When it has a magnitude-decreasing modifier such as quite applied to it, the resulting compound becomes bounded. The phrase quite hot only has limited applicability before hot or very hot become more appropriate descriptions. To say the surface of the Sun is quite hot would be acceptable only in a non-literal, sarcastic context, or perhaps when comparing with even hotter stars.
As mentioned by Hampton [
The non-uniform warping of space caused by the context changes in Figure 4 cause hot and warm to shift more than cool or cold. The limit adjectives boiling and freezing do not move at all unless they are used for exaggeration as when describing the weather, in which case they adopt the graded category membership of scalar adjectives. They could also shift if applied to more exotic contexts such as chemicals other than water.
Certain concepts are not shown in figure 4 and are not necessarily available in all contexts. For
example, mild is largely restricted to describing the weather, lukewarm to describing food, and tepid to
describing water [
Not only can context shift the position of concepts within a space, it can also project them from one space to another. This occurs when words from one conceptual space are used metaphorically to describe entities from a diferent conceptual space. Temperature adjectives are commonly used metaphorically for example in descriptions of spicy food or of people.
But in metaphorical usages, only a subset of temperature adjectives are imported into their new context: food’s degree of spiciness can be described as hot or mild, but not warm or cold; people’s behaviour to others can be described as warm, lukewarm, or cold, but not hot or mild; and people’s attractiveness can be described only as hot (or not).
Adjectives can be used and re-used metaphorically. If someone were to describe the weather as lukewarm, this would be an odd use of words since lukewarm is usually reserved for descriptions of food. But the description could also be interpreted as a personification of the weather into a person who is lukewarm, this in turn being a metaphorical description of someone with an unfriendly disposition.
Temperature is an everyday topic and its perception is fundamental to human existence. Ways of describing temperature indicate a variety of types of conceptual behaviour that need to be covered by a representation based on conceptual spaces, even in the case of a simple, unidimensional space. Flexible prototype locations Adjectival concepts have prototype values, but these must be capable of changing (potentially by orders of magnitude) depending on the context where they are used.
Certain adjectives, such as lukewarm are also unavailable or less available in given contexts. Flexible concept behaviour Not all adjectives within the same space have the same behaviour. In temperature, most adjectives are scalar, but there are also limit adjectives. Limit adjectives can adopt the behaviour of scalar or extreme adjectives if they are used metaphorically, for example boiling weather has graded membership, whereas boiling water does not.
Concept polarity As well as a prototype value, concepts should also be associated with a polarity, or direction. This helps to determine how comparatives are formed and how modifiers afect meaning. There are at least three types of behaviour in relation to comparatives and modifiers: for positive adjectives (e.g. warm) more and very indicate higher temperatures; for negative adjectives (e.g. cool) more and very indicate lower temperatures; for neutral adjectives (e.g. mild) at the conceptual space’s origin, more and very indicate temperatures closer to the origin. The applicability of Voronoi tessellation The existence of potentially overlapping concepts (e.g. cold and cool or cool and mild), the introduction of compounds (e.g. quite cold, very cool), and the metaphorical use of concepts (e.g. freezing) means that there is a plethora of ways to describe the same temperature. It is therefore unclear if it is useful to divide a space into disjoint regions belonging to separate concepts.
Odd behaviour of non-native concepts A space can gain non-native concepts, even without metaphorical usage, just as temperature space gains mild and extreme. These concepts or compounds involving them can occupy non-convex regions and therefore have unusual behaviour compared with other concepts.
Exaggeration and metaphor Words can be used non-literally within a conceptual space for exaggerated efect ( boiling weather); metaphorically in a foreign conceptual space (a warm person); and can be projected metaphorically out of and back into their home conceptual space (lukewarm weather).
Other issues not considered in this paper include the problem of negation, and the association of concepts across conceptual spaces, for example the varying association in diferent contexts between hot or cold and good or bad. These important issues no doubt cause further complications.
This research has been supported by EPSRC grant EP/R513106/1. Many thanks also to the attendees of the CAOS workshop for their questions and feedback.