A non-phenomenological model of Conceptual Spaces [*Best Explanation*]

One of the most serious problems connected with G¨ardenfors’ conceptual spaces is that these have, for him, a phenomenological connotation. In other words, if, for example, we take, the conceptual space of colours this, according to G¨ardenfors, must be able to represent the geometry of colour concepts in relation to how colours are given to us. Now, since we believe that this type of approach is bound to come to grief as a consequence of the well-known problem connected with the subjectivity of the so-called ‘qualia’, e.g., the specific and incommunicable quality of my visual perception of the rising Sun or of that ripe orange etc. etc., we have chosen a non phenomenological approach to conceptual spaces in which we substitute the expression ‘measurement’ for the expression ‘perception’, and consider a cognitive agent which interacts with the environment by means of the measurements taken by its sensors rather than a human being. 

Of course, we are well aware of the controversial nature of our non phenomenological approach to conceptual spaces. But, since our main task in this paper is characterizing a rational agent with the view of providing a model for artificial agents, it follows that our non-phenomenological approach to conceptual spaces is justified independently of our opinions on qualia and their possible representations within conceptual spaces Although the cognitive agent we have in mind is not a human being, the idea of simulating perception by means of measurement is not so far removed from biology. 

To see this, consider that human beings, and other animals, to survive need to have a fairly good ability to estimate distance. The frog unable to determine whether a fly is ‘within reach’ or not is, probably, not going to live a long and happy life. Our CA is provided with sensors which are capable, within a certain interval of intensities, of registering different intensities of stimulation. For example, let us assume that CA has a visual perception of a green object h. 

If CA makes of the measure of the colour of h its present stereotype of green then it can, by means 6 Actually, we do not agree with G¨ardenfors when he asserts that: Properties. . . form a special case of concepts. [4], chapter 4, §4.1, p. 101. 7 A set S is convex if and only if whenever a, b ∈ S and c is between a and b then c ∈ S. 27 of a comparison of different measurements, introduce an ordering of gradations of green with respect to the stereotype; and, of course, it can also distinguish the colour of the stereotype from the colour of other red, blue, yellow, etc. objects. In other words, in this way CA is able to introduce a ‘green dimension’ into its colour space, a dimension within which the measure of the colour of the stereotype can be taken to perform the rˆole of 0. 

The formal model of a conceptual space that at this point immediately springs to mind is that of a metric space, i.e., it is that of a set X endowed with a metric. However, since the metric space X which is the candidate for being a model of a conceptual space has dimensions, dimensions the elements of which are associated with coordinates which are the outcomes of (possible) measurements made by CA, perhaps a better model of a conceptual space might be an ndimensional vector space V over a field K like, for example, Rn (with the usual inner product and norm) on R. 

Although this suggestion is very interesting, we cannot help noticing that an important disanalogy between an n-dimensional vector space V over a field K, and the ‘biological conceptual space’ that V is supposed to model is that human, animal, and artificial sensors are strongly non-linear. In spite of its cogency, at this stage we are not going to dwell on this difficulty, because: (1) we intend to examine the ‘ideal’ case first; and because (2) we hypothesize that it is always possible to map a perceptual space into a conceptual space where linearity is preserved either by performing, for example, a small-signal approach, or by means of a projection onto a linear space, as it is performed in kernel systems