A standard way of producing a classi cation is that obtained by means of sharp concepts. A concept C is sharp if and only if C refers to a property P such that, if D is the domain of quanti cation, there exists a class A such that A = fx j P (x)g; A \ A = ;; and A [ A = D: For example, if D = N; the concept x is prime is sharp. And we say that `the number 2 falls under the concept x is prime' or, more simply, that `2 is prime.'

As is well known, when we are dealing with sharp concepts the law of excluded middle holds, whereas this is not the case with concepts that are not sharp like x is a heap. Within ordinary language we make an extensive, and productive, use of fuzzy concepts like x is a heap, y is bald, z is tall, etc. without even being aware of their problematic logical status.

However, besides the kind of classi cation we obtain by means of sharp/fuzzy concepts, there is a di erent type of classi cation which is not based on the existence of one and the same property common to all the members of a class A. But, it, rather, relies on similarities between the objects belonging to a certain class A and the elements of a subclass AS of A; elements that we are going to call `stereotypes.' By way of example, take As to contain two elements: a shark and a mullet. Starting from As we could generate A by taking some similarities between our stereotypes and other things.

Note that here by `similarity between a and b' we mean a property common to a and b; where a and b belong to D; e.g. being an animal, having ns, having scales, etc. Of course, more are the properties that a and b have in common the more similar a and b are to one another. The limiting case being that expressed by the identity a = b where the objects denoted by a and b have the same properties, that is, when a and b denote the same object.

It is not di cult to see how, from such similarities with our stereotypes, we can generate a concept of sh given in terms of: animal living in water having either ns or scales or both. And, of course, from such a way of characterising the concept of sh it would follow that whales, dolphins, etc. are sh. But, we know that contemporary zoology has successfully challenged the above mentioned use of the term ` sh' by introducing a distinction between mammals and sh, a distinction according to which whales and dolphins are not sh, but mammals.

The quasi-accidental, purely phenomenical nature of the classi cations obtained by means of brute correlations, such as those arising from mere similarities between objects and a set of stereotypes, has led us to call the cognitive representatives of these classi cations `proto-concepts.' We are going to use the term `concepts' only for the cognitive representatives of all the other kinds of classi cation.

The main aim of this article is showing that, if a proto-concept is given simply in terms of the ability to make the appropriate distinctions, then there are stimulus-response cognitive systems1 | whose way of manipulating information is based on Neural Networks (NN) | able to make the appropriate distinctions typical of proto-concepts in the absence of high-level cognitive features such as consciousness, understanding, representation, and intentionality. This, of course, implies that either proto-concepts cannot be given simply in terms of the ability to make the appropriate distinctions, or that we need to modify our traditional conception of mind, because the induction-like procedure followed by a NN in producing its classi cations, far from being the ultimate product of a `linguistic mind,' is, rather, inscribed in the nuts and bolts of the system's biology/electronics to which the NN belongs.

The present paper is a follow up to `Wittgenstein, Turing, and Neural Networks' by G. Oliveri and S. Gaglio where, among other things, the authors endeavour to bring out the genuine cognitive character of Neural Networks (NN), cognitive character exhibited, primarily, by their ability to learn and being trained to perform a certain task. 2

Concepts have always played a central r^ole in philosophy and, especially, in logic. One of the most important philosophical disputes in which medieval philosophers engaged | the so-called `dispute about universals' | was directly related to the attempt to provide a plausible explanation of the classifying power of concepts. In fact, the realists argued, the reason why the concept red is so useful in classifying certain objects, separating them out from all the others, is that to such a concept there corresponds a property, a universal, that is present in all and only those things of which we correctly say that they are red.

On the other hand, the nominalists thought that, in contrast with what asserted by the realists, the universals do not exist. For if you take two red things a and b you, immediately, realise that the shade of red of a is di erent from that of b and that, therefore, `red' is just a word, a name, to which no universal property corresponds. Therefore, according to the medieval nominalist, `red' is a name whose usefulness in classifying boils down to the possibility of putting together all and only those things that are similar to one another with respect to (a certain) colour.

1See on this [ 13 ], Chapters 2 and 3.

Concerning the importance of concepts in logic, we can mention here, by way of example, the peculiar relation that, in pre-Fregean logic, a concept/predicate was supposed to have to the subject in a judgment, a relation exploited by Kant in the Critique of Pure Reason to draw the important distinction between analytic and synthetic judgments.

However, the modern theory of concepts starts with Frege for whom a concept is not an object, but a function2 that takes proper names (or expressions performing the ro^le of proper names) as arguments, and truth-values as values. From this it, immediately, follows that, according to Frege, x is bald, y is a heap, z is tall, etc. are not concepts, because the expressions `a is bald,' `b is a heap,' `c is tall,' etc. may not have a truth-value for certain proper names a; b; c:

Although in Frege we do not nd the analytical philosopher's commitment to the idea that only a theory of language can provide a safe basis for the construction of a theory of thought,3 nevertheless, for Frege, concepts have an essential function in thought that consists in presiding over the formulation and justi cation of judgments. It is only with the advent of Gestalt psychology,4 Husserl's phenomenology,5 and the philosophy of psychology of the later Wittgenstein | embodied, in particular, in the study of the phenomenon known as seeing-as6 | that some non-idealist philosophers and psychologists discovered the very important ro^le performed by concepts in perception.

Consider the Necker cube given in Figure 1. Now, apart from the well known possibility of shifting from perceiving face ABCD as `coming forward' to seeing, instead, face 1234 as `coming forward' | depending on which of the two faces you are focussing your attention | the really interesting thing here is that one of the necessary conditions for you to see the object in Figure 1 as a cube is having the concept of cube.

In fact, although a young child with no knowledge of mathematics would be able to perceive the face-shifting phenomenon, and draw a fairly resembling picture of the object in Figure 1; if he were asked to say what he sees the object as, he would probably reply `a box,' `a lump of sugar,' `a brick,' `a wire frame,' etc. but, certainly, not `a cube,' because he does not know what a cube is.

If what we have said so far is correct, concepts have at least three di erent 2See [ 4 ] and [ 5 ]. 3See on this [ 3 ], especially Chapter 10.

4See on this [ 20 ] especially in relation to the impact that Gestalt psychology has on what he calls `productive thinking.' 5See on this [ 7 ], Part Three, Chapter Three. 6See on this [ 22 ], Part II, Chapter XI. important functions: classifying, being integral part of judgments, a ecting some of our perceptions. Of these three functions of concepts two of them | being integral part of judgments, a ecting some of our perceptions | presuppose the existence of a cognitive system able to produce judgments/thoughts and representations of objects. The classifying function, instead, seems to us to be somewhat independent of the existence of such a cognitive system. For, although, when a competent speaker of English says `Socrates is a man' the very meaningfulness of the assertion, and of the thought expressed by it, presuppose the speaker's ability to classify some of the objects of his domain of quanti cation as men; and that when someone sees something as a cube, the very possibility of his perception depends on his ability to classify certain objects of his domain of quanti cation as cubes; the ability, for example, to classify something as a sh (see x1) may not presuppose either judging or seeing-as (representing).

Perhaps, some light on these matters will be obtained from considering how concepts are given to us, because in so doing we might come across concepts that are given to us as means of classi cation and not as instruments for judging or representing. 3

In investigating how concepts are given to us, one of the obvious things to look at is language. Two of the main concept-producing devices present in language are the so-called `prototypes,'7 and `stereotypes.'8 A prototype is an individual belonging to the domain of quanti cation that has certain features `at their best.' Take, by way of example, D to be the set of the elements of the colour spectrum projected on to a wall by a prism when this is hit by a pencil of light rays (see Figure 2). The colour spectrum D; with the Euclidean distance de ned on it, is a metric space (D; d):

Now, for each colour band i present in the colour spectrum, where i 2 R; choose the middle element of the band as the prototype of that colour, and call it `pi:' If k; p 2 D; where p represents a prototype, as is shown by the colour spectrum, the shorter is the distance between k and p the more similar the colour represented by k is to the colour represented by p: Clearly, if pi is the prototype of the red colour band then the set R = fx j d(x; pi) < d(x; pj); for any j such that j 6= ig can be considered as a classi cation of elements of D which can, eventually, be turned into the extension of a concept and, precisely, of the concept x is red.

A stereotype, on the other hand, is an individual s belonging to the domain of quanti cation that appears to have certain features, but such features are not necessarily given at their best in s. Take (D; d) to be, as above. If k; s 2 D; where s represents a stereotype then, as above, the smaller is the distance between k and s the more similar the colour represented by k is to the colour represented by s: As colour stereotypes s consider the colours of the paints present on the palette of a Renaissance painter.

Assuming that on the palette of a Renaissance painter there could be a nite number of colour stereotypes s1; : : : ; sn; and that si is the only stereotype 7On a related concept of prototype see [ 6 ], especially Chapter 3, x3.9. See also [ 10 ] on prototypes and theory-like representations; and [ 18 ] on incorporating prototype theory in Convolutional Neural Networks.

8On a linguistic concept of stereotype see [ 16 ]. of red, we have that Ci = fx j d(x; si) < d(x; sj); for any j such that j 6= ig is a classi cation of elements of D which can be, eventually, turned into the extension of the concept x is red.

Although, as we have seen in the examples above, both prototypes and stereotypes generate potential extensions of concepts, there are some analogies and di erences existing between these two types of objects that deserve some attention.

One of the main di erences between prototypes and stereotypes is that there is only one prototype of something, but you could have several stereotypes of the same kind of thing. Indeed, the prototype of red is that electromagnetic wave having a wavelength of 700 nanometers (see Figure 2), whereas the so-called `Titian red' and `Pompeian red' are two di erent possible stereotypes of red. Of course, if there is more than one stereotype of, say, red the classi cation induced by all the stereotypes of red available is going to be the union of the classi cations induced by each single stereotype, and the larger is the number of di erent stereotypes of red the better are the chances of producing a correct classi cation of red objects.

Secondly, the very idea of `features at their best,' present in the de nition of prototype, requires some form of theorising and judging to distinguish, for example, between a n at its best and a n that is not at its best. On the other hand, when it comes to producing stereotypes, the situation looks rather di erent.

To see this consider the well known ethological phenomenon of imprinting. Imprinting is that type of learning that takes place only within a certain number of hours from birth. As Konrad Lorenz has shown,9 if, within a certain number of hours from its birth, a gosling is exposed to a human being, rather than to a goose, it ends up considering the human being as its parent following him, etc. (see Figures 3 and 4). In other words, imprinting is that form of learning whereby geese, and other animals, form a stereotype not only of their mother/parent, but also of the species to which they belong. This has two very important consequences for us. First, the phenomenon of imprinting, clearly, shows that the formation of some stereotypes is brute, that is, it does not take place through theorising, or any sort of reasoning or representation in uenced by previously acquired concepts.

Secondly, there are stereotypes, some of which play a crucial r^ole in producing very important, basic classi cations, that are independent of a linguistic mind.

But, be as it may with regard to the connection between imprinting and stereotypes formation, we believe that, if su ciently many samples are provided, then the classi cation obtained of the elements of the domain of quanti cation D can be turned into the extension of the corresponding protoconcept by means of CNNs. Although the following section gestures in this direction, substantiating this claim will be one of the objects of our future investigations. 4

Traditional Feed-Forward Neural Network architectures receive a single vector as an input and process it through a series of hidden layers. Each hidden layer is constituted by a set of arti cial neurons, where each unit is fully connected to all the units belonging to the previous layer. The neural units belonging to the same layer make their computation in parallel with the other units of the same layer. Furthermore, the neurons of the same layer do not share any connections. Traditional feed-forward architectures do not perform well on image recognition and image segmentation tasks. In the last years a category of Neural Network architectures, known in the literature as Convolutional Neural Networks (CNNs) and inspired by the mammalian visual system [ 23 ][Fukushima,1980] [ 25 ], have proven to be very e ective in performing tasks like image recognition and segmentation. The very rst convolutional neural network architecture was LeNet, developed by Yann LeCun and it was e ectively used for character recognition tasks. This kind of architectures gave rise to the general paradigm of Deep Learning.

The main operations performed in Convolutional Neural Network are Convolution, Pooling or Sub Sampling, and Classi cation.

Traditionally, ConvNets assume that the kind of inputs are images. A typical representation of a convolutional network is shown in Figure 5.

The network processes the original image layer by layer from the original pixel values to the nal classi cation output. The input layer speci es the dimensions of the input images. CNNs derive their name from the \convolution" operator. The main goal of Convolution in this kind of architectures is to extract features from the input image. The Convolution operation allows the network to learn image features using small portions of input data. A set of parameterized kernels constitutes the convolution layer. Every kernel is spatially small, and it is applied to the whole image through a spanning process.

The input image is convolved with these multiple learned kernels which exploit shared weights. This operation generates a bidimensional activation map that gives the responses of that lter at every spatial position. The size of the kernels gives rise to the locally connected structure and produces a set of feature maps. Then, the pooling layer reduces the size of the image, attempting to maintain the information.

The combination of the convolution and the pooling layers realizes the feature extraction part. Subsequently, the features are weighted and combined in the fully-connected layer, which constitutes the classi cation layer of the network [ 26 ].

Convolutional Neural Networks (CNNs) are powerful models capable of achieving outstanding results in particular for image classi cation and segmentation tasks. Nowadays this kind of neural architectures has been extremely successful in identifying faces, objects, tra c signs and are widely used in vision for robots and self-driving cars. Moreover, ConvNets have been e ectively used also in several Natural Language Processing tasks as well. Furthermore, they are capable to e ectively extract features from images: pre-trained models are used as generic feature extractors[ 29 ]. This goal is reached by removing the last layer which gives the output classi cation scores. The activations from the last fully connected layer de ne the features extracted from the input image [31].

These kinds of features extracted from pre-trained CNN have been successfully used in computer vision tasks such as scene recognition or object attribute detection, yielding better results concerning traditional handcrafted features [ 29 ].

Moreover, Athiwaratkun et al. [ 30 ] have also demonstrated that Random Forest and SVM can be used with features extracted from CNN to obtain better a prediction accuracy compared to the original CNN.

Recent results indicate that very deep networks achieve even better results on various benchmarks [ 27 ], [ 28 ]. One drawback of this trend, however, is the time required to train such kind of neural architecture.

Summing up we can say that CNNs can be used to perform a generalization/classi cation based on the stereotypes belonging to the training set and, as discussed in this section, with considerable improvements with regard to other traditional types of neural networks (even if the limitations typical of neural networks, relating to the size of the training set and the curse of dimensionality, still remain). This is a good baseline to investigate, in future works, if the classi cation obtained by a CNN can be turned into the extension of corresponding proto-concepts. 5

This is a philosophy and ethology inspired paper relating to cognition. Starting from a discussion of various kinds of classi cation, we are led to distinguishing between classi cations operated on the basis of concepts, and classi cations driven by what we have called `proto-concepts.' The di erence between the two di erent kinds of classi cations being that whereas concepts appeal to the existence of a property common to all the objects falling under them, proto-concepts, instead, derive their classi cation power from a set of what we call `stereotypes' and the relevant similarities existing between these stereotypes and the objects falling under the proto-concepts.

Having discussed the di erence between prototypes and stereotypes and their ro^le in producing classi cations | classi cations that are presented as potential extensions of proto-concepts | we discuss the ethological phenomenon of imprinting discovered and studied by Konrad Lorenz. As is well known, imprinting is that cognitive phenomenon whereby goslings, if exposed to a certain object K within a certain time from birth | a goose, a human being, etc. | elect K as a stereotype (in our sense) of parent/representative of their species and behave accordingly following K, etc. This, of course, implies that goslings subject to imprinting classify K, and themselves, as belonging to the same class K that becomes the potential extension of a proto-concept.

We then engage in a discussion of a particular type of Neural Network, the so-called `Convolutional Neural Network' (CNN). What we intend to show in our discussion of CNNs is that cognitive agents that operate on the basis of CNNs are able to produce classi cations typical of proto-concepts in the absence of high-level cognitive features such as consciousness, understanding, representation, and intentionality. On the basis of this result we ask ourselves whether this means that proto-concepts cannot be given simply in terms of the ability to make the appropriate distinctions or that we should, instead, modify our traditional conception of mind. [31] Garcia-Gasulla, Dario, Ferran Pars, Armand Vilalta, Jonatan Moreno, Eduard Ayguad, Jess Labarta, Ulises Corts, and Toyotaro Suzumura. "On the Behavior of Convolutional Nets for Feature Extraction." Journal of Arti cial Intelligence Research 61 (2018): 563-592.