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'closed form', while still believing it possible to define a 'closed form' [74]. He was
an advocate of proposing criteria based on necessary characteristics and suggested
the optional selection of those of them which, after a given theory was proven, could
serve effectively. Various phases can be distinguished in J. S. Mill's works, in which
he professed various beliefs and preferred various methodologies. During the period
in which he supported utilitarianism, he detailed and adjusted the concept of the em-
pirical theory of cognition, several thoughts and theories from which have become
the foundations of today's mathematical logic. The most important of them undoubt-
edly include the first attempt to narrow down the concepts of traditional Aristotelian
logic. Mill presented the rules, which in Aristotle's works had the form of unclear
narrations, as diagrams, schematics and symbolic representations. In addition, he
proposed to stop elevating logic and mathematics to a superior level and instead
view both of these sciences as languages which can be freely, but in a non-
contradictory way, adjusted to the needs of their users. Important achievements of
Mill's also include the attempt to structure inductive reasoning rules into so-called
Mill's principles: methods, whose use led to producing categorical judgements by
way of induction.
Mill's methods are as follows: the method of agreement, the method of differ-
ence, the method of residue, the method of concomitant variations and the joint
method of agreement and difference. These subjects are too broad and too de-
tailed to discuss them in this topic, but should be suggested to the Reader as the
direction of further study. The progress in Mill's thinking and views meant that
this philosopher later moved closer to naturalistic views of the theory of cognition,
while in ethics he went in the direction of neo-Kantian theories. However, this
subject goes far beyond the limits of this topic.
For centuries, philosophy and science struggled, with greater or smaller suc-
cess, with attempts that were to produce a prototype learning system and at the
same time to create a general cognitive system that could be used to describe the
human process of acquiring knowledge, but without losing the ability to describe
similar processes taking place in an artificial, computer system. Every great scien-
tist tried to relate to the theories and theses of his equally great predecessors as
well as extend and improve those theories, whose aim was to get better knowledge
of not just the human being, but also the entire world around it.
Over the last centuries, methods aimed at the formalised writing of definitions,
logical methods, fuzzy methods and prototype building have developed greatly.
All the above methods are aimed at defining words, concepts, meanings etc. Even
though they had already been proposed in the 19
th
century, they were then effec-
tively implemented also in modern computer systems.
The progress of civilisation, particularly the western one, has for many years
been accompanied by the belief that only correctly conducted scientific research
could ensure progress in knowledge. This conviction also applied to the processes
of cognition (and understanding) treated as the subject of research at the time
when the toolbox already developed for science was applied to studies of these
phenomena. The pioneers of this approach were Hermann von Helmholtz [44],
