  With his research on Aristotelian syllogistic, Jan {\L}ukasiewicz (1929;
1934; 1939; 1951/57) initiated a branch of logic called the calculus of names.
This section deals with axiomatic systems that analyze various fragments of the
logic of names, i.e., one that study various forms of names and functors acting
on them, as well as logical relationships between sentences in which these
names and functors occur. In this work, we want not only to present the genesis
of the calculus of names and its first system created by {\L}ukasiewicz. We
also want to deliver systems that extend the first. In this work, we will also
show that, from the point of view of modern logic, {\L}ukasiewicz's approach to
syllogistic is not the only possible one. However, this does not diminish
{\L}ukasiewicz's role in the study of syllogism. We believe that the calculus
of names is indisputably {\L}ukasiewicz's legacy.
