We discuss a theory presented in a posthumous paper by Alfred Tarski
entitled “What are logical notions?”. Although the theory of these logical
notions is something outside of the main stream of logic, not presented in
logic textbooks, it is a very interesting theory and can easily be
understood by anybody, especially studying the simplest case of the four
basic logical notions. This is what we are doing here, as well as
introducing a challenging fifth logical notion. We first recall the context
and origin of what are here called Tarski-Lindenbaum logical notions. In
the second part, we present these notions in the simple case of a binary
relation. In the third part, we examine in which sense these are
considered as logical notions contrasting them with an example of a nonlogical
relation. In the fourth part, we discuss the formulations of the four
logical notions in natural language and in first-order logic without
equality, emphasizing the fact that two of the four logical notions cannot
be expressed in this formal language. In the fifth part, we discuss the
relations between these notions using the theory of the square of
opposition. In the sixth part, we introduce the notion of variety
corresponding to all non-logical notions and we argue that it can be
considered as a logical notion because it is invariant, always referring to
the same class of structures. In the seventh part, we present an enigma: is
variety formalizable in first-order logic without equality? There follow
recollections concerning Jan Woleński. This paper is dedicated to his 80th
birthday. We end with the bibliography, giving some precise references
for those wanting to know more about the topic.