Studia humana (SH) is a multi-disciplinary peer reviewed journal publishing valuable

contributions on any aspect of human sciences such as...

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Studia humana (SH) is a multi-disciplinary peer reviewed journal publishing valuable

contributions on any aspect of human sciences such as...

read more...

Issue: 9:3/4 (the thirty fifth/sixth issue)

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.

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.