Abstract
Abstract 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.
Highlights
The present paper is based on a posthumous piece by Tarski entitled “What are logical notions?” [47]
Tarski had a great many original ideas. He is very famous among philosophical logicians for his theory of truth, and among mathematical logicians for the development of model theory, many of his ideas and works are still not well-known
At the end of the 1920s, Tarski developed the theory of the consequence operator, and for many years this theory was hardly known outside of Poland. The idea of this theory appeared for the first time in a two-page paper published in French in Poland in 1929 [43]. It was translated into English by Robert Purdy and Jan Zygmunt only in 2012, and it was published with a presentation by Jan Zygmunt in the Anthology of Universal Logic [58]
Summary
I.e., relations between two objects, elements, things. There are many such relations and it is possible to prove that any n-ary relation can be expressed/reduced to a binary relation. Tarski says the following about logical notions in case of binary relations:. Alice may point out that this formula describes the following configuration (CONF1b) To reply to this question, we have to introduce model theory to Alice, a theory developed by Alfred Tarski himself. If we allow only formulas with no specific names, no constants, only variables, the answer to question (2) is negative This is not necessarily a problem because these two models are considered to be isomorphic: we can establish a one-to-one correspondence between the two that preserves the given structure of this configuration, which in model theory is called a structure. (A2) If a binary relation can be described by a categorical formula, is it sufficient to consider it to be a logical notion?. (A3) Is a binary relation considered to be a logical notion only if it can be described by a categorical formula?.
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