During his brief life, the Polish mathematician and logician Adolf Lindenbaum (1904–1941) contributed to mathematical logic, among other things, by several significant achievements. Some results of Lindenbaum’s, which bear his name, were published without proofs by other people from the Lvov–Warsaw School and the proofs later were provided by some others, though the authorship of Lindenbaum has never been challenged. Many may have heard about Lindenbaum’s lemma, asserting the existence of Lindenbaum’s extension, and LindenbaumTarski algebra; less known is Lindenbaum’s logical matrix. This tutorial is devoted to the two last concepts rather than the first one. However, the latter can be understood in a purely algebraic fashion, if one employs the notion of Lindenbaum-Tarski algebra. In general, the notions of Lindenbaum matrix and Lindenbaum-Tarski algebra have paved a way to further algebraization of logic, which had been begun by George Boole in the 19th century, as well as to a new branch of logic, model theory. For this reason, the present tutorial is also a gentle introduction to algebraic logic.

A uniting idea of the aforementioned concepts is a special view on the formal judgments of a formal language. It is this view we call the Lindenbaum method. Although Lindenbaum expressed merely a starting viewpoint in the tradition of Polish logic of the time, this viewpoint became a standard ever since and its development goes on until this day, continuing to shape the field of algebraic logic. Our main objective is to demonstrate how this view gave rise to formulating the aforementioned concepts and how it opens door to unexplored paths.

I. Lindenbaum’s logical matrix
/The idea to interpret symbolic judgments in mathematical structures goes back to George Boole. It was Lindenbaum who took for an interpretation of judgments the judgments themselves. But prior to this, he started treating the entire class of judgments as an abstract algebra, nowadays known as a formula algebra. Some experts call this Lindenbaum’s move a milestone in the history of algebraic logic and universal algebra.
The turning point distinguishing the Boole-DeMorgan-Schr¨oder tradition in algebraic logic from modern tradition is the algebraization of formal deduction. The first step in this direction is the introduction of the notion of deductive system and that of consequence relation. There are a few standard ways to define a deductive system; in this part, we focus on two of them — rules of inference and logical matrix. On the one hand, the Lindenbaum logical matrix is just a special case, on the other, it characterizes all formal theorems (or theses) of any deductive system (Lindenbaum theorem)

II. Characterization of deductive systems
Powerful enough to characterize any class of theses, the notion of Lindenbaum matrix does not suffice to determine any deductive system. This part will address the question of characterization of deductive systems. Two W´ojcicki’s theorems will be discussed. The first deals with the notion of a bundle of logical matrix ; in terms of the latter any deductive system can be determined. The second theorem finds conditions under which a deductive system can be characterized by a single logical matrix.
The theorems of Lindenbaum’s and W´ojcicki’s were merely first steps towards algebraization of deduction based on sentential formal language. Next came analysis of matrices and algebras which “separate” premises from not derived from them sentences in deductive systems, thereby introducing the conception of separating means. This in turn has led to the notion of Lindenbaum-Tarski algebra. The latter often is obtained by a transformation of a Lindenbaum matrix with the use of a special congruence. In unital deductive systems the LindenbaumTarski algebra of such a system is adequate for the set of its theses.


Adolf Lindenbaum, September, 1922

Adolf Lindenbaum: Notes on his Life, with Bibliography and Selected References

III. Effectiveness issues
The idea of effectiveness (in a broad sense of the word) and its importance had gradually established itself by the middle of the 20th century, when the notion of cardinal number and that of effective method, that is computability, were fully realized. The problems like the followingwere raised and solved: whether a deductive system formulated in a countable language can always have a finite logical matrix adequate for its theorems (J.C.C. McKinsey and A. Tarski); whether it is effectively decidable that any two finite logical matrices or any two finite bundles have the same set of theorems (J. Kalicki for matrices, A. Citkin for bundles); whether any deductive system formulated in a countable formal language can be determined by a single denumerable matrix (A. Wro´nski). Some of these problems and related to them, as well as the finite model property of a system, will be discussed in this part

Bibliography
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