STUDIA
LOGICA
Tom XIX
-
1966
K A Z I MIERZ AJDUKIEWICZ
FROM THE METHODOLOGY OF THE DEDUCTIVE SCIENCES (Translated by Jerzy Giedymin) E d i t o r ' s n o t e : K. Ajdukiewicz's book From the methodology of the deductive sciences* appeared in 1920 (post-dated.) I t consists of three essays: 1) The logical concept of proof, 2) On proofs of consistency of axioms, 3) On the notion of existence in deductive sciences. T h e first of these essays contains defirLitions of some fundamental concepts of the methodology of the deductive sciences, scil. definitions of the concepts of proof, of logical thesis, of a thesis of a deductive theory and a syntactical definition of the concept of consequence. All these definitions differ b u t slightly in their basic content from contemporary formulations. It should be emphasized that the definition of the fundamental syntactical concept of consequence has been formulated by Ajdukiewicz in a way which to-day would be called purely metalogical. A n d yet it was a period w h e n implication ,,p ~ q" was read as p entails q, which gave rise to various misunderstandings and caused L e w i s to construct -- in order to define the concept of consequence -- his system of strict implication in which again the expression of this system ,,p -< q" is read meta-linguisticaUy: q is deducible from p. T h e definition given by Ajdukiewicz avoids these misunderstandings and difficulties. I n the second essay the A u t h o r analyses, in a fashion conforming with our contemporary approach, proofs of consistency of axioms by interpretation, discusses the problem of existence of absolute consistency proofs, examines the proofs of consistency of logic and of arithmetic given by H i l b e r t a n d refutes arguments claiming t h a t every proof of consistency of the axioms of logic would have to be circular. T h e Author employs here concepts defined in the first essay, in particular the concept of consequence. W h e n analyzing consistency proofs by interpretation he gives an interesting formulation implied by the definition of this concept. I n the t h i r d essay, dealing with the concept of existence in deductive sciences, the Author is not concerned with the problem which was in those days the main interest of philosophers, scil. the problem what kind of existence is attributable to the objects of deductive sciences. He is concerned instead with a different problem of greater relevance, so it seems, from the point of view of contemporary theory of the deductive sciences, scil. with the problem what is the mean;ng of the word "exist" in deductive sciences. It is shown there that existence in deductive sciences cannot be identified with consistency, neither can consistency be regarded as a sufficient condition of existence. I n formulating his own answer to the question u n d e r discussion, the A u t h o r employs semantical concepts of satisfaction, t r u t h and the domain of a theory, which -in a considerably later period -- would be attentively studied in logic. * KAZIMIERZ AJDUKIEWICZ: Z metodologii nauk dedukcyjnych, Lw6w 1921, Naldadem Polskiego Towarzystwa Filozoficznego (Wydawnictwo Polskiego Towarzystwa Filozoficznego we Lwowie, T o m X), Pp. 63. [91
10
[2]
K. Ajdukicwicz
T h e book From the methodology of the deductive sdences was a pioneering work which b y several years preceded in Poland t h e main t r e n d o f the studies in the methodoIogy of the deductive sciences. As has been said b y the A u t h o r himself in the preface to the first volume o f his selected writings Language and knowledge, the. essay The logical concept of proof contained in the book, "was the first Polish contribution to the methodology of the deductive sciences written u n d e r t h e influence of mathematical logic. I t initiated, at least in Poland, the structural m e t h o d o f defining methodological concepts (e.g. the concepts o f p r o o f a n d of logical entailmeat), which later o n was to play a n essential role in the striking development of the theory of deductive systems, to which the n a m e metamathematics was given". T h e essay The logical concept of proof played a historically i m p o r t a n t role in the discovery of the deduction theorem. T h e discovery was made by A. T a r s k i in 1921 in connection -- as Tarski himself explains -- with the analysis contained in t h a t essay. W h e n defining the concept of p r o o f Ajdukiewicz refers to substitutions in the logical axioms a n d to substitutions in deftnitions b u t does not introduce a rule of substitution as a rule leading to new steps in the p r o o f T h i s is why deduction theorem is essential for his definition of the concept of consequence Moreover there are in Ajdukiewicz's book some formulations equlform (though not equlsignificant) with the contemporary formulation of the deduction theorem. T h e y m i g h t have suggested a formulation and a p r o o f of this theorem given a different definition of the concept of consequence which occurred in them. Some of the more important results contained in the book h a d been presented in abstracts of lectures given at the meetings of the Philosophical Society in Lw6w in 1919 and in 1920 and published in the fifth volume (1919-- 1920) of ,,Ruch Filozoficzny". Particularly interesting is the abstract of the lecture A definition of the logical concept of proof because of a certain formulation of the definition of the concept of consequence. As one of the earliest contributions to logic a n d methodology in the interwar period in Poland Ajclukiewicz's book ought to be made accessible to those interested in the history of contemporary logic and this is just why it now appears in an English translation.
Ludwik Borkowski
