Jerzy Słupecki

S. Leśniewski’s Calculus of Names

The only primitive term in Leśniewski’s system of the Calculus of Names is the verb ‘is’ for which the participle ‘being’ corresponds to the Greek ‘oν’1 (gen. ‘oντoζ’). This was by no means the only reason for Leśniewski’s use for his system a name indicating one of the main branches of philosophy. Thus in Leśniewski’s article “On the Foundations of Mathematics”2 we read: … I used the term ‘ontology’ for the theory I developed, as this was not opposed to my ‘linguistic intuition’, just in view of the fact that I formulated in that theory a sort of ‘general principles of being’.

A logical system based on rules and its application in teaching mathematical logic

As it is known, the contemporary formal logic has sufficient means to for? malize each proof, in particular each mathematical proof. But it is also known that the formalization of proofs by means of logical laws gives often very long and burdensome constructions even in the case of simple intuitive proofs. Thus some divergences arise between the theoretical and practical scope of logic in this domain. Hence there appears the tendency to introduce such logical means which would enable us to eliminate these divergences and to formalize proofs in a practically simple and didactically easy manner. To bring logic as near as possible to the practice of proofs was one of the aims of the authors of the logical systems based on rules.1 It seems however that the existing systems of logic based on rules are only to a small extent pra? ctically and didactically exploited. In the present article we shall describe a logical system based on rules2. For some years we were using it in the lectures of formal logic and in some branches of mathematics. The practice showed that students easily and quickly learn the rules of the system and ? without the help of the teachers ? perform

St. Leśniewski’s Protothetics

In this paper I am discussing one of the three logical systems formulated by the late Professor Stanislaw Leśniewski, Ph.D., namely, that which he at first named logistics and later protothetics 1; on the two other systems, i.e. ontology and mereology I am working at present. As Professor Leśniewski’s manuscripts were destroyed during the Warsaw Uprising in 1944, the main source for this paper constitute the notes taken by students on his lectures, in the first place by the late Mr. Jerzy Billich of the Warsaw University. ‡ The initiative of collecting those notes which escaped destruction and of arranging systematically the result of Professor Leśniewski’ research work, was taken by Professor Tadeusz Kotarbinski, who asked me as one of Professor Leśniewski’s students to work on these notes. Professor Leśniewski left a great part of the results of his research work unpublished. Those which he did publish are often fragmentary and thus do not represent a complete picture of the systems he created. Besides, some of his papers are styled in an extremely difficult manner, while some others are at present almost unavailable. These are the reasons that Professor Leśniewski’s works are not much known, even in this country, and therefore subject to falling into oblivion, notwithstanding the fact that according to unanimous opinion of those who know them, they are of lasting value.

A variant of the proof of the completeness of the first order functional calculus

It will be proved that every true formula of the first order functional cal? culus is valid in it. The first proof of this theorem known as the theorem of comple? teness of the functional calculus of first order was given by Kurt G?del in 1930. The present proof1 follows those involving the concept of prime ideal, especially the proof given by Juliusz Reichbach in [I]2. The characteristic feature of the present proof is that the general theory of deductive systems constructed by A. Tarski in [3] is essential for it, and espe? cially Lindenbaums theorem on complete supersystems. The proof is similar to the proof of completeness of sentential calculus presented in [2]. 1. We assume that the letters X, Y, Z, ... in the axioms of Tarskis theory designate subsets of the set of all meaningful formulae, and the letters a, ?, y,... the elements of this set. This convention will be observed in further remarks on Tarskis theory. Al. ?< Ko A2. X C CnX c 5 A3. If XcY, then CnX C CnY A4. CnCnX C CnX

Proof of axiomatizability of full many-valued systems of calculus of propositions

By a matrix3 we shall mean every ordering which ascribes a value of propositional function to an arbitrary set of values of its arguments. A system of calculus of propo? sitions will be said to have a matrix presentation if there are given matrices of all its primitive terms and the meaningful propositions which are numbered into the system are exactly these ones which for arbitrary sets of values of their arguments take dis? tinguished values only, according to matrices of functors which appear in the propo? sitions. Or ? to make it short ? the propositions numbered into the system are these and only these which are satisfied by appropriate matrices. If the primitive terms of a given system of calculus of propositions allow to define

О правилах исчисления предложе ий

РезюмеСущественной переменной выражения исчисления предложений мы называем переме ную, значение которой при некотором сочетании значений других переменных влияет на значение целого выражения.