This paper has four parts. In the first part, I present Leniewski's protothetics and the complete system provided for that logic by Henkin. The second part presents a generalized notion of partial functions in propositional type theory. In the third part, these partial functions are used to define partial interpretations for protothetics. Finally, I present in the fourth part a complete system for partial protothetics. Completeness is proved by Henkin's method [4] using saturated sets instead of maximally saturated sets. This (...) technique provides a canonical representation of a partial semantic space and it is suggested that this space can be interpreted as an epistemic state of a non-omniscient agent. (shrink)

The volume collects many of the most significant commentaries on, and contributions to, Protothetic. A Protothetic Bibliography is included.

"Lesniewski defined ontology, one of his three foundational systems, as 'a certain kind of modernized 'traditional logic' [On the foundations of mathematics (FM), p. 176]. In this respect it is worth bearing in mind that in the 1937-38 academic year Lesniewski taught a course called "Traditional 'formal logic' and traditional 'set theory' on the ground of ontology"; cf. Srzednicki and Stachniak, S. Lesniewski's Systems. Protothetic, 1988, p. 180. On this see Kotarbinski Gnosiology. The scientific approach to the theory of (...) knowledge, 1966, pp. 253-54 [the Polish original was published in 1929], which Lesniewski praised in [FM]: see in particular pp. 373 ff. Kotarbinski noted that Lesniewski "calls his system 'ontology' in harmony with certain terms used earlier (as in.. (shrink)

It was noted recently in [4] that the definitions in a certain system of protothetic cease to be creative if one extends the rule of substitution so that direct substitution of the so called incomplete expressions for functor variables becomes possible. However, precise formulations were omitted in [4]. Here, we describe the new rule in some details and formulate some relevant theorems. For more circumstantial discussion, see [5].

Nondeterministic exponential time complexity bounds are established for recognizing true propositional formulas with partially ordered quantifiers on propositional variables.

Stanisław Leśniewski was one of the two originators and drivers of the Warsaw School of logic. This article describes his work chronologically, from his early philosophical work in Lvov to his highly original logical systems of protothetic, ontology and mereology. His struggles to overcome logical antinomies, his absolute commitment to logical clarity and precision, and his antipathy towards set theory made his nominalistic approach to logic among the most original of the twentieth century, while his early death and the (...) loss of his papers meant his work was only gradually discovered and appreciated outside Poland. (shrink)

Due to the current availability of the English translation of almost all of Lesniewski's works it is now possible to give a clear and detailed picture of his ideas. Lesniewski's system of the foundation of mathematics is discussed. In abrief ouüine of his three systems Mereology, Ontology and Protothetics his positions conceming the problems of the forms of expression, proper names, synonymity, analytic and synthetic propositions, existential propositions, the concept of logic, and his views of theory of science and metaphysics (...) are sketched. The influence of Mill, Lukasiewicz, Austrian philosophy and especially Petrazycki on his thinking is evaluated and an interpretation is suggested setting him squarely in a tradition of classical Aristotelian logic. (shrink)

According to the author, no body of theories deserves to be called philosophy unless some of these theories and problems fall within the province of metphysics. No problems deserves the name of metaphysics unless some of them add up to constitute ontology. The author presents the reistic version of the science of being as the union of Protothetic, Ontology, Mereology and Chronology. The next possible step in the construction of the reistic ontology will be Stereology (a kind of reistic (...) geometry). (shrink)

The most difficult problem that Leniewski came across in constructing his system of the foundations of mathematics was the problem of defining definitions, as he used to put it. He solved it to his satisfaction only when he had completed the formalization of his protothetic and ontology. By formalization of a deductive system one ought to understand in this context the statement, as precise and unambiguous as possible, of the conditions an expression has to satisfy if it is added (...) to the system as a new thesis. Now, some protothetical theses, and some ontological ones, included in the respective systems, happen to be definitions. In the present essay I employ Leniewski's method of terminological explanations for the purpose of formalizing ukasiewicz's system of implicational calculus of propositions, which system, without having recourse to quantification, I first extended some time ago into a functionally complete system. This I achieved by allowing for a rule of implicational definitions, which enabled me to define any propositionforming functor for any finite number of propositional arguments. (shrink)

The publication of this book constitutes a real service to students of logic and of the foundations and philosophy of mathematics. Here, "under one roof," are translations of seventeen of the most important papers on logic and metalogic by Ajdukiewicz, Chwistek, Jaskowski, Jordan, Lesniewski, Lukasiewicz, S upecki, Sobocinski, and Wajsberg. All but two of them appear in English for the first time. Notably absent are papers by Alfred Tarski, but this omission is fully justified in view of the publisher's well-known (...) companion volume devoted exclusively to this most important logician. Within the contents of this volume, the major topics covered are various systems of sentential logic ; philosophically-motivated foundational studies ; miscellaneous papers on logical syntax, intuitionism and metalogic; and historical studies of the development of Polish logic. One cautionary note: the protothetic notation of Lesniewski is extremely difficult to read. Before undertaking that author's articles, the reader will do well to turn to the explanatory footnotes in the subsequent articles by Sobocinski. The book has a bibliography, but no index. In general, this is a worthwhile and long overdue book.—H. P. K. (shrink)

History and Philosophy of Logic, Volume 32, Issue 2, Page 191-193, May 2011.