Lesniewski's Ontology and Godel's Incompleteness Theorem: A Dissertation Submitted to the Graduate School of the University of Notre Dame in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy. John Thomas Canty.

Lesniewski's Ontology and Godel's Incompleteness Theorem: A Dissertation Submitted to the Graduate School of the University of Notre Dame in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

Notre Dame, Indiana: Department of Philosophy, 1967. Large Hardcover. Near Fine / No Jacket. Item #2326327

No jacket. Two title pages.

314 pp. "Leśniewski 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 Leśniewski taught a course called "Traditional 'formal logic' and traditional 'set theory' on the ground of ontology"; cf. Srzednicki and Stachniak, Leśniewski'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 Leśniewski praised in [FM]: see in particular pp. 373 ff. Kotarbinski noted that Leśniewski "calls his system 'ontology' in harmony with certain terms used earlier (as in the 'ontological principle of contradiction')", and in strict relation to the Greek root of 'ontology' as the participle of the verb 'to be'. Leśniewski's 'ontology' is therefore "closely connected with traditional Aristotelian formal logic, of which it is an extension and an improvement, while on the other hand it is a terminal point in the attempt to construct a calculus of names in the area of logistic ... If in spite of these reasons we do not use the word 'ontology' here as a name for the calculus of names, this is only because of the fear of a misunderstanding. Misunderstanding could arise from the fact that this name has its roots already in another role, i.e., it has been long agreed to call 'ontology' the enquiry 'on the general principles of existence' conducted in the spirit of certain parts of Aristotelian 'metaphysical' books. It has to be admitted however, that if the Aristotelian definition of the main theory (prote filosofia) discussed in those books is interpreted in the spirit of a 'general theory of objects', then both the word and its meaning, can be applied to the calculus of names of Leśniewski", Kotarbinski 1966, pp. 373-374. Leśniewski commented on Kotarbinski's remarks thus: "I used the name 'Ontology' to characterize the theory I was developing, without offence to my 'linguistic instincts' because I was formulating in that theory a certain kind of 'general principles of existence"' [FM, 374]. Given these premises, we gain clearer understanding of his interest in the principles of non-contradiction [PC] and excluded middle [EM], as well as his references to the theory of conversion (p. 68 ff), of the suppositio (p. 18) and of the validity of the syllogism (p. 71 ff). This inquiry was encouraged by his interest in the history of logic and in the formal treatment of the problems of classical philosophy by the Lvov-Warsaw school. Jan Łukasiewicz's (1886-1939) research into the history of propositional calculus, the Aristotelian syllogistic and the principle of non-contradiction are well known. (...) Twardowski, the founder of the school, was also interested in traditional logic. As a lecturer at the University of Lvov, for many years he taught a course on Attempts to reform traditional logic, in which he outlined the theories of Bolzano, Brentano, Boole and Schröder; cf. Dambska François Brentano et la pensée philosophique en Pologne: Casimir Twardowski et son École, Grazer philosophischen Studien, 5, 1978, p. 123." (pp. 187-188). From: Roberto Poli and Massimo Libardi, "Logic, Theory of Science, and Metaphysics According to Stanislaw Leśniewski", Grazer Philosophische Studien, 57, 1999, pp. 183-219.--Ontology. Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The theorems are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible. The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.e., an algorithm) is capable of proving all truths about the arithmetic of natural numbers. For any such consistent formal system, there will always be statements about natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, an extension of the first, shows that the system cannot demonstrate its own consistency. Employing a diagonal argument, Gödel's incompleteness theorems were the first of several closely related theorems on the limitations of formal systems. They were followed by Tarski's undefinability theorem on the formal undefinability of truth, Church's proof that Hilbert's Entscheidungsproblem is unsolvable, and Turing's theorem that there is no algorithm to solve the halting problem.--Wikipedia

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