Systems of Leśniewski's ontology with the functor of weak inclusion as the only primitive term

Studia Logica ◽  
1977 ◽  
Vol 36 (4) ◽  
pp. 323-349 ◽  
Author(s):  
Czesław Lejewski
Keyword(s):  
Primitive Term

Computational logic

1949 ◽  
Vol 14 (3) ◽  
pp. 167-172 ◽  
Author(s):  
Nathan P. Levin
Keyword(s):  
Computational Logic ◽  
Elementary Algebra ◽  
Sentential Calculus ◽  
Primitive Term ◽  
Arbitrary Sign ◽  
Image Position ◽  
Binary Combination

We make certain notational conventions. These are referred to as CL (Computational Logic). The relation of interchangeability is introduced as the basic connection between logical formulas. This approach lends perspicuity to the results of sentential calculus. With its technical devices CL is able to rephrase logical theorems in rather succinct manner. Our exposition tries to steer a middle course between informality and strict rigor.It is felt that the method of CL offers advantages for the teaching of logic. Proofs are algorithmic and resemble those of elementary algebra. All inferences are reversible and practically non-tentative.Sign is a primitive term of CL. Its denning property is a capacity for entering into binary combination with other signs, or with itself, according to this convention:x and y are signs if, and only if, (x y) is a sign. Bracketing is looked upon as an operation on signs in terms of which other operations are definable. The practice of some authors in classifying brackets under the heading of symbols, seems to us questionable; for unlike symbols or signs brackets are never used to denote anything. Brackets enter into the composition of signs, not to denote a grouping, but rather to exhibit it, in the manner of a diagram.An unending list of letters, with or without subscripts,serve to denote arbitrary signs. An arbitrary sign may or may not have other signs as parts. The numeral “2” is used as a constant. Bracketing abbreviation is as follows:and so on. Outermost brackets will ordinarily be omitted. This kind of bracketing may be termed left-associative. For convenience, these bracketing conventions are crystallized into a rule.

A revision of the fundamental rules of combinatory logic

1941 ◽  
Vol 6 (2) ◽  
pp. 41-53 ◽  
Author(s):  
Haskell B. Curry
Keyword(s):  
Traditional Method ◽  
General Type ◽  
Original System ◽  
Combinatory Logic ◽  
Full System ◽  
Primitive Term ◽  
Rules Of Procedure ◽  
Simple Character

The purpose of this paper is to put on record some theorems relating to improvements in the primitive frame of combinatory logic. These improvements were, for the most part, suggested by the work of Rosser, who formulated a weakened system of combinatory logic in which the rules had a simple character not possessed by those of the original system. In the latter the rules B, C, W, K were in reality axiom-schemes, and their postulation amounted to assuming infinitely many axioms. Rosser had rules of procedure such that no propositions were deducible from them except in combination with axioms (or previously proved propositions); moreover, the conclusion of each rule was uniquely determined by the premises. He also eliminated equality as a primitive term, defining it (essentially) according to the traditional method. This paper shows that these and related advantages apply to certain formulations of the full system of combinatory logic, so far as it concerns the theory of combinators.The method of procedure is as follows. Instead of setting up a primitive frame at the start and then deriving its properties, I begin (after some preliminary explanations in §2) by stating in §3 the properties which it is desired to establish. The next few sections are devoted to the formulation and proof of certain general theorems concerning possible bases for the system of §3. These theorems are, perhaps, more general than is necessary for the immediate purpose, but they are of some interest on their own account. A formulation in terms of the primitives of original system (i.e., B, C, W, and K), which is of the same general type as Rosser's formulation, is obtained at the end of §6. In §7 are discussed the changes in this formulation which are sufficient in order to base it on the primitive combinators S and K of Schönfinkel.

Substantival Self: A Primitive Term for a Sociological Psychology1

1975 ◽  
Vol 5 (1) ◽  
pp. 43-62 ◽  
Author(s):  
Andrew J. Weigert
Keyword(s):  
Primitive Term

On the Primitive Term of Logistic

1998 ◽  
pp. 43-68
Author(s):  
Alfred Tajtelbaum-Tarski
Keyword(s):  
Primitive Term

Reviews - J. H. Woodger. Translator's preface. Logic, semantics, metamathematics, papers from 1923 to 1938.Oxford at the Clarendon Press, London1956, pp. vii–ix. - Alfred Tarski. Author's acknowledgments.Logic, semantics, metamathematics, papers from 1923 to 1938.Oxford at the Clarendon Press, London1956, pp. xi–xii. - Alfred Tarski. On the primitive term of logistic. Modified English translation based on 2852–4. Logic, semantics, metamathematics, papers from 1923 to 1938.Oxford at the Clarendon Press, London1956, pp. 1–23. - Alfred Tarski. Foundations of the geometry of solids.Logic, semantics, metamathematics, papers from 1923 to 1938.Oxford at the Clarendon Press, London1956, pp. 24–29. (Translated, with additions, from Księga Pamiątkowa Pierwszego Polskiego Zjazdu Matematycznego, supplement to Annales de la Société Polonaise de Mathématique, Cracow 1929, pp. 29-33.) - Alfred Tarski. On some fundamental concepts of metamathematics. English translation of 2857. Logic, semantics, metamathematics, papers from 1923 to 1938.Oxford at the Clarendon Press, London1956, 30–37. - Jan Łukasiewicz and Alfred Tarski. Investigations into the sentential calculus. English translation of 4077, with added footnotes. Logic, semantics, metamathematics, papers from 1923 to 1938.Oxford at the Clarendon Press, London1956, pp. 38–59. - Alfred Tarski. Fundamental concepts of the methodology of the deductive sciences. English translation of 2858, with added footnotes. Logic, semantics, metamathematics, papers from 1923 to 1938.Oxford at the Clarendon Press, London1956, pp. 60–109. - Alfred Tarski. On definable sets of real numbers. English translation of 28510, with additions in the text by the author as well as added footnotes. Logic, semantics, metamathematics, papers from 1923 to 1938.Oxford at the Clarendon Press, London1956, pp. 110–142. - Kazimierz Kuratowski and Alfred Tarski. Logical operations and projective sets. English translation of 4321, with added footnotes. Logic, semantics, metamathematics, papers from 1923 to 1938.Oxford at the Clarendon Press, London1956, pp. 143–151. - Alfred Tarski. The concept of truth in formalized languages. English translation of 28516, with added footnotes. Logic, semantics, metamathematics, papers from 1923 to 1938.Oxford at the Clarendon Press, London1956, pp. 152–278.

1969 ◽  
Vol 34 (1) ◽  
pp. 99-106 ◽  
Author(s):  
W. A. Pogorzelski ◽  
S. J. Surma
Keyword(s):  
English Translation ◽  
Real Numbers ◽  
Sentential Calculus ◽  
Alfred Tarski ◽  
Primitive Term ◽  
Logical Operations ◽  
Definable Sets ◽  
Concept Of Truth

Systems of the propositional and of the functional calculus based on one primitive term

Studia Logica ◽  
1957 ◽  
Vol 6 (1) ◽  
pp. 7-55 ◽  
Author(s):  
Ludwik Borkowski
Keyword(s):  
Functional Calculus ◽  
Primitive Term

Formalization of functionally complete propositional calculus with the functor of implication as the only primitive term

Studia Logica ◽  
1989 ◽  
Vol 48 (4) ◽  
pp. 479-494 ◽  
Author(s):  
Czes?aw Lejewski
Keyword(s):  
Propositional Calculus ◽  
Primitive Term

Relation of Leśniewski's mereology to Boolean algebra

1974 ◽  
Vol 39 (4) ◽  
pp. 638-648 ◽  
Author(s):  
Robert E. Clay
Keyword(s):  
Boolean Algebra ◽  
Final Stage ◽  
Propositional Calculus ◽  
Ordinary Language ◽  
Complete Boolean Algebra ◽  
Primitive Term ◽  
Logical Foundation ◽  
Formal Relations

It has been stated by Tarski [5] and “proved” by Grzegorczyk [3] that: (A) The models of mereology and the models of complete Boolean algebra with zero deleted are identical.Proved has been put in quotes, not because Grzegorczyk's proof is faulty but because the system he describes as mereology is in fact not Leśniewski's mereology.Leśniewski's first attempt at describing the collective class, i.e. mereology, was done in ordinary language with no rigorous logical foundation. In describing the collective class, he needed to use the notion of distributive class. So as to clearly distinguish and expose the interplay between the two notions of class, he introduced his calculus of name (name being the distributive notion), which is also called ontology, since he used the primitive term “is.” At this stage, mereology included ontology. Then, in order to have a logically rigorous system, he developed as a basis, a propositional calculus with quantifiers and semantical categories (types), called protothetic. At this final stage, what is properly called mereology includes both protothetic and ontology.What Grzegorczyk describes as mereology is even weaker than Leśniewski's initial version. To quote from [3]: “In order to emphasize these formal relations let us consider the systems of axioms of mereology for another of its primitive terms, namely for the term “ingr” defined as follows:A ingr B.≡.A is a part B ∨ A is identical to B.The proposition “A ingr B” can be read “A is contained in B” or after Leśniewski, “A is ingredient of B”.”

scholarly journals The Zahl-Anzahl Distinction in Gottlob Frege: Arithmetic of Natural Numbers with Anzahl as a Primitive Term

Axioms ◽  
2019 ◽  
Vol 9 (1) ◽  
pp. 6
Author(s):  
Eugeniusz Wojciechowski
Keyword(s):  
Natural Number ◽  
Gottlob Frege ◽  
Natural Numbers ◽  
Primitive Term ◽  
Starting Point ◽  
Elementary Ontology

The starting point is Peano’s expression of the axiomatics of natural numbers in the framework of Leśniewski’s elementary ontology. The author enriches elementary ontology with the so-called Frege’s predication scheme and goes on to propose the formulations of this axiomatic, in which the original natural number (N) term is replaced by the term Anzahl (A). The functor of the successor (S) is defined in it.

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