The idea of rejection originated by Aristotle. The notion of rejection was introduced into formal logic by Łukasiewicz [20]. He applied it to complete syntactic characterization of deductive systems using an axiomatic method of rejection of propositions [22, 23]. The paper gives not only genesis, but also development and generalization of the notion of rejection. It also emphasizes the methodological approach to biaspectual axiomatic method of characterization of deductive systems as acceptance (asserted) systems and rejection (refutation) systems, introduced by Łukasiewicz (...) and developed by his student Słupecki, the pioneers of the method, which becomes relevant in modern approaches to logic. (shrink)

This paper examines relations between reism, the metaphysical theory invented by Tadeusz Kotarbi?ski, and Le?niewski's calculus of names. It is shown that Kotarbi?ski's interpretation of common nouns as genuine names, i.e. names of things is essentially based on Le?niewski's logical ideas. It is pointed out that Le?niewskian semantics offers better prospects for nominalism than does semantics of the standard firstorder predicate calculus.

In [6] I tried to show how an objection to “the nominalist's” analysis of “This is red” and “That is red” on the basis of “the doctrine of common names” might be overcome. The objection is that “the nominalist,” attempting to analyze and by construing the pronouns in these sentences as two different proper names and “red” as a common name, is forced thereby to construe the copula in both sentences as the “is” of identity, and hence this and that (...) are identical, i.e., that there is only one red spot and not two. I attempted to show that by using Leśniewski's original axiom of ontology “the nominalist” could construe the pronouns in and as proper names, and “red” as a common name without taking the copula to express identity; he would not be forced to identify this with that. (shrink)

In some systems of Legniewskian Ontology were introduced as a toolkit for Husserl's and Meinong's theory of objects. Here such consi- deration is extended to Brentano-Husserl's theory of time. So-called antidiodo- rean logics are used as the foundations of the approach undertaken.

Sobocinski in his paper on Leśniewski's solution to Russell's paradox (1949b) argued that Leśniewski has succeeded in explaining it away. The general strategy of this alleged explanation is presented. The key element of this attempt is the distinction between the collective (mereological) and the distributive (set-theoretic) understanding of the set. The mereological part of the solution, although correct, is likely to fall short of providing foundations of mathematics. I argue that the remaining part of the solution which suggests a specific (...) reading of the distributive interpretation is unacceptable. It follows from it that every individual is an element of every individual. Finally, another Leśniewskian-style approach which uses so-called higher-order epsilon connectives is used and its weakness is indicated. (shrink)

A theory of definitions which places the eliminability and conservativeness requirements on definitions is usually called the standard theory. We examine a persistent myth which credits this theory to Leśniewski, a Polish logician. After a brief survey of its origins, we show that the myth is highly dubious. First, no place in Leśniewski's published or unpublished work is known where the standard conditions are discussed. Second, Leśniewski's own logical theories allow for creative definitions. Third, Leśniewski's celebrated ‘rules of definition’ lay (...) merely syntactical restrictions on the form of definitions: they do not provide definitions with such meta-theoretical requirements as eliminability or conservativeness. On the positive side, we point out that among the Polish logicians, in the 1920s and 1930s, a study of these meta-theoretical conditions is more readily found in the works of Łukasiewicz and Ajdukiewicz. (shrink)

Let EOA be the elementary ontology augmented by an additional axiom S (S S), and let LS be the monadic second-order predicate logic. We show that the mapping which was introduced by V. A. Smirnov is an embedding of EOA into LS. We also give an embedding of LS into EOA.

A structure A for the language L, which is the first-order language (without equality) whose only nonlogical symbol is the binary predicate symbol , is called a quasi -struoture iff (a) the universe A of A consists of sets and (b) a b is true in A ([p) a = {p } & p b] for every a and b in A, where a(b) is the name of a (b). A quasi -structure A is called an -structure iff (c) {p (...) } A whenever p a A. Then a closed formula in L is derivable from Leniewski's axiom x, y[x y u (u x) u; v(u, v x u v) u(u x u y)] (from the axiom x, y(x y x x) x, y, z(x y z y x z)) iff is true in every -structure (in every quasi -structure). (shrink)

This paper discusses the resolution principle in the context of non-classical logics.

LetEO be the elementary ontology of Leniewski formalized as in Iwanu [1], and letLS be the monadic second-order calculus of predicates. In this paper we give an example of a recursive function , defined on the formulas of the language ofEO with values in the set of formulas of the language of LS, such that EO A iff LS (A) for each formulaA.

In [6] I tried to show how an objection to “the nominalist's” analysis of (a) “This is red” and (b) “That is red” on the basis of “the doctrine of common names” might be overcome. The objection is that “the nominalist,” attempting to analyze (a) and (b) by construing the pronouns in these sentences as two different proper names and “red” as a common name, is forced thereby to construe the copula in both sentences as the “is” of identity, and (...) hence (it is claimed)thisandthatare identical, i.e., that there is only one red spot and not two. I attempted to show that by using Leśniewski's original axiom ofontology“the nominalist” could construe the pronouns in (a) and (b) as proper names, and “red” as a common name without taking the copula to express identity; he would not be forced to identifythiswiththat. (shrink)

In [6] I tried to show how an objection to “the nominalist's” analysis of “This is red” and “That is red” on the basis of “the doctrine of common names” might be overcome. The objection is that “the nominalist,” attempting to analyze and by construing the pronouns in these sentences as two different proper names and “red” as a common name, is forced thereby to construe the copula in both sentences as the “is” of identity, and hence this and that (...) are identical, i.e., that there is only one red spot and not two. I attempted to show that by using Leśniewski's original axiom of ontology “the nominalist” could construe the pronouns in and as proper names, and “red” as a common name without taking the copula to express identity; he would not be forced to identify this with that. (shrink)

Stanisław Leśniewski developed a system of logic and foundations of mathematics that considerably differs from Russell and Whitehead’s system. The difference between these two approaches to logic is significant primarily in the case of Leśniewski’s calculus of names, Ontology, and the concept of names that it contains. Russell’s theory of descriptions played a much more important role than Leśniewski’s concept of names in the history of philosophy. In response to that, several researchers aimed to approximate Leśniewski’s concept of names to (...) Russell’s theory. There are several such attempts. This paper presents an interpretation that was suggested by Arthur Prior. Prior interpreted Leśniewskian names as class names. This interpretation was rejected by such prominent Leśniewskian scholars as Peter Simons or Rafal Urbaniak. According to Simons, class names oppose Leśniewski’s ontological views. Urbaniak claimed that Prior’s interpretation of Leśniewskian names as class names is unclear. The aim of this paper is to demonstrate that from the formal point of view, Prior’s interpretation could be justified. (shrink)

A first-order formulation of Leśniewski's ontology is formulated and shown to be interpretable within a free first-order logic of identity extended to include nominal quantification over proper and common-name concepts. The latter theory is then shown to be interpretable in monadic second-order predicate logic, which shows that the first-order part of Leśniewski's ontology is decidable.

This paper is composed of two independent parts. The first is concerned with Russell’s early philosophy of mathematics and his quarrel with Poincaré about the nature of their opposition. I argue that the main divergence between the two philosophers was about the nature of definitions. In the second part, I briefly present Le!niewski’s Ontology and suggest that Le!niewski’s original treatment of definitions in the foundations of mathematics is the natural solution to the problem that divided Russell and Poincaré.

SummaryInterpreted distributively the sentence‘Indiana is a member of the class of American federal states’means the same as‘Indiana is an American federal state’. In accordance with the collective sense of class expressions the sentence can be understood as implying that Indiana is a part of the country whose capital city is Washington. Neither interpretation appears to accommodate all the intuitions connected with the informal notion of class. A closer accommodation can be achieved, it seems, if class expressions are interpreted as verb‐like (...) expressions of a certain kind available within the framework of Lesniewski's Ontology. (shrink)

A calculus of names is a logical theory describing relations between names. By a pure calculus of names we mean a quantifier-free formulation of such a theory, based on classical propositional calculus. An axiomatisation of a pure calculus of names is presented and its completeness is discussed. It is shown that the axiomatisation is complete in three different ways: with respect to a set theoretical model, with respect to Leśniewski's Ontology and in a sense defined with the use of axiomatic (...) rejection. The independence of axioms is proved. A decision procedure based on syntactic transformations and models defined in the domain of only two members is defined. (shrink)

When Russell argued for his ontological convictions, for instance that there are negative facts or that there are universals, he expressed himself in English. But Wittgenstein must have noticed that from the point of view of Russell's ideal language these ontological statements appear to be pseudo-propositions. He believed therefore that what these statements pretend to say, could not really be said but only shown. Carnap discovered a way out of this mutism: what in the material mode of speech of the (...) object language looks like a pseudo-proposition can be translated into a perfectly meaningful proposition in the formal mode of speech (in the metalinguistic mode of speech of the logical syntax of language). But is this ascent into the metalanguage necessary? Taking advantage of Lésniewski's logical system there exists another way outwe can expand the number of categories of our ideal language. But Leniewski's formulas raise another profound problem, the problem of semantical muteness (cf. W. G. Lycan Semantic Competence and Funny Functors Monist 64 (1979), 209–222). (shrink)

In this paper, we shall give another proof of the faithfulness of Blass translation of the propositional fragment \ of Leśniewski’s ontology in the modal logic \ by means of Hintikka formula. And we extend the result to von Wright-type deontic logics, i.e., ten Smiley-Hanson systems of monadic deontic logic. As a result of observing the proofs we shall give general theorems on the faithfulness of B-translation with respect to normal modal logics complete to certain sets of well-known accessibility relations (...) with a restriction that transitivity and symmetry are not set at the same time. As an application of the theorems, for example, B-translation is faithful for the provability logic \ ), that is, \ \ \ \supset \Box \phi \). The faithfulness also holds for normal modal logics, e.g., \, \, \, \. We shall conclude this paper with the section of some open problems and conjectures. (shrink)

In this paper, we shall show that the following translation \(I^M\) from the propositional fragment \(\bf L_1\) of Leśniewski's ontology to modal logic \(\bf KTB\) is sound: for any formula \(\phi\) and \(\psi\) of \(\bf L_1\), it is defined as (M1) \(I^M(\phi \vee \psi) = I^M(\phi) \vee I^M(\psi)\), (M2) \(I^M(\neg \phi) = \neg I^M(\phi)\), (M3) \(I^M(\epsilon ab) = \Diamond p_a \supset p_a. \wedge. \Box p_a \supset \Box p_b.\wedge. \Diamond p_b \supset p_a\), where \(p_a\) and \(p_b\) are propositional variables corresponding to (...) the name variables \(a\) and \(b\), respectively. In the last, we shall give some comments including some open problems and my conjectures. (shrink)

With material on his early philosophical views, his contributions to set theory and his work on nominalism and higher-order quantification, this book offers a uniquely expansive critical commentary on one of analytical philosophy’s great ...

The idea of rejection originated by Aristotle. The notion of rejection was introduced into formal logic by Łukasiewicz [20]. He applied it to complete syntactic characterization of deductive systems using an axiomatic method of rejection of propositions [22, 23]. The paper gives not only genesis, but also development and generalization of the notion of rejection. It also emphasizes the methodological approach to biaspectual axiomatic method of characterization of deductive systems as acceptance (asserted) systems and rejection (refutation) systems, introduced by Łukasiewicz (...) and developed by his student Słupecki, the pioneers of the method, which becomes relevant in modern approaches to logic. (shrink)