An overview of contemporary part-whole theories, with reference to both their axiomatic developments and their philosophical underpinnings.

During 1927-1931 Leśniewski published a series of articles (169 pages) entitled 'O podstawach matematyki' [On the Foundations of Mathematics] in the journal Przeglad Filozoficzny [Philosophical Review], and an abridged English translation of this series is presented here. With the exception of this work, all of Leśniewski's publications appearing after the first World War were written in German, and hence accessible to scholars and logicians in the West. This work, however, since written in Polish, has heretofore not been accessible to most (...) Western readers, and it is hoped that this translation will encourage both the study of Leśniewski's works as well as the further development of his theories. (shrink)

An axiomatic treatment of the relation part of is shown to lead naturally to an account of the ways in which parts of things are matched. The determination of matchings by the properties of parts and by the relations between parts is discussed and shown to be relevant to certain classificatory problems in science. The connexions between matchings and symmetries of parts are explored, and a general account is given of the ways in which ambiguities in the matching of parts (...) may be resolved. (shrink)

This is the first, out of two papers, devoted to Andrzej Grzegorczyk’s point-free system of topology from Grzegorczyk :228–235, 1960. https://doi.org/10.1007/BF00485101). His system was one of the very first fully fledged axiomatizations of topology based on the notions of region, parthood and separation. Its peculiar and interesting feature is the definition of point, whose intention is to grasp our geometrical intuitions of points as systems of shrinking regions of space. In this part we analyze separation structures and Grzegorczyk structures, and (...) establish their properties which will be useful in the sequel. We prove that in the class of Urysohn spaces with countable chain condition, to every topologically interpreted representative of a point in the sense of Grzegorczyk’s corresponds exactly one point of a space. We also demonstrate that Tychonoff first-countable spaces give rise to complete Grzegorczyk structures. The results established below will be used in the second part devoted to points and topological spaces. (shrink)

Mereological theories are theories based on a binary predicate ‘being a part of’. It is believed that such a predicate must at least define a partial ordering. A mereological theory can be obtained by adding on top of the basic axioms of partial orderings some of the other axioms posited based on pertinent philosophical insights. Though mereological theories have aroused quite a few philosophers’ interest recently, not much has been said about their meta-logical properties. In this paper, I will look (...) into whether those theories are decidable or not. Besides, since theories of Boolean algebras are in some sense upper bounds of mereological theories which can be found in the literature, I shall also make some observations about the possibility of getting mereological theories beyond Boolean algebras. (shrink)

This article is concerned to formulate some open problems in the philosophy of space and time that require methods characteristic of mathematical traditions in the foundations of geometry for their solution. In formulating the problems an effort has been made to fuse the separate traditions of the foundations of physics on the one hand and the foundations of geometry on the other. The first part of the paper deals with two classical problems in the geometry of space, that of giving (...) operationalism an exact foundation in the case of the measurement of spatial relations, and that of providing an adequate theory of approximation and error in a geometrical setting. The second part is concerned with physical space and space-time and deals mainly with topics concerning the axiomatic theory of bodies, the operational foundations of special relativity and the conceptual foundations of elementary physics. (shrink)

Stanislaw Lesniewski’s interests were, for the most part, more philosophical than mathematical. Prior to taking his doctorate at Jan Kazimierz University in Lvov, Lesniewski had spent time at several continental universities, apparently becoming relatively attached to the philosophy of one of his teachers, Hans Comelius, to the chapters of John Stuart Mill’s System of Logic that dealt specifically with semantics, and, in general, to studies of general grammar and philosophy of language. In these several early interests are already to be (...) found the roots of the work that was to occupy Lesniewski’s life: a search for a definitive doctrine of what sorts of things there are in the world, or better, of what language must be like if it is adequately and efficiently to represent the world. (shrink)

Rough sets are used in numerous knowledge representation contexts and are then empowered with varied ontologies. These may be intrinsically associated with ideas of rationality under certain conditions. In recent papers, specific granular generalisations of graded and variable precision rough sets are investigated by the present author from the perspective of rationality of approximations (and the associated semantics of rationality in approximate reasoning). The studies are extended to ideal-based approximations (sometimes referred to as subsethood-based approximations). It is additionally shown that (...) co-granular or point-wise approximations defined by σ-ideals/filters (for an arbitrary relation σ) fit easily into the entire scheme. Concepts of the rationality of objects (vague or crisp) and their types are introduced and are shown to be applicable to most general rough sets by the present author. Surprising results on these are proved on these by her in this part of the research paper. The present paper is the first of a three part study on the topic. (shrink)

Tadeusz Kotarbinski is one of towering figures in contemporary Polish philosophy. He was a great thinker, a great teacher, a great organizer of philosophical and scientific life, and, last but not least, a great moral authority. He died at the age of 96 on October 3, 1981. Kotarbinski was active in almost all branches of philosophy. He made many significant contributions to logic, semantics, ontology, epistemology, history of philosophy, and ethics. He created a new field, namely praxiology. Thus, using an (...) ancient distinction, he contributed to theoretical as well as practical philoso~hy. Kotarbinski regarded praxiology as his major philosophical "child". Doubtless, praxiology belongs to practical philosophy. This collection, howewer, is mainly devoted to Kotarbinski' s theoretical philosophy. Reism - Kotarbinski' s fundamental idea of ontology and semantics - is the central topic of most papers included here; even Pszczolowski' s essay on praxiology considers its ontological basis.,Only two papers, namely that of Zarnecka-Bialy and that of Wolenski, are not linked with reism. However, both fall under the general label "Kotarbinski: logic, semantics and ontology". The collection partly consists of earlier published papers. (shrink)

The signature of the formal language of mereology contains only one binary predicate P which stands for the relation “being a part of”. Traditionally, P must be a partial ordering, that is, ${\forall{x}Pxx, \forall{x}\forall{y}((Pxy\land Pyx)\to x=y)}$ and ${\forall{x}\forall{y}\forall{z}((Pxy\land Pyz)\to Pxz))}$ are three basic mereological axioms. The best-known mereological theory is “general extensional mereology”, which is axiomatized by the three basic axioms plus the following axiom and axiom schema: (Strong Supplementation) ${\forall{x}\forall{y}(\neg Pyx\to \exists z(Pzy\land \neg Ozx))}$ , where Oxy means ${\exists (...) z(Pzx\land Pzy)}$ , and (Fusion) ${\exists x\alpha \to \exists z\forall y(Oyz\leftrightarrow \exists x(\alpha \land Oyx))}$ , for any formula α where z and y do not occur free. In this paper, I will show that general extensional mereology is decidable, and will also point out that the decidability of the first-order approximation of the theory of complete Boolean algebras can be shown in the same way. (shrink)

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)

In this paper, using a propositional modal language extended with the window modality, we capture the first-order properties of various mereological theories. In this setting, $\Box \varphi $ reads all the parts (of the current object) are $\varphi $, interpreted on the models with a whole-part binary relation under various constraints. We show that all the usual mereological theories can be captured by modal formulas in our language via frame correspondence. We also correct a mistake in the existing completeness proof (...) for a basic system of mereology by providing a new construction of the canonical model. (shrink)

Leśniewski is known for his pedantry and idiosyncratic notation, which make it extremely difficult to read and follow. As reading comes before understanding, this paper therefore attempts only one...

Leibniz’s juvenile work De arte combinatoria of 1666 included the “Proof for the Existence of God.” This proof bears a mathematical character and is constructed in line with Euclid’s pattern. I attempted to logically formalize it in 1982. In this text, on the basis of then analysis and the contents of the proof, I seek to show what concept of substance Leibniz used on behalf of the proof. Besides, Leibnizian conception of the whole and part as well as Leibniz’s definitional (...) method have been reconstructed here. (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)

A 1971 paper by Roman Suszko, ‘Identity Connective and Modality’, claimed that a certain identity-free schema expressed the condition that there are at most two objects in the domain. Section 1 here gives that schema and enough of the background to this claim to explain Suszko’s own interest in it and related conditions—via non-Fregean logic, in which the objects in question are situations and the aim is to refrain from imposing this condition. Section 3 shows that the claim is false, (...) and suggests a diagnosis as to why it might have been made, in terms of quantifier scope distinctions as they bear on schematic formulations. In between, Section 2 discusses an issue brought up by this discussion but not involving the mistaken claim itself, and shows that Suszko was familiar with two different ways to set an upper or lower finite bound to the domain—a contrast which some contemporary readers may associate with David Lewis. (shrink)

The signature of the formal language of mereology contains only one binary predicate which stands for the relation “being a part of” and it has been strongly suggested that such a predicate must at least define a partial ordering. Mereological theories owe their origin to Leśniewski. However, some more recent authors, such as Simons as well as Casati and Varzi, have reformulated mereology in a way most logicians today are familiar with. It turns out that any theory which can be (...) formed by using the reformulated mereological axioms or axiom schemas is in a sense a subtheory of the elementary theory of Boolean algebras or of the theory of infinite atomic Boolean algebras. It is known that the theory of partial orderings is undecidable while the elementary theory of Boolean algebras and the theory of infinite atomic Boolean algebras are decidable. In this paper, I will look into the behaviors in terms of decidability of those mereological theories located in between. More precisely, I will give a comprehensive picture of the said issue by offering solutions to the open problems which I have raised in some of my papers published previously. (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)

Summary of the talk given to the 22nd Conference on the History of Logic, Cracow (Poland), July 5–9, 1976.

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)

ABSTRACT Leśniewskian Ontology (LO) is a system in which the basic subject-predicate formula takes the form of a b and express one-argument predication, e.g. John is a student. In LO’s language, there is no many-argument form of predication given that would allow for the structural expression of, for example, the sentence John is Anne’s son. In this article, a simple and natural extension of LO is suggested to encompass many-argument predication. The system thus obtained corresponds to polyadic second-order logic.