On the Ontological Commitment of Mereology

Carrara, Massimiliano; Martino, Enrico

doi:10.1017/s1755020309090157

Cited by 6 publications

(4 citation statements)

References 7 publications

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“…In classical RST, 'supplementation' does not hold, while the weaker version (¬Pxy −→ ∃z(Pzx ∧ ¬Ozy)) is trivially satisfied due to the existence of the empty object (∅). Proper selection of semantic domains is essential for avoiding possible problems of ontological innocence [82], wherein the 'sum' operation may result in non existent objects relative the domain. A similar operation results in 'plural reference' in [65,26], and related papers.…”

Section: Remarkmentioning

confidence: 99%

Dialectics of Counting and the Mathematics of Vagueness

Mani

2012

Transactions on Rough Sets XV

100

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New concepts of rough natural number systems are introduced in this research paper from both formal and less formal perspectives. These are used to improve most rough set-theoretical measures in general Rough Set theory (RST) and to represent rough semantics. The foundations of the theory also rely upon the axiomatic approach to granularity for all types of general RST recently developed by the present author. The latter theory is expanded upon in this paper. It is also shown that algebraic semantics of classical RST can be obtained from the developed dialectical counting procedures. Fuzzy set theory is also shown to be representable in purely granule-theoretic terms in the general perspective of solving the contamination problem that pervades this research paper. All this constitutes a radically different approach to the mathematics of vague phenomena and suggests new directions for a more realistic extension of the foundations of mathematics of vagueness from both foundational and application points of view. Algebras corresponding to a concept of rough naturals are also studied and variants are characterised in the penultimate section.

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Section: Remarkmentioning

confidence: 99%

Dialectics of Counting and the Mathematics of Vagueness

Mani

2012

Transactions on Rough Sets XV

100

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“…9zðPzx^:OzyÞÞ is trivially satisfied due to the existence of the empty object ð;Þ. Proper selection of semantic domains is essential for avoiding possible problems of ontological innocence [3], wherein the 'sum' operation may result in non existent objects relative the domain. A similar operation results in 'plural reference' in [31,30], and related papers.…”

Section: Definitionmentioning

confidence: 99%

Choice inclusive general rough semantics

Mani¹

2011

Information Sciences

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“…Of course, Lewis' claim, literally understood, would be contradictory. We think that it is to be understood in the sense that whoever accepts the existence of the Xs, commits oneself automatically to the existence of their sum (for a more detailed discussion of the innocence of mereology see Carrara & Martino, 2009).…”

Section: Definition 315 a Relation Between Two Separate Pluralitiesmentioning

confidence: 99%

On the Infinite in Mereology With Plural Quantification

Carrara

Martino

2010

Rev. symb. logic

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Abstract. In "Mathematics is megethology," Lewis reconstructs set theory using mereology and plural quantification (MPQ). In his recontruction he assumes from the beginning that there is an infinite plurality of atoms, whose size is equivalent to that of the set theoretical universe. Since this assumption is far beyond the basic axioms of mereology, it might seem that MPQ do not play any role in order to guarantee the existence of a large infinity of objects. However, we intend to demonstrate that mereology and plural quantification are, in some ways, particularly relevant to a certain conception of the infinite. More precisely, though the principles of mereology and plural quantification do not guarantee the existence of an infinite number of objects, nevertheless, once the existence of any infinite object is admitted, they are able to assure the existence of an uncountable infinity of objects. So, if-as Lewis maintains-MPQ were parts of logic, the implausible consequence would follow that, given a countable infinity of individuals, logic would be able to guarantee an uncountable infinity of objects. §1. Introduction. In "Mathematics is megethology" Lewis (1993) reconstructs set theory using mereology and plural quantification (hereafter: MPQ). Lewis assumes from the beginning that there is an infinite plurality of atoms, whose size is equivalent to that of the set theoretical universe. So, it might seem that MPQ, though they supply a suitable language for reconstructing set theory, nevertheless they do not play any role in order to guarantee the existence of a large infinity of objects.On the contrary, we intend to demonstrate that MPQ are, in some ways, particularly relevant to a certain conception of the infinite. More precisely, though the principles of MPQ do not guarantee the existence of an infinite number of objects, nevertheless, once the existence of any infinite object is admitted, they are able to assure the existence of an uncountable infinity of objects. 1 We divide our paper into the following sections. In Section 2, we will briefly review the crucial assumptions underlying Lewis' reconstruction of set theory. In Section 3 we will construct a mereological model of the power set of any plurality of pairwise nonoverlapping individuals and show that any infinite individual has uncountably many parts. In Section 4, we will make some final remarks on the alleged innocence of MPQ, on its significance for the conception of the infinite and on the problem of impredicativity of the uncountably infinite. §2. Lewis' megethology. "Mereology," literally the "science of parts," analyzes the relation ". . . is a part of . . . ." There are different formulations of mereology depending on

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On the Ontological Commitment of Mereology

Cited by 6 publications

References 7 publications

Dialectics of Counting and the Mathematics of Vagueness

Dialectics of Counting and the Mathematics of Vagueness

Choice inclusive general rough semantics

On the Infinite in Mereology With Plural Quantification

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