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Nonclassical mereology and its application to sets. (English) Zbl 1049.03003
The paper consists of two parts. The author himself qualifies the first part as “the case against classical mereology”; it contains some objections to the latter and some motivation to what is called there Heyting mereology. The second part shows how this alternative system of mereology provides us with “all the sets we need”.
The systems of mereology discussed in the paper are based on first-order logic with identity and the definite description operator; the non-logical primitives are the proper part relation and a name for the “fictious null thing”. (However, the null thing is admitted in the scope of quantified variables and thus, formally, is not fictious.) Therefore, the classical mereology is reduced in the paper to the elementary part of Leśniewski’s original system. The author argues that the axiom of the classical mereology which declares uniqueness of mereological classes (called fusions in the paper) is a careless extrapolation from the finite to the infinite case. However, he does not reject the principle that things have a unique sum. The position of the author can be explained more precisely as follows.
A model of a mereology is a complete lattice with zero deleted. In such a lattice, the fusion of a subset $$F$$ is an upper bound $$z$$ of $$F$$ such that any $$x$$ disjoint from every element of $$F$$ is disjoint from $$z$$. The sum of $$F$$ is a fusion which happens to be the join of $$F$$. In classical mereology, $$F$$ has to have a unique fusion; hence the notions of fusion, join and sum coincide there. In the author’s nonclassical mereology, uniqueness of fusions is not required, but, in order to obtain a mathematically well-behaved mereology, it is assumed that meet distributes over arbitrary joins (the latter condition is known to be equivalent to the one that the lattice is Heyting). By the way, this assumption provides that the join of $$F$$ is always a fusion.
In contrast to classical mereology, the sum of proper parts of a thing $$x$$ in Heyting mereology may itself be a proper part, which is then the unique maximal proper part (mpp) of $$x$$. If this is the case, $$x$$ is said to be an atom. The author uses the term ‘gunk’ for a thing that has no atomic parts, and discusses the problem of measuring gunks. A sufficient motivation is found here for the hypothesis that a model of merology should be a Heyting lattice.
Heyting mereology provides tools for treating sets or, more accurately, pseudosets. Let $$x A y$$ mean that $$x$$ is the mpp of $$y$$. The pseudomembership relation $$E$$ is defined by $$x E y$$ iff $$x A y$$ and there is no $$u$$ such that $$x A u$$ and $$u A y$$. A pseudoset is then any $$y$$ such that $$x E y$$ and $$y E z$$ for some $$x$$ and $$z$$. The author shows, shortly, in what sense his mereology has the resources to perform the work done in a usual set theory by simply founded sets ($$z$$ is simply well-founded if the membership relation of $$\in$$ restricted to $$z$$ is a tree), pure sets or sets with urelements.
[Reviewers remark: The formal definition of a fusion, HMD11, evidently contains some misprint. The second argument of conjunction there should read as $$\forall w(\forall z(Fz \supset \sim w \circ z) \supset \sim w \circ x)$$.]

##### MSC:
 03A05 Philosophical and critical aspects of logic and foundations 03E70 Nonclassical and second-order set theories
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##### References:
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