Mereology

In logic, philosophy and related fields, mereology ( (root: , ''mere-'', 'part') and the suffix ''-logy'', 'study, discussion, science') is the study of parts and the wholes they form. Whereas set theory is founded on the membership relation between a set and its elements, mereology emphasizes the meronomic relation between entities, which—from a set-theoretic perspective—is closer to the concept of inclusion between sets.

Mereology has been explored in various ways as applications of predicate logic to formal ontology, in each of which mereology is an important part. Each of these fields provides its own axiomatic definition of mereology. A common element of such axiomatizations is the assumption, shared with inclusion, that the part-whole relation orders its universe, meaning that everything is a part of itself ( reflexivity), that a part of a part of a whole is itself a part of that whole ( transitivity), and that two distinct entities cannot each be a part of the other ( antisymmetry), thus forming a poset. A variant of this axiomatization denies that anything is ever part of itself (irreflexivity) while accepting transitivity, from which antisymmetry follows automatically.

Although mereology is an application of mathematical logic, what could be argued to be a sort of "proto-geometry", it has been wholly developed by logicians, ontologists, linguists, engineers, and computer scientists, especially those working in artificial intelligence. In particular, mereology is also on the basis for a point-free foundation of geometry (see for example the quoted pioneering paper of Alfred Tarski and the review paper by Gerla 1995).

In general systems theory, mereology refers to formal work on system decomposition and parts, wholes and boundaries (by, e.g., Mihajlo D. Mesarovic (1970), Gabriel Kron (1963), or Maurice Jessel (see Bowden (1989, 1998)). A hierarchical version of Gabriel Kron's Network Tearing was published by Keith Bowden (1991), reflecting David Lewis's ideas on gunk. Such ideas appear in theoretical computer science and physics, often in combination with sheaf theory, topos, or category theory. See also the work of Steve Vickers on (parts of) specifications in computer science, Joseph Goguen on physical systems, and Tom Etter (1996, 1998) on link theory and quantum mechanics.

History

Informal part-whole reasoning was consciously invoked in metaphysics and ontology from Plato (in particular, in the second half of the ''Parmenides'') and Aristotle onwards, and more or less unwittingly in 19th-century mathematics until the triumph of set theory around 1910. Metaphysical ideas of this era that discuss the concepts of parts and wholes include divine simplicity and the classical conception of beauty.

Ivor Grattan-Guinness (2001) sheds much light on part-whole reasoning during the 19th and early 20th centuries, and reviews how Cantor and Peano devised set theory. It appears that the first to reason consciously and at length about parts and wholes was Edmund Husserl, in 1901, in the second volume of '' Logical Investigations'' – Third Investigation: "On the Theory of Wholes and Parts" (Husserl 1970 is the English translation). However, the word "mereology" is absent from his writings, and he employed no symbolism even though his doctorate was in mathematics.

Stanisław Leśniewski coined "mereology" in 1927, from the Greek word μέρος (''méros'', "part"), to refer to a formal theory of part-whole he devised in a series of highly technical papers published between 1916 and 1931, and translated in Leśniewski (1992). Leśniewski's student Alfred Tarski, in his Appendix E to Woodger (1937) and the paper translated as Tarski (1984), greatly simplified Leśniewski's formalism. Other students (and students of students) of Lesniewski elaborated this "Polish mereology" over the course of the 20th century. For a good selection of the literature on Polish mereology, see Srzednicki and Rickey (1984). For a survey of Polish mereology, see Simons (1987). Since 1980 or so, however, research on Polish mereology has been almost entirely historical in nature.

A. N. Whitehead planned a fourth volume of '' Principia Mathematica'', on geometry, but never wrote it. His 1914 correspondence with Bertrand Russell reveals that his intended approach to geometry can be seen, with the benefit of hindsight, as mereological in essence. This work culminated in Whitehead (1916) and the mereological systems of Whitehead (1919, 1920).

In 1930, Henry S. Leonard completed a Harvard Ph.D. dissertation in philosophy, setting out a formal theory of the part-whole relation. This evolved into the "calculus of individuals" of Goodman and Leonard (1940). Goodman revised and elaborated this calculus in the three editions of Goodman (1951). The calculus of individuals is the starting point for the post-1970 revival of mereology among logicians, ontologists, and computer scientists, a revival well-surveyed in Simons (1987), Casati and Varzi (1999), and Cotnoir and Varzi (2021).

Axioms and primitive notions

Reflexivity: A basic choice in defining a mereological system, is whether to consider things to be parts of themselves. In naive set theory a similar question arises: whether a set is to be considered a "subset" of itself. In both cases, "yes" gives rise to paradoxes analogous to Russell's paradox: Let there be an object O such that every object that is not a proper part of itself is a proper part of O. Is O a proper part of itself? No, because no object is a proper part of itself; and yes, because it meets the specified requirement for inclusion as a proper part of O. In set theory, a set is often termed an ''improper'' subset of itself. Given such paradoxes, mereology requires an axiomatic formulation.

A mereological "system" is a first-order theory (with identity) whose universe of discourse consists of wholes and their respective parts, collectively called ''objects''. Mereology is a collection of nested and non-nested axiomatic systems, not unlike the case with modal logic.

The treatment, terminology, and hierarchical organization below follow Casati and Varzi (1999: Ch. 3) closely. For a more recent treatment, correcting certain misconceptions, see Hovda (2008). Lower-case letters denote variables ranging over objects. Following each symbolic axiom or definition is the number of the corresponding formula in Casati and Varzi, written in bold.

A mereological system requires at least one primitive binary relation ( dyadic predicate). The most conventional choice for such a relation is parthood (also called "inclusion"), "''x'' is a ''part'' of ''y''", written ''Pxy''. Nearly all systems require that parthood partially order the universe. The following defined relations, required for the axioms below, follow immediately from parthood alone:
*An immediate defined predicate is "x is a proper part of ''y''", written ''PPxy'', which holds (i.e., is satisfied, comes out true) if ''Pxy'' is true and ''Pyx'' is false. Compared to parthood (which is a partial order), ProperPart is a strict partial order.
:$PPxy \leftrightarrow (Pxy \land \lnot Pyx).$ 3.3
:An object lacking proper parts is an ''atom''. The mereological universe consists of all objects we wish to think about, and all of their proper parts:
*Overlap: ''x'' and ''y'' overlap, written ''Oxy'', if there exists an object ''z'' such that ''Pzx'' and ''Pzy'' both hold.
:Oxy \leftrightarrow \exists z zx \land Pzy 3.1
:The parts of ''z'', the "overlap" or "product" of ''x'' and ''y'', are precisely those objects that are parts of both ''x'' and ''y''.
*Underlap: ''x'' and ''y'' underlap, written ''Uxy'', if there exists an object ''z'' such that ''x'' and ''y'' are both parts of ''z''.
:Uxy \leftrightarrow \exists z xz \land Pyz 3.2
Overlap and Underlap are reflexive, symmetric, and intransitive.

Systems vary in what relations they take as primitive and as defined. For example, in extensional mereologies (defined below), ''parthood'' can be defined from Overlap as follows:
:Pxy \leftrightarrow \forall z zx \rightarrow Ozy 3.31

The axioms are:
*Parthood partially orders the universe:
:M1, Reflexive: An object is a part of itself.
:$\ Pxx.$ P.1
:M2, Antisymmetric: If ''Pxy'' and ''Pyx'' both hold, then ''x'' and ''y'' are the same object.
:$(Pxy \land Pyx) \rightarrow x = y.$ P.2
:M3, Transitive: If ''Pxy'' and ''Pyz'', then ''Pxz''.
:$(Pxy \land Pyz) \rightarrow Pxz.$ P.3
*M4, Weak Supplementation: If ''PPxy'' holds, there exists a ''z'' such that ''Pzy'' holds but ''Ozx'' does not.
:PPxy \rightarrow \exists z zy \land \lnot Ozx P.4

*M5, Strong Supplementation: If ''Pyx'' does not hold, there exists a ''z'' such that ''Pzy'' holds but ''Ozx'' does not.
:\lnot Pyx \rightarrow \exists z zy \land \lnot Ozx P.5

*M5', Atomistic Supplementation: If ''Pxy'' does not hold, then there exists an atom ''z'' such that ''Pzx'' holds but ''Ozy'' does not.
:\lnot Pxy \rightarrow \exists z zx \land \lnot Ozy \land \lnot \exists v [PPvz. P.5'

*Top: There exists a "universal object", designated ''W'', such that ''PxW'' holds for any ''x''.
:$\exists W \forall x [PxW].$ 3.20
:Top is a theorem if M8 holds.

*Bottom: There exists an atomic "null object", designated ''N'', such that ''PNx'' holds for any ''x''.
:$\exists N \forall x [PNx].$ 3.22

*M6, Sum: If ''Uxy'' holds, there exists a ''z'', called the "sum" or "fusion" of ''x'' and ''y'', such that the objects overlapping of ''z'' are just those objects that overlap either ''x'' or ''y''.
:Uxy \rightarrow \exists z \forall v vz \leftrightarrow (Ovx \lor Ovy) P.6
*M7, Product: If ''Oxy'' holds, there exists a ''z'', called the "product" of ''x'' and ''y'', such that the parts of ''z'' are just those objects that are parts of both ''x'' and ''y''.
:Oxy \rightarrow \exists z \forall v vz \leftrightarrow (Pvx \land Pvy) P.7
:If ''Oxy'' does not hold, ''x'' and ''y'' have no parts in common, and the product of ''x'' and ''y'' is undefined.
*M8, Unrestricted Fusion: Let φ(''x'') be a first-order formula in which ''x'' is a free variable. Then the fusion of all objects satisfying φ exists.
:\exists x phi(x)\to \exists z \forall y yz_\leftrightarrow_\exists_x[\phi_(x)_\land_Oyx._P.8
:M8_is_also_called_"General_Sum_Principle",_"Unrestricted_Mereological_Composition",_or_"Universalism"._M8_corresponds_to_the_set_builder_notation.html" ;"title="phi_(x)_\land_Oyx.html" ;"title="yz \leftrightarrow \exists x yz_\leftrightarrow_\exists_x[\phi_(x)_\land_Oyx._P.8
:M8_is_also_called_"General_Sum_Principle",_"Unrestricted_Mereological_Composition",_or_"Universalism"._M8_corresponds_to_the_set_builder_notation">principle_of_unrestricted_comprehension_of_naive_set_theory_

Naive_set_theory_is_any_of_several_theories_of_sets_used_in_the_discussion_of_the__foundations_of_mathematics.
Unlike__axiomatic_set_theories,_which_are_defined_using__formal_logic,_naive_set_theory_is_defined_informally,_in__natural_language._It_...