Draft - not proofed - from Casati and Varzi - Holes and Other Superficialities, MIT Press, posted with permission, for non-commercial, content purposes only

 

Appendix: Outline of a Theory

In the chapters above we presented our views in a rather infor­mal set­ting, trying to show the philosophical importance and conse­quences of a realist attitude toward holes rather than spelling out a full-fledged theory of holes.

In this appendix we attempt to address this task more di­rectly by summarizing some basic tenets of our account in a rather systematic – though by no means complete – fashion. For convenience, we divide the presentation into four main sec­tions:

(1)      a preliminary ontological part, which introduces the basic binary relation “is a hole in (or through)” along with some rel­evant facts;

(2)      a mereo­logical part, which systematizes some funda­men­tal prin­ci­ples governing the interplay between the host-hole and the part-whole relations;

(3)      a topological part, which summa­rizes some basic facts con­cern­ing surfaces and the taxonomy of holes; and

(4)      a morphological part, focusing on the fact that ob­jects with holes constitute – as we have put it – the morphologi­cal mani­fold of fil­lable things.

This organization does not ex­actly parallel the order we fol­lowed in the text, where mereo­logical facts were examined only after – and somehow on the basis of – a topo-morphological char­ac­terization of holes. In­deed, the interplay among these dif­fer­ent domains (as well as be­tween these and other domains, such as kine­matics or causality, considered in the text but not in­cluded here) is an in­ter­esting issue in it­self, but it would take us too far afield to address it here. For the mod­est pur­poses of this Ap­pendix, suffice it to note that the order fol­low­ed here permits a rather nice and intuitively simple step-by-step con­struc­tion.

The exposition proceeds more geometrico. Each section be­gins with basic definitions, followed by some basic principles, or axioms, followed by a list of a few noteworthy conse­quences or theorems. (Numbers in braces indicate the axiom or defini­tion from which a theorem is derived.) It is understood that there is no pre­tense of com­ple­te­ness or logical elegance. In fact we have been quite relaxed in our choice of axioms and theo­rems, as our aim is first and fore­most perspicuity. An informal spel­ling of every formula, some­times with a brief comment, is also pro­vided (the reader may want to skip the formulas and just focus on these informal ren­derings).

In the formalization, we assume basic logical notions and notation. We use Ø, Ù, Ú, ®, and « as connectives for negation, conjunction, disjunction, material implication, and material equivalence re­spectively; " and $ for the universal and exis­ten­tial quantifiers; and i for the definite descriptor. To sim­plify readability, we dis­pense with quotation devices as far as is prac­ticable and rely on standard conventions to minimize the use of parentheses. (In particular, we assume our con­nec­tives to bind their argu­ments in decreasing order of strength as listed, so that negation binds the strongest and equi­va­lence the weak­est.)

The underly­ing logic is deliberately left vague, as after all we think holes are ut­terly neutral in this respect. A preferred al­ternative is some sort of a free logic, where im­proper de­scrip­tions and other pos­sibly empty expres­sions can be admitted bona fide; however, every­thing that follows could in prin­ciple be dealt with within the frame­work of a standard first-order logic, with i treated as an im­proper symbol. In ad­di­tion, we as­sume familiarity with some ba­sic princi­ples of ex­ten­sional mereo­logy and topol­ogy.

1. Ontology

The main thesis is that a hole is an immaterial body located at the sur­face (or at some surface) of a material object. Since the notion of a surface is essentially a topological one, and since the property of being im­material is reflected in the morphological property of being fill­able, the ontological basis is concerned first and foremost with the general dependence of a hole on its host.

Notation

N1.1         Hxy = x is a hole in (or through) y.

This is our primitive relation. We need a binary relation to express the basic intuition that holes are dependent en­ti­ties. A hole is al­ways in (or through) some object.

Definition

D1.1         Hx =df $yHxy.

We write ‘Hx’ for “x is a hole”. Since every hole is onto­lo­g­i­cally de­pendent on its host, being a hole is defined as being a hole in (or through) something.

Basic Axiom

A1.1         Hxy ® ØHy.

The host of a hole is not a hole.

Some Theorems

T1.1         Hxy ® ØHyx.

Being a hole in (or through) is an asymmetric relation: a hole can­not host its own host.

T1.2         ØHxx.

Being a hole in (or through) is an irreflexive relation: a hole can­not host itself.

T1.3         Hx ® Ø$yHyx.

Holes do not have holes: they cannot host one another (though they can have holes as proper parts).

T1.4         $xHx ® $xØHx.

Holes cannot be the only things around.

2. Mereology

As immaterial bodies, holes have parts and can bear part-whole relations to one another (though not to their hosts). The main principles con­cerning these relations can be formulated within the frame­work of classical ex­tensional mereology sup­plemented with some specific axioms on the behavior of the on­tological relation ‘H’.

Notation

N2.1         x≤y = x is a part of y.

This is one of many possible mereological primitives: a re­flexive, anti­sym­metric, transitive relation (i.e., a par­tial ordering), accord­ing to classical mereol­ogy.

Definitions

D2.1         x<y =df x≤y Ù Øx=y.

‘x<y’ means “x is a proper part of y”; i.e., x is a part of y other than y it­self. This is a transi­tive and asym­metric (hence irreflexive) rela­tion.

D2.2         xoy =df $z(z≤x Ù z≤y).

‘xoy’ means “x overlaps y”; i.e., x and y have some parts in com­mon. This is a reflexive and symmetric (but not tran­sitive) rela­tion.

D2.3         SxAx =df iz"y(yoz « $w(Aw Ù yow)).

‘SxAx’ stands for “the fusion of all x such that Ax”. The exis­tence of such an object is always assumed in classical mereology, pro­vided there is some x such that Ax. We do not as­sume it unless otherwise specified.

D2.4         x»y =df Sw(w≤x Ú w≤y).

‘x»y’ stands for “the sum of x and y”, i.e., the smallest thing whose parts are either parts of x or parts of y. This is an idempotent, com­muta­tive, dis­tribu­tive operation.

D2.5         x«y =df Sw(w≤x Ù w≤y).

‘x«y’ stands for “the product of x and y”, i.e., the largest thing whose parts are both parts of x and parts of y. This too is an idempotent, commuta­tive, distributive operation, though one that is de­fined only if xoy.

D2.6         x–y =df Sw(w≤x Ù Øwoy).

‘x–y’ stands for “the difference of x and y”, i.e., the largest thing con­tained in x that has no part in common with y.

D2.7         xp y =df Hx Ù x≤y.

‘xp y’ means “x is a hole-part of y”, i.e., a hole that is a part of y. This is a partial ordering, like ≤; it applies only when y is itself a (part of a) hole.

D2.8         xp y =df xp y Ù Øx=y.

‘xp y’ means “x is a proper hole-part of y”, i.e., a hole that is a proper part of y. This is a transi­tive, asym­metric, ir­re­­flexive relation, like <.

Basic Axioms

A2.1         Hxy ® Øxoy.

No hole overlaps its own host (though the sum of a hole and its host may be a legitimate host for different holes: e.g. the sum of a dough­nut y and its hole x – if such a sum exists – will not be a host of x, but it will be a host of, say, a cavity that may be hidden inside y).

A2.2         Hxy Ù Hxz ® $w(w<y«z Ù Hxw).

Any two hosts of a hole have a common proper part that en­tirely hosts the hole. (Of course, intuitively a hole has one host; but if we al­low for mereological sums or split­tings, then every hole has a vir­tu­ally in­finite class of hosts, partially ordered by <. See D3.2 below.)

A2.3         Hxy Ù Hzy ® "w(wp x»z ® Hwy).

Any host of a hole entirely hosts all common hole-parts of any sum involving that hole.

A2.4         Hxy Ù y≤z ® xoz Ú Hxz.

Any object that includes the host of a hole is a host of that hole, unless its parts also include parts of that very hole.

A2.5         Hxy Ù Hzw Ù xoz ® yow.

Overlapping holes have overlapping hosts. (However, two holes may occupy the same region, or part of the same region, with­out shar­ing any parts. Holes are immaterial, and can penetrate one another; mereo­logical over­lap­ping is not implied by spatial co-loca­liza­tion.)

A2.6         Hx ® $z(z<x).

No hole is atomic (though holes need not have proper hole-parts; oth­erwise every hole would corre­spond to a pile of infinitely many, grad­ually smaller holes).

Some Theorems

T2.1         Hxy ® Øx≤y.

Holes are not parts of their hosts (although the hosts of a hole may have different holes as parts) {A2.1}.

T2.2         Hxy ® Øxp y.

Holes are not hole-parts of their hosts (although they can be hole-parts of parts of holes) {A2.1}.

T2.3         Hxy ® Øy≤x.

The host of a hole is not part of it {A2.1}.

T2.4         Hxy Ù y≤z ® ØHz.

The host of a hole is not part of any hole {A1.1 + A2.4–A2.5}. This is a gener­al­iza­tion of the ontological axiom (A1.1) and of the thesis that a hole a cannot host any­thing (T1.3).

T2.5         Hxy Ù Hxz ® yoz.

Any two hosts of the same hole are overlapping; i.e., a hole cannot have two discrete hosts {A2.2}.

T2.6         Hxy ® $z(z<y Ù ØHxz).

Not every part of a hole’s host is a host of the hole (though it could host a different hole) {A2.2}.

T2.7         Hxy ® $z(z<y Ù Hxz).

Hosting a hole is having some proper part that entirely hosts the hole; i.e., there is no minimal host for a given hole {A2.2}.

T2.8         Hxy ® "z(zp x ® Hzy).

Hosting a hole is hosting any proper parts of the hole that are holes them­selves {A2.3}. (Note that the same does not hold relative to <.)

T2.9         Hxy Ù Hxz ® Hx(y»z).

The sum of any two hosts of a hole is itself a host of that hole {A2.1 + A2.4}.

T2.10       Hxy Ù Hxz ® Hx(y«z).

The product of any two hosts of a hole is itself a host of that hole {A2.1 + A2.2 + A2.4}.

T2.11       Hxy Ù Hxz ® ØHx(y–z).

The difference of two hosts of a hole is not itself a host of that hole {A2.2}.

T2.12       Hxy ® "z(Øxoz ® Hx(y»z)).

A hole in an object is also a hole in any mereological sum including that object, provided the sum does not include any parts of the hole it­self {A2.4}.

T2.13       Hxy ® $z(Øxoz Ù Hx(y«z)).

A hole in an object is also a hole in some mereological product in­volving that object, provided the product does not include any parts of the hole it­self {A2.2}.

T2.14       Hxy ® "zØHx(z–y).

The difference between an object and the host of a hole is not a host of that hole {A2.2}.

T2.15        ØH(Sz.z=z).

If the universal individual exists, it surely isn’t a hole {A1.1 + A2.1}.

T2.16        ØH(Sz.z≠z).

The null individual (if it exists) cannot be a hole {A2.6}.

T2.17       Ø$z.z<y ® Ø$xHxy.

Atoms are holeless {A2.2}.

T2.18       Hx ® Hx(SzHxz)

The fusion of all hosts of a hole is a host of that hole {A2.1 + A2.4}.

3. Topology

Topology constitutes in many ways a natural next step after mereology, although various mereological notions could for­mal­ly be de­fin­ed in terms of topological ones. Particularly in a theory of holes, topological notions are important to account for the fact that every hole is located at some surface of its host as well as for taxonomic purposes.

Notation

N3.1         xcy = x is connected with y.

This is a reflexive and symmetric relation capturing the intuitive notion of touching or being co-local­ized at some point. We take it to satisfy the clause that what is con­nected with a part is also connected with the whole, so that x≤y im­plies "z(zcx ® zcy). (We do not, how­ever, as­sume the con­verse, which would have the effect of reduc­ing mereology to topology.)

N3.2         gx = the genus of x.

Intuitively, the genus of an object is the maximum num­ber of simul­taneous cuts that can be made without sepa­rating the object into two unconnected pieces (0 if it is a sphere, 1 if it is a torus, etc.). This no­tion could be de­fined in terms of c, but that would lead us too far afield.

Definitions

D3.1         Cx =df "y"z(x=y»z ® ycz).

‘Cx’ means “x is (self-)connected”; i.e., x does not consist of two or more dis­connected parts.

D3.2         hx =df Sy(Hxy Ù Cy).

‘hx’ stands for “the principal host of x”, i.e., x’s maxi­mally con­nected host (a notion that we take to be defined only when x is a hole). We may in­tuitively re­gard this as the host of the hole, every other host being either a topo­logi­cally scat­tered mereological aggre­gate including the principal host or a po­tential part of this latter (see above ad A2.2).

D3.3         x›‹y =df xcy Ù Øxoy.

‘x›‹y’ means “x is externally connected with y”; i.e., x is connected with y but does not over­lap it. This is an irreflexive and symmetric relation.

D3.4         x<y =df x≤y Ù "z(z›‹x ® Øz›‹y).­

‘x<y’ means “x is an interior part of y”; i.e., x is a part of y that is exter­nally connected only with things that are not so connected with y itself. This is a transitive relation in­cluded in ≤ and closed under both » and «.

D3.5         x‹‹y =df x≤y Ù Ø$z.z<x.

‘x‹‹y’ means “x is a superficial part of y”; i.e., x is a part of y that has no inte­rior parts of its own (or, intuitively, that only overlaps those parts of y that are externally con­nected with the geometric com­ple­ment of y). This too is a transi­tive relation closed under » and «.

D3.6         Sxy =df x‹‹y Ù Cx Ù "z(z‹‹y Ù Cz ® (zcx ® z<x)).

‘Sxy’ means “x is a surface of y”; i.e., x is a maximally con­nected su­per­fi­cial part of y.

D3.7         Hcavxy =df Hxy Ù $z(Szy Ù "w(w≤z « x›‹w)).

‘Hcavxy’ means “x is a cavity in y”; i.e., x is an internal hole en­veloped by an en­tire host surface. A cavity is a topo­logically non-erasable disconti­nuity.

D3.8         Htunxy =df Hxy Ù "z(z≤y Ù Cz Ù Hxz ® gz≠0).

‘Htunxy’ means “x is a tunnel (or a perforation) through y”. This is also a topo­logi­cally non-erasable hole, charac­ter­ized by the fact that its host has no connected part of genus 0 entirely hosting the hole. (Note that a hole may at once be a tunnel and a cavity: it may be a cavity-tunnel.)

D3.9         Hholxy =df Hxy Ù ØHcavxy Ù ØHtunxy.

‘Hholxy’ means “x is a hollow (or a depression) in y”; i.e., x is a hole in y which is neither a tun­nel in y nor a cavity in y. This is always an external, topologically erasable dis­turbance, characterized by the fact that the relevant host must have a part of genus 0 entirely hosting the hole.

Basic Axioms

A3.1         Hx ® Cx.

Holes are self-connected; i.e., there is no scattered hole.

A3.2         Hxy ® xcy.

Holes are connected with their hosts.

A3.3         Hx ® $y(Hxy Ù Cy).

Every hole has some self-connected host.

A3.4         Hx Ù ypx ® $z(z›‹x Ù Øz›‹y).

No hole can have a proper hole-part that is externally connected with exactly the same things as the hole itself.

Some Theorems

T3.1         Hxy ® x›‹y.

Holes are only externally connected with their hosts; i.e., a hole and its host touch each other, but have no parts in common {A2.1 + A3.2}.

T3.2         Hxy ® $z(z‹‹y Ù xcz).

Every hole is connected with some superficial part of its hosts {A2.1 + A3.2}. (Holes are super­fi­cial entities; they go hand in hand with sur­faces.)

T3.3         Hxy ® "z(z<y ® Øxcz).

No hole is connected with any interior parts of its hosts {A2.1 + A3.2}. (Making a hole is transforming interior parts of an object into su­perficial ones.)

T3.4         Hxy Ù zpx ® z›‹y.

The hole-parts of a hole are all externally connected with the host of that hole, i.e., they are all located along the “walls” of that hole {A2.1 + A2.3 + A3.2}.

T3.5         Hxy Ù zpx Ù w≤y ® (z›‹w ® x›‹w).

A hole is externally connected with every part of its host that is so con­nected with some hole-part of that hole {A2.1 + A3.2}.

T3.6         Hx Ù zpx Ù Hzw ® xcw.

A hole is sure to be connected (externally or not) with the hosts of its own hole-parts {A2.1 + A3.2}.

T3.7         Hx Ù Hy ® (Hx»y ®xcy).

Only holes that are connected with each other can join to form a hole {A3.1}.

T3.8         Hxy ® $z(z<y Ù Hxz Ù Cz).

Hosting a hole is having some proper self-connected part that en­tirely hosts the hole {A2.2 + A3.3}.

T3.9         Hxy Ù Hxz ® ycz.

Any two hosts of a hole are connected with each other {A2.2 + A3.3}.

T3.10       Hxy Ù Hxz ® Øy›‹z.

Two hosts of a hole cannot be externally connected {A2.2 + A3.3}.

T3.11       Hxy Ù Hxz ® $w(w≤y«z Ù Hxz Ù Cz).

Any two hosts of a hole have a common self-connected part that en­tirely hosts the hole {A2.2 + A3.3}.

T3.12       Hx ® Chx.

Principal hosts are always sure to be self-connected {A2.1 + A2.2 + A2.4 + A3.3}.

T3.13       Hx ® Hxhx.

Every hole is a hole in its principal host {A2.1 + A2.2 + A2.4 + A3.3}.

T3.14       Hx Ù zpx ® hx=hz.

The principal host of a hole is also the principal host of any hole-parts of that hole {A2.3 + A3.3}.

T3.15       Hcavxy Ù Hzy ® Øx›‹z.

No hole can be externally connected with an internal cav­ity (though it may well overlap one) {A2.1}.

T3.16       Hcavxy Ù Hzy ® Øx<z.

Cavities are maximal holes; i.e., no hole can include a cavity as a proper part {A2.1 + A3.3 + A3.4}.

T3.17       Hcavxy Ù Hzy Ù xoz ® z≤x.

Cavities are maximally connected holes; i.e., a cavity in­cludes ev­ery hole with which it is connected {A2.1 + A3.3 + A3.4}.

T3.18       Htunxy Ù Hholzy ® Øx≤z.

A hole that qualifies as a tunnel with respect to a hollow’s host cannot be part of that hollow {A2.3}.

T3.19       Htunxy Ù Hholxz ® Øz≤y.

A hole cannot qualify as a hollow with respect to any part of a host relative to which it qualifies as a tunnel {A2.3}.

4. Morphology

Topological concepts make it possi­ble to distinguish and classify ob­jects with dif­ferent types of holes, but we need morphology in order to account for the feeling that blind hollows, perforating tunnels, and internal cavities are all part of a single fam­ily. The rele­vant notion operating here is that of fil­ling: objects with holes – or, better, holes in objects – constitute the morphologi­cal mani­fold of fil­lable entities.

Notation

N4.1         Fxy = x is (perfectly) filled by y.

Holes can be filled, and we mean here perfectly filled. They can be filled (without losing their status of holes) in­sofar as they determine a (partially) concave discontinu­ity in the surface of their host.

Definitions

D4.1         Fx =df u$zFxz.

‘Fx’ means “x is fillable”; i.e., x can be perfectly filled by some­thing. Here and in the following we use u and n as modal connec­tives for pos­sibility and necessity, respec­tively.

D4.2         Fcomxz =df $w(w≤z Ù Fxw).

‘Fcomxz’ means “x is completely filled by z”; i.e., there is some part of z that perfectly fills x. This is a monotonic re­lation, in the sense that Fcomxy Ù y≤z ® Fcomxz .

D4.3         Fparxz =df $w(w≤x Ù Fcomwz).

‘Fparxz’ means “x is partially filled by z”; i.e., there is some part of x that is completely fil­led by z. This too is a mono­tonic relation, in the sense that Fparxy Ù y≤z ® Fparxz. Note that a partial filler need not be wholly inside a hole (it may “stick out”), which means that every com­plete filler also qualifies as (a limit cases of) a partial one.

D4.4         Fproxz =df $w(w≤x Ù Fwz).

‘Fproxz’ means “x is properly (though perhaps incom­pletely) filled by z”; i.e., some part of x is perfectly fil­led by z. This is the dual of Fcom and is related to Fpar by the equiv­alence Fproxz « "w(w≤z ® Fparxw). (Thus, every perfect filler is both complete and proper in this sense.)

D4.5         sx =df Sz(z‹‹hx Ù x›‹z).

‘sx ’ stands for “the skin of x”, i.e., the fusion of those su­per­fi­cial parts of x’s principal host with which x is exter­nally connected (a no­tion that is meant to apply only when x is a hole). This is a slight de­par­ture from the text, where the skin is defined as the part of the filler’s surface that is in contact with the host, or as that part of the host’s sur­face that is in contact with the filler. However, these def­ini­tions are es­sentially equi­v­­alent and serve the same purpose: the topol­ogy of the skin re­flects the mor­pho­­lo­gi­cal com­plexity of the hole.

D4.6         fwzx =df w‹‹z Ù Øwcsx.

‘fwzx’ means “w is a free superficial part of z relative to x”; i.e., w is a super­fi­cial part of z that is not con­nected with x’s host(s). (This no­tion is meant to apply only when x is a hole and z a correspond­ing perfect fil­ler.)

Axioms

A4.1         Fx « $y(Hy Ù x≤y).

Something is fillable just in case it is part of a hole; i.e., fillability is an exclusive property of holes and their parts.

A4.2         Fxy Ù Fz ®Øyoz.

Perfect fillers and fillable entities have no parts in com­mon (rather, they may occupy the same spatial region).

A4.3         Fcomxy Ù zcx ®zcy.

A complete filler of (a part of) a hole is connected with ev­ery part of (that part of) that hole .

A4.4         Fproxy Ù zcy ®zcx.

Every part of a proper filler of (a part of) a hole is con­nected with (that part of) that hole.

A4.5         Fxy Ù z<x ® Fcomzy.

A perfect filler of (a part of) a hole completely fills every proper part of (that part of) that hole.

A4.6         Fxy Ù z<y ® Fproxz.

Every proper part of a perfect filler of (a part of) a hole properly fills (that part of) that hole.

Some Theorems

T4.1         Hx ® Fx.

Every hole can be filled (and continue to be a hole) {A4.1}.

T4.2         Hxy ® ØFy.

One cannot fill the host of a hole (except in a loose way of speaking: we can, e.g., say that something material is fil­lable meaning that it is the host of some fill­able entity proper) {A1.1 + A2.4–A2.5 + A4.1}.

T4.3         Fxy ® ØHy.

No filler can be a hole (though it can be holed, like the filler of an in­ternal tunnel-cavity) {A4.1 + A4.2}.

T4.4         Fxy ® ØHxy.

A filler cannot host the thing it fills {A2.1 + A4.3}.

T4.5         ØFxx.

Being filled is an irreflexive relation: holes or parts of holes can­not fill themselves {A4.2}.

T4.6         Fxz ® ØFzx.

Being filled is an asymmetric relation: holes or parts of holes can­not fill their own fillers {A4.2}.

T4.7         Fxz ® Øzox.

Fillers do not have any parts in common with the holes they fill (though some of their parts are spa­tio-temporally co-localized with parts of the holes, holes being immate­rial) {A4.2}.

T4.8         Fxy Ù y≤z ® ØFz.

A filler is not part of any fillable entity (just as a host is not part of any hosted entity: compare T2.4) {A4.2}.

T4.9         Fxy Ù z≤y ® ØFz.

A perfect filler cannot have any fillable part (though a complete fil­ler, or even a partial filler, can) {A4.2}.

T4.10       Fxz Ù Hxy Ù w≤y Ù xcw ® z›‹w.

A hole’s perfect filler is externally connected with every part of the hole’s hosts with which the hole itself is con­nected (i.e., it perfectly “fits” the hole) {A4.2 + A4.3}.

T4.11       Fxz Ù w≤hx ® z›‹w « x›‹w.

A hole’s perfect filler is externally connected with exactly those parts of the principal host with which the hole itself is con­nected {A4.2 + A4.3}.

T4.12       Fxy Ù z≤sx ® z›‹y.

A hole’s perfect filler is externally connected with every part of the hole’s skin {A4.2 + A4.3}.

T4.13       Hxy ® sx‹‹y.

The skin of a hole is a superficial part of each and every host of that hole {A3.1 + A4.2 + A4.3}.

T4.14       ØHxsx.

Nothing is a hole in its own skin; i.e., the skin of a hole envelops the hole but does not qual­ify as a host of it {A2.2 + A3.1 + A4.2 + A4.3}.

T4.15       Fcomxy Ù w≤sx ® wcy.

A complete filler of a hole is connected with every part of the hole’s skin {A4.2 + A4.3}.

T4.16       Fxy Ù z<y ® Fparxz.

Every proper part of a perfect filler of (a part of) a hole partially fills (that part of) that hole {A4.5}.

T4.17       Fcomxy Ù z≤x ® Fcomzy.

Any part of a completely filled entity is also completely filled (by the same filler) {A4.5}.

T4.18       Fproxy Ù z≤y ® Fproxz.

Any part of a proper filler is also a proper filler (of the same entity) {A4.6}.

T4.19       Hcavxy Ù Fcomxz Ù Cz ® Fxz.

An internal cavity admits of no imperfect complete self-connected fillers: it can only be completely filled in a per­fect way {A4.3}.

T4.20       Hcavxy Ù Fparxz Ù Cz ® Fproxz.

An internal cavity admits of no improper partial self-con­nected fillers: it can only be partially filled in a proper way {A4.3 + A4.4}.

T4.21       Hcavxy Ù Fxz ® Ø$w.fwzx.

An internal cavity admits of no perfect fillers with free superfi­cial parts {A4.3 + A4.4}.

T4.22       Hholxy Ù Fxz ® $w(fwzx Ù "v(fvzx ® v≤w)).

The filler of a hollow is sure to have exactly one maximal free su­perfi­cial part {A4.3}.

T4.23       Hcavxy « Hxy Ù n"z(Fxz ® Ø$w.fwzx).

A cavity is a hole whose fillers cannot have any free su­perfi­cial parts {A4.3}.

T4.24       Hholxy « Hxy Ù g(sx)=0 Ù n"z(Fxz ® $w.fwzx).

A hollow is a hole whose skin has genus 0 and cannot be connected with every superficial part of a perfect filler; i.e., the filler of a hol­low must have free superficial parts {A4.3}.

T4.25       Hxy ® (u$z Fparxz Ù u$z Fproxz Ù u$z Fcompxz).

Every hole can be filled partially, properly, or completely {A4.1}.