Intuitionistic Mereology
Paolo Maffezioli and Achille C. Varzi
Final version published in Synthese 198:S18 (2021), 4277–4302
Abstract
Two mereological theories are presented based on a primitive apartness
relation along with binary relations of mereological excess and weak excess, respectively. It is shown that both theories are acceptable from the
standpoint of constructive reasoning while remaining faithful to the spirit
of classical mereology. The two theories are then compared and assessed
with regard to their extensional import.
Keywords: Mereology; Intuitionism; Apartness; Excess; Extensionality
1
Introduction
Like any formal theory, classical mereology consists of logical axioms and proper
axioms. Over the years, philosophical reasons have motivated interest in mereological theories that depart from the classical framework with regard to some
of its proper axioms—initially composition and supplementation axioms (see
[47, §§3–4]), more recently ordering axioms such as the antisymmetry of parthood [9, 10] or its transitivity [32]. On the other hand, logical axioms have been
challenged, too. For instance, many-valued logics [36, 42], free logic [11, 41, 44],
and plural quantification [28] have been considered as sensible alternatives to
classical first-oder logic and, more recently, mereological theories based on paraconsistent logic have also been proposed [48]. The notion of a ‘non-classical mereology’ is thus ambivalent, as the epithet ‘classical’ may itself be understood with
reference to one sort of axioms or the other.1 Mereologists may go non-classical
without leaving the terra firma of classical logic, or they may get non-classical
for reasons that have little or even nothing to do with their views about specific
properties of the part-whole relation as such and stem instead from general concerns about the underlying logic. In the latter case, it is indeed an interesting
question whether the shift to a different logical framework calls for a parallel
1 Actually there is some ambiguity also in what counts as classical mereology tout court.
Here we are thinking of the familiar theory based on classical first-order logic [47]. Historically,
however, this theory came to us in different guises. Leśniewski’s ‘Mereologia’ [26, 27] was based
on Ontology and Protothetic; Leonard and Goodman’s ‘Calculus of Individuals’ [25] made use
of quantification over classes. Such systems are not elementarily axiomatizable [33] and are,
therefore, strictly stronger than their first-order approximation, which is due to Goodman [16].
1
rethinking of our basic mereological tenets. Can we simply change the logic and
leave the rest of classical mereology as is? This is not a question that admits a
general answer, let alone a uniform one; it must be addressed on a case-by-case
basis. In this paper we aim to do so, if only partially, with regard to a theory that
so far has received little attention in the literature: intuitionistic mereology.2
Our purpose is twofold. First, we want to outline the salient features of such
a theory—a theory of the relations of part to whole, and of part to part within
a whole, that is acceptable according to the principles of constructive reasoning.
We prove that a constructively acceptable counterpart of classical mereology
requires more than a mere revision of the underlying logic; it requires genuine
revisions at the level of primitive notions and proper axioms. Importantly, simply
adding the proper axioms of classical mereology on top of intuitionistic logic
would result in a theory that fails to be fully extensional, i.e. to validate the
following general principles:
If x and y are part of each other, then x = y (Extensionality of Parthood);
If x and y overlap the same things, then x = y (Extensionality of Overlap);
If x and y are composite things with the same proper parts, then x = y
(Extensionality of Proper Parthood).
Except for Extensionality of Parthood (which is just the antisymmetry axiom),
Extensionality of Overlap and of Proper Parthood can be derived only using
classical logic.
Our second purpose is to show how such principles can nonetheless be recovered when the shift from classical to intuitionistic logic is accompanied by a
corresponding revision of the basic notions. Of course extensionality is by itself a
contentious issue. The thought that mereological indiscernibility is sufficient for
identity sits well with Goodman’s principle of nominalism (“no distinction of
entities without a distinction of content” [16, p. 26]), but it isn’t intrinsic to our
understanding of the part-whole relation and one may want to weaken the underlying logic precisely to relinquish this feature of classical mereology.3 However
this is not to say that an intuitionist is perforce committed to this move. At least
in principle, a nominalist-extensionalist stance should be compatible with the
demands of constructive reasoning, so an intuitionist should be able to preserve
the spirit, if not the letter, of the three principles listed above. We argue that
this is indeed a viable option, provided we understand the relevant mereological
vocabulary in an intuitionistically friendly way. As we change the logic, we also
need to revisit the fundamental concepts of the theory—and the proper axioms
that govern them—in terms of more primitive, constructively justified notions.
2 There is a literature on so-called ‘Heyting mereologies’, as in [12], [29], and [37]. Despite
the name, however, such theories are based on classical logic and differ from classical mereology
with regard to their proper axioms (they lack Weak Supplementation), so they are nonclassical in the first sense introduced above. Essentially, they deliver a parthood relation
whose structure is not a Boolean but a Heyting algebra (about which see [22]).
3 For a review of the arguments, see [24, pt. ii]. For an explicit defense of extensionality on
behalf of classical mereology, see [46].
2
2
Intuitionistic Primitives
Building on previous work in constructive mathematics, we focus on the following three primitive relations:
a 6= b
a b
a≮b
a is apart from b
a exceeds b
a weakly exceeds b
Apartness was originally introduced by Brouwer [3, 4, 5] (with a different
notation) to express inequality between real numbers in the constructive analysis
of the continuum: whereas saying that two real numbers a and b are unequal
only means that the assumption a = b is contradictory, to say that a and b are
apart4 expresses the constructively stronger requirement that their distance on
the real line can be effectively measured, i.e. that | a − b | > 0 has a constructive
proof. Classically, inequality and apartness coincide, but intuitionistically two
real numbers can be unequal without being apart. An early application of the
theory of apartness is Heyting’s work on projective geometry, where a basic
relation A ω B between two points A and B is read as: point A is away from
(entfernt von) point B (cf. axiom IIa in [18, p. 493]). Later [19, p. 49], Heyting
defined apartness as a relation # on a species S such that, for all a, b in S:
If a # b, then b # a;
If a # b, then not a = b;
If not a # b, then a = b;
If a # b, then, for any element c of S, either a # c or b # c.
The notation ‘6=’ to denote the apartness relation was eventually introduced
by Scott [39] and adopted by later authors, and here we shall follow suit. It is
important, however, that it be treated as primitive (rather than shorthand for
the negation of ‘=’.)
The excess relation was introduced by von Plato in [34, 35] to express in a
constructive way the negation of a partial order 6 and has been further investigated by Negri in [30] using sequent calculi. Its basic axiomatization treats
as irreflexive and closed under co-transitivity:5
Not a a;
If a b, then, for all c, either a
c or c
b.
Moreover, apartness can be defined in terms of excess as a
b∨b
a. Here
we are going to treat both 6= and
as primitives, but their intended reading
will correspond to von Plato’s. Intuitively, to say that a exceeds b will mean
4 In Dutch: ‘verwijderd’, ‘plaatselijk verschillend’ [4, §2]; in German: ‘entfernt’ [5, p. 254],
‘örtlich verschieden’ [3, p. 3].
5 Similar axioms for constructive ordering relations may be found in Bridges [2]; cf. also
Scott [39, §1].
3
that a outstrips b of a non-zero quantity that is effectively computable. Thus, in
particular, a will exceed b only if a is not (intuitionistically) part of b. However,
intuitionistically a may fail to be part of b without exceeding it.
Weak excess is introduced here in a parallel fashion to express the constructive negation of a strict partial order <. It is called ‘weak’ insofar as it covers
equality as a limit case: whereas excess requires apartness, weak access does not.
Thus, intuitively, a will weakly exceed b whenever a is not (intuitionistically) a
proper part of b. In this sense, the relationship between and ≮ may be seen
as dual to the relationship between parthood and proper parthood in classical
mereology, though only insofar as equality may be understood as intuitionistic
non-apartness.
Now, the familiar relations of equality, parthood, and proper parthood need
not of course disappear altogether. The idea is rather that such relations—hence
any other mereological relation that can be defined in terms of these, such as
overlap6 —are not fundamental. In the following we shall show that they can
indeed be recovered from our constructive primitives (suitably axiomatized) in
such a way as to satisfy the main properties they have in classical mereology,
including extensionality. More precisely, we shall see that a mereological theory
based on excess (and apartness) is deductively powerful enough to guarantee
Extensionality of Parthood and, given Strong Supplementation, Extensionality
of Overlap, though not Extensionality of Proper Parthood. On the other hand,
a theory based on weak excess (and apartness) can derive all three forms of
extensionality. Thus it is weak excess, we shall argue, that provides the best
resources for a natural intuitionistic counterpart of classical mereology.
3
Mereologies based on excess and apartness
We begin by introducing the first sort of theory, IM1. Its language, L1 , comprises the usual first-order logical operators along with the two binary predicates
(excess) and 6= (apartness) treated as primitives. These predicates are governed by the following proper axioms, where all free variables are tacitly assumed
to be universally quantified.
¬x x
x y→x
z∨z
(1)
(2)
y
¬ x 6= x
(3)
x 6= y → y 6= x
x 6= y → x 6= z ∨ z 6= y
(4)
(5)
x 6= y → x
x
y∨y
(6)
x
(7)
y → x 6= y
6 Classically, overlap is defined as sharing of a common part, and here we shall go along
with that definition. We leave it to future work to investigate the possibility of adopting a
notion of overlap with greater constructive appeal, such as the intuitionistic overlap relation
developed by Ciraulo et al. [7, 8] in the context of Sambin’s ‘overlap algebra’ [38].
4
In the terminology of constructive orders, axiom (1) and (2) state that
is
irreflexive and co-transitive, whereas axioms (3)–(5) state that 6= is irreflexive
and co-transitive as well as symmetric. Axiom (6) may be thought of as the
contrapositive of the antisymmetry principle of classical parthood: it states that
is co-antisymmetric or, less clumsily, weakly linear. Finally, (7) states that
excess implies apartness.
Now let AX1 = NI + (1)–(7), where NI (also known in the literature as NJ) is
Gentzen’s natural deduction system of first-order intuitionistic logic [14]. Thus
AX1 is a variant of von Plato’s excess theory mentioned above, whose only
primitive obeys axioms (1) and (2); we shall see below that his definition of
x 6= y as x
y∨y
x yields a system that is essentially equivalent to the
result of adding our axioms (3)–(7). Moreover, in AX1 there is a natural way of
defining the familiar mereological relations of parthood, equality, and overlap
as well as four distinct relations of proper parthood.
x6y
x=y
x◦y
x <i y
x
x
x
x
is part of y
is equal to y
overlaps y
is a properi part of y, for i ∈ {1, 2, 3, 4}
The definitions are as follows:
y
(8)
x = y := ¬ x 6= y
(9)
x 6 y := ¬ x
x ◦ y := ∃z(z 6 x ∧ z 6 y)
x <1 y := x 6 y ∧ ¬ x = y
(10)
(11)
x <2 y := x 6 y ∧ ¬ y 6 x
(12)
x <3 y := x 6 y ∧ x 6= y
x <4 y := x 6 y ∧ y x
(13)
(14)
(The first two definitions of x <i y parallel the classical definitions of < in terms
of non-equality or non-parthood; the last two are their natural counterparts in
terms of 6= and .) These definitions will be collectively referred to as DF1. Our
first mereological theory, IM1, is the result of adding DF1 to AX1.
Here are some important features of IM1. First, it follows immediately that
6, =, ◦, and each <i has the properties one would expect.
Theorem 1. In IM1 , (i) 6 is reflexive, antisymmetric, and transitive (a partial
order); (ii) = is reflexive, symmetric, and transitive (an equivalence relation);
(iii) ◦ is reflexive and symmetric (a tolerance relation); and (iv) each <i is irreflexive and transitive (a strict partial order). Moreover, (v) 6 and = are stable,
i.e. satisfy the biconditionals x 6 y ↔ ¬¬ x 6 y and x = y ↔ ¬¬ x = y.
Proof. The reflexivity of 6, =, and ◦ follows immediately from axioms (1) and
(3) and definitions (8)–(10). The remaining proofs for (i)–(iv) are routine. As an
5
illustration we give a derivation of the antisymmetry of 6. To this end, here and
below it will be convenient to write out derivations by treating proper axioms
and definitions as rules of inference, writing (n) to label an inference step that
results from an application of the rule corresponding to axiom or definition (n).
For example, in place of definition (8) we shall use the following rules:
x6y
¬x y
¬x y
x6y
(8)
(8)
(Labels corresponding to the applications of purely logical rules will be omitted.)
On this understanding, the derivation of 6-antisymmetry in IM1 goes as follows.
2
1
[x 6= y]
x
y∨y
x
(6)
[x
3
3
[x 6 y ∧ y 6 x]
x6y
[x 6 y ∧ y 6 x]
y6x
¬x
y]
y
1
(8)
[y
⊥
¬y
x]
x
(8)
⊥
⊥
2
¬ x 6= y
x = y (9)
x6y∧y 6x→x=y
1
3
The other derivations are similar and left to the reader. As for (v), both biconditionals follow immediately from definitions (8) and (9) owing to the intuitionistic
validity of ¬¬-introduction and ¬¬¬-reduction.
Second, in IM1 the relations and 6= are irreducibly distinct from the negations of 6 and =. Specifically, the conditionals
y → ¬x 6 y
(15)
x 6= y → ¬ x = y
(16)
x
are derivable in IM1, but the converse conditionals
y
(17)
¬ x = y → x 6= y
(18)
¬x 6 y → x
are not. This can be checked with standard semantic techniques, e.g. Kripke
models.7 For (17) it suffices to consider a model with two worlds α, β and two
objects a, b in the domain of α such that (i) β is accessible from α, and (ii) a b
holds only at β.
a
α
b
β
7 We assume familiarity with Kripke models for intuitionistic logic, referring to Kripke’s
original article [23] and to [45] for a systematic presentation. Specific applications to intuitionistic theories may be found in [43] and, with special reference to order theories, in [17].
6
In such a model all the axioms of IM1 are valid. Moreover, we have that ¬ a b
holds at neither world, and hence ¬¬ a b holds at α. By (8), this means that
¬ a 6 b holds at α. Yet a b does not hold at α by assumption. Therefore the
conditional ¬ a 6 b → a b does not hold either, showing the invalidity of (17).
The invalidity of (18) is shown in a perfectly similar way, assuming a 6= b holds
only at β.
Another principle that is not derivable in IM1 is the following classically
looking conditional, which fails for each i ∈ {1, 2, 3, 4}.
(19)
x 6 y → x <i y ∨ x = y
This can be seen by considering slight variants of the previous model, with just
two worlds α, β and β accessible from α. For instance, for i = 4 it is enough to
assume that b a holds only at β.
b
α
a
β
Then again the axioms of IM1 are valid but x 6 y → x <4 y ∨ x = y is not. For,
on the one hand, since b
a does not hold at α, neither does a 6 b ∧ b
a.
Hence a <4 b does not hold at α by (14). Moreover, since b a holds at β, so
does b 6= a by (7), from which it follows by (4) that a 6= b also holds at β; and
since β is accessible from α, the latter fact implies that ¬ a 6= b does not hold
at α. Hence a = b does not hold at α by (9). So neither a <4 b nor a = b holds
at α, and this is enough to conclude that (i) the disjunction a <4 b ∨ a = b
does not hold at α. On the other hand, the model is such that a
b holds
neither at α nor at any other world accessible from it, which means that ¬ a b
holds at α. Hence by (8) we also have that (ii) a 6 b holds at α. Given (i) and
(ii), it follows that a 6 b → a <4 b ∨ a = b does not hold at α, showing that
(19) is not valid for i = 4. For i = 1, 2, 3, the invalidity of (19) can be shown in
similar ways.
A third important fact about IM1 regards precisely the plurality of proper
parthood predicates defined in (11)–(14). We saw that (11) and (12) are patterned after the classical definitions, using = and 6, whereas (13) and (14)
characterize proper parthood directly in terms of 6= and . As it tuns out, each
pair is redundant. Given the co–antisymmetry axiom (6), the former definitions
are equivalent to each other (as they are in classical mereology) and are implied
by the latter, which are also equivalent to each other.
Theorem 2. Each the following is derivable in IM1 : (i) x <1 y ↔ x <2 y;
(ii) x <3 y → x <1 y; (iii) x <4 y → x <2 y; (iv) x <3 y ↔ x <4 y.
Proof. We prove explicitly the last claim, considering each direction of the biconditional separately. The other claims are proved similarly. The derivation of
left-to-right direction of (iv) is as follows.
7
2
2
1
[x 6 y ∧ x 6= y]
x 6= y
2
[x 6 y ∧ x 6= y]
x6y
x
y∨y
[x 6 y ∧ x 6= y]
x6y
[x
y
y
x6y∧y
y
(8)
⊥
(6)
x
¬x
y]
1
[y
x
x]
1
x
x
x 6 y ∧ x 6= y → x 6 y ∧ y
x
2
For the right-to-left direction, the derivation is even simpler.
3
[x 6 y ∧ y
y
1
[x 6 y ∧ y x]
y 6= x
x6y
x 6= y
x 6 y ∧ x 6= y
x6y∧y
x]
x
(7)
(4)
x → x 6 y ∧ x 6= y
1
The content of Theorem 2 is reproduced in the diagram below, where the two additional implications (dashed arrows) follow directly from the others by the
transitivity of derivability in NI.
<3
<4
<1
<2
These facts concerning proper parthood are especially important when it
comes to the intuitionistic counterparts of other principles of classical mereology, beginning with so-called supplementation principles. Consider Weak Supplementation, which says that whenever something has a proper part, it has
another part disjoint from the first. In L1 we can formulate four variants of this
principle, one for each <i .
x <1 y → ∃z(z 6 y ∧ ¬ z ◦ x)
x <2 y → ∃z(z 6 y ∧ ¬ z ◦ x)
(WS1 )
(WS2 )
x <3 y → ∃z(z 6 y ∧ ¬ z ◦ x)
x <4 y → ∃z(z 6 y ∧ ¬ z ◦ x)
(WS3 )
(WS4 )
Given the equivalences in Theorem 2, it’s clear that WS1 and WS2 coincide,
as do WS3 and WS4 , so really there are only two different ways of recasting
Weak Supplementation in IM1. Moreover, the implications in Theorem 2 immediately give us that the equivalent formulations based on <1 and <2 , i.e.
8
WS1 and WS2 , entail the other formulations, WS3 and WS4 . This, in turn, is
relevant to the relationship between Weak Supplementation and Strong Supplementation, which again can be formulated either indirectly using 6 in the
antecedent or directly using .
(SS1)
(SS2)
¬ y 6 x → ∃z(z 6 y ∧ ¬ z ◦ x)
y x → ∃z(z 6 y ∧ ¬ z ◦ x)
Since y
x implies ¬ y 6 x, the first of these principles is stronger than the
second; and while it entails both forms of Weak Supplementation, SS2 entails
only the weaker form expressed by WS3 and WS4 . The following theorem and
diagram summarize these facts.
Theorem 3. Each the following is derivable in IM1 : (i) WS1 → WS3 ; (ii) WS2
→ WS4 ; (iii) WS1 ↔ WS2 ; (iv) WS3 ↔ WS4 ; (v) SS1 → WS1 ; (vi) SS2 → WS3 ;
(vii) SS1 → SS2.
SS1
SS2
WS3
WS4
WS1
WS2
To be sure, the overall picture is more complex. Classically Weak Supplementation is sometimes formulated using < in the consequent instead of 6,8 so
one could do the same in IM1. Each WSi would have four variants, each one
obtained by replacing 6 with <j for some j ∈ {1, 2, 3, 4}, for a total of sixteen
instances of the following schema:
(WSij )
x <i y → ∃z(z <j y ∧ ¬ z ◦ x)
And while the two classical formulations are equivalent, at least so long as
6 and < behave classically, one may wonder how things are in IM1. The answer
8 This is actually Simons’ original formulation in [40]; the formulation with 6 is from [6].
See also [20] and [47, §3.1].
9
is that, in light of Theorem 2, the sixteen principles in question reduce to just
four principles, namely:
x <1 y → ∃z(z <1 y ∧ ¬ z ◦ x)
(WS11 )
x <1 y → ∃z(z <3 y ∧ ¬ z ◦ x)
(WS13 )
x <3 y → ∃z(z <1 y ∧ ¬ z ◦ x)
x <3 y → ∃z(z <3 y ∧ ¬ z ◦ x)
(WS31 )
(WS33 )
Moreover, since <3 implies <1 , these four principle are partially ordered in terms
of logical strength, with WS13 at the top and WS31 at the bottom. And since
any <i implies 6, it follows immediately that WS31 in turn implies the weakest
of the supplementation principles mentioned earlier, namely WS3 (equivalently:
WS4 ). These implications are summarized in the following theorem and diagram.
Theorem 4. Each the following is derivable in IM1 : (i) WS11 → WS31 ; (ii)
WS13 → WS33 ; (iii) WS13 → WS11 ; (iv) WS33 → WS31 ; (v) WS31 → WS3 .
WS13
WS11
WS33
WS31
WS3
Let us also stress that all these supplementation principles involve a notion of
overlap that is well-behaved.9 By itself, the definition in (10) does not prevent ◦
from holding trivially because of a “null individual” that is part of everything.
However, precisely the existence of such an individual is ruled out (in nondegenerate models) by the principles in question. This is how things are in
classical mereology, too. There, the non-existence of a null individual is usually
expressed by the following thesis, which follows from Weak Supplementation:
(20)
∃x∃y ¬ x = y → ¬∃z∀u z 6 u
Here the same idea may be captured by (20) itself, or by its counterpart in terms
of apartness:
(21)
∃x∃y x 6= y → ¬∃z∀u z 6 u
9
Thanks to a referee for pressing us on this point.
10
And it’s easy to see that both conditionals can be derived in IM1 from any of
the supplementation principles charted above.
Theorem 5. In IM1 , both (20) and (21) are derivable from SS1, from SS2,
and from each WSi and each WSij (i, j ∈ {1, 2, 3, 4}).
Proof. Since WS3 and WS4 are entailed by all other supplementation principles, and since (21) is an immediate consequence of (20) (because x 6= y implies
¬ x = y), it will suffice to show that (20) can be derived from WS3 or, equivalently, from WS4 . We give the derivation from WS3 .
4
[x 6= y]
5
[¬ x = y]
6
[∃y ¬ x = y]
9
[∃x∃y ¬ x = y]
8
WS3 ∀u z 6 u
.. ..
..
D.
.
..
⊥
4
¬ x 6= y
(9)
x=y
⊥
5
⊥
6
[∃z∀u z 6 u]
⊥
7
⊥
8
¬∃z∀u z 6 u
9
∃x∃y ¬ x = y → ¬∃z∀u z 6 u
Here D stands for the following derivation of ⊥ from x 6= y, WS3 , and ∀u z 6 u,
3
3
[x 6= z]
x 6= y
x 6= z ∨ z 6= y
WS3 ∀u z 6 u
.. ..
..
E.
.
.
.
⊥
⊥
WS3 ∀u z 6 u
.. ..
..
F
.
.
.
.
⊥
[z 6= y]
3
where, in turn, E and F are perfectly parallel derivations of ⊥ from x 6= z and
z 6= y, respectively. We just give the details for E , where it will be clear that
the overall proof depends on the reflexivity of 6, as in classical mereology.
∀u z 6 u ∀z z 6 z
z6v
z6z
z
6
v
∧
z
6z
1
[v 6 x ∧ ¬ v ◦ z] ∃w(w 6 v ∧ w 6 z)
¬v ◦ z
v◦z
⊥
∀u z 6 u
z6x
x 6= z
2
[z 6 x] z 6= x
(13)
WS3
z <3 x
∃v(v 6 x ∧ ¬ v ◦ z)
⊥
¬z 6 x
⊥
11
1
2
(10)
Lastly, it is worth noting that although IM1 treats both
tives, the following biconditional is provable:
x 6= y ↔ x
y∨y
and 6= as primi(22)
x
From left to right, this is just the co-antisymmetry axiom (6). In the other direction, we have that x y ∨ y x → x 6= y is intuitionistically equivalent to
(x y → x 6= y) ∧ (y x → x 6= y), which is an immediate consequence of (7)
and (4). It follows, therefore, that in IM1 apartness is definable in terms of .
This takes us back to von Plato’s treatment of the excess relation, which adopts
the definition explicitly along with just (1) and (2), the irreflexivity and cotransitivity axioms for . That axioms (3)–(5) are derivable from (22) is proved
in [35, thm. 3.1], and it’s easy to verify that (6) and (7) are also derivable. So
the two systems are essentially equivalent. It should be noted, however, that von
Plato’s actual axiomatization, as well as the further developments in [30], do
not quite coincide with the system NI + (1) + (2) + (22), since they are based
on a quantifier-free logic. By contrast, mereology requires quantification theory.
This is clear from the formulation of such theses as (20) and (21), or of any
supplementation principle, as well as from the definition of such concepts as
overlap. So we need the full strength of L1 .
4
Mereologies based on weak excess and apartness
We now move to the second sort of theory, whose language L2 has ≮ (weak excess) and 6= (apartness) as primitives. The axioms are:
x≮x
x≮y →x≮z∨z ≮y
(23)
(24)
¬ x 6= x
(25)
x 6= y → y 6= x
x 6= y → x 6= z ∨ z 6= y
(26)
(27)
x ≮ y → x ≮ z ∨ z 6= y
(28)
x ≮ y → x 6= z ∨ z ≮ y
(29)
Axioms (23) and (24) state that ≮ is reflexive and co-transitive, whereas (25)–
(27) coincide with the apartness axioms of AX1, (3)–(5). As for (28) and (29),
these axioms may be thought of as contrapositives of the Leibniz principles
x < z ∧ z = y → x < y and x = z ∧ z < y → x < y.
Let AX2 = NI + (23)–(29). It turns out that this is quite a strong theory, as
it derives two forms of linearity along with instances of the De Morgan laws.
Theorem 6. Each of the following is derivable in AX2 : (i) x ≮ y ∨ y ≮ x;
(ii) x ≮ y ∨ x 6= y; (iii) (¬ x ≮ y ∨ ¬ y ≮ x) ↔ ¬(x ≮ y ∧ y ≮ x); (iv) (¬ x ≮ y ∨
¬ x 6= y) ↔ ¬(x ≮ y ∧ x 6= y).
12
Proof. Regarding (i), consider x ≮ x → x ≮ y ∨ y ≮ x. This is an instance of (24).
Given (23), x ≮ y ∨ y ≮ x follows by modus ponens. The proof of (ii) is similar.
Regarding (iii), one direction is an instance of ¬A ∨ ¬B → ¬(A ∧ B), which is
a theorem of NI. For the other direction we have the following derivation, where
A is ¬(x ≮ y ∧ y ≮ x) and B is ¬ x ≮ y ∨ ¬ y ≮ x. We helpfully use (i).
3
[x ≮ y]
x≮y∨y ≮x
2
1
[y ≮ x]
1
x≮y∧y ≮x
[A]
⊥
¬y ≮ x
1
B
B
4
A→B
[x ≮ y]
3
[y ≮ x]
x≮y∧y ≮x
⊥
¬x ≮ y
B
4
[A]
1
3
The proof of (iv) is similar and is left to the reader.
In AX2 we can again introduce defined predicates for parthood (6), equality
(=), and overlap (◦) as well as a predicate for proper parthood (<), which now
can be characterized directly in terms of ≮.
x < y := ¬ x ≮ y
(30)
x 6 y := ¬(x ≮ y ∧ x 6= y)
(31)
x = y := ¬ x 6= y
x ◦ y := ∃z(z 6 x ∧ z 6 y)
(32)
(33)
Of course, given (31) one could still rely on 6 to introduce four additional
properi parthood predicates <i as in (11)–(14). Alternatively, one could consider
splitting 6 into four parthoodj predicates
x 61 y := ¬(x ≮ y ∧ x 6= y)
x 62 y := ¬(x ≮ y ∧ ¬ x = y)
(34)
(35)
x 63 y := ¬(¬ x < y ∧ x 6= y)
x 64 y := ¬(¬ x < y ∧ ¬ x = y)
(36)
(37)
and define a properi parthoodj predicate for each i, j ∈ {1, 2, 3, 4}. This is just
combinatorics and we shall not pursue the details, focusing on the simple predicates < and 6. The definitions in (30)–(33) will be collectively referred to as DF2
and we shall take our second theory, IM2, to be the result of adding DF2 to AX2.
As with IM1, it’s easy to verify that in IM2 6 is still a partial order, = an
equivalence relation, and ◦ a tolerance relation. Moreover < is a strict partial
order and is stable, like 6 and =.
Theorem 7. In IM2 (i) 6 is reflexive, antisymmetric, and transitive; (ii) = is
reflexive, symmetric, and transitive; (iii) ◦ is reflexive and symmetric; (iv) < is
irreflexive and transitive; (v) 6, =, and < are stable.
13
Proof. Again, the proofs are routine. We give derivations only for the reflexivity
and antisymmetry of 6, which will be used later. Here is reflexivity.
1
[x
x ∧ x 6= x]
x 6= x
¬ x 6= x
⊥
¬(x ≮ x ∧ x 6= x)
x6x
(25)
1
(31)
For the proof of antisymmetry, we use again the linearity property x ≮ y ∨ y ≮ x
established in Theorem 6(i), which we cite as L for space limitations.
2
3
[x 6= y]
[x 6 y ∧ y 6 x]
1
[y ≮ x] y 6= x
[x ≮ y] [x 6= y]
x6y
(31)
x ≮ y ∧ x 6= y
¬(x ≮ y ∧ x 6= y)
y ≮ x ∧ y 6= x
⊥
⊥
2
¬ x 6= y
x = y (32)
3
x6y∧y 6x→x=y
1
L
2
3
(26)
⊥
[x 6 y ∧ y 6 x]
y6x
¬(y ≮ x ∧ y 6= x)
(31)
1
We also note that IM2 is strong enough to prove the following conditional,
whose <i -counterpart (19) was not derivable in IM1 for any i.
(38)
x6y →x<y∨x=y
For a proof, assume x 6 y. By definition (31), this amounts to ¬(x ≮ y ∧ x 6= y).
By Theorem 6(iv), the latter is equivalent to ¬ x ≮ y ∨ ¬ x 6= y, and hence to
x < y ∨ x = y by definitions (30) and (32). Thus, x 6 y → x < y ∨ x = y.
Given the relation of proper parthood defined in (30), both Weak and Strong
Supplementation can be formulated in IM2 as in classical mereology, and it is
easy to see that, as in classical mereology, the latter principle implies the former
by the ordering properties of < (using (38) to derive ¬ y 6 x from x < y).
(WS)
x < y → ∃z(z 6 y ∧ ¬ z ◦ x)
(SS)
¬ y 6 x → ∃z(z 6 y ∧ ¬ z ◦ x)
We leave it to the reader to check that both principles also rule out the existence
of a null individual, as expressed by (20) or (21).10
5
Excess vs. weak excess
What exactly is the relationship between excess and weak excess? In Section 2 we
said it is analogous to the relationship between parthood and proper parthood
in classical mereology, and we know those relations are interdefinable. But this
10
This result also follows from Theorem 8 below.
14
should not suggest that and ≮ are themselves interdefinable intuitionistically.
In fact they are not, at least not straightforwardly. We can prove that the system
based on ≮ allows for a natural way to define a relation satisfying the axioms
of AX1. However, in the system based on it is hard to find an equally natural
definition of ≮ that yields the axioms of AX2 as theorems.
We start with the first claim. The natural way to define in AX2 is to think
of excess as ‘proper’ weak excess: x y iff x ≮ y ∧ x 6= y. Under this definition,
we can show that AX1 is indeed interpretable in AX2. More precisely, let τ be a
translation from L1 to L2 such that:
τ (x
y) := x ≮ y ∧ x 6= y
τ (x 6= y) := x 6= y
τ (⊥)
:= ⊥
τ (A ∧ B) := τ (A) ∧ τ (B)
τ (A ∨ B) := τ (A) ∨ τ (B)
τ (A → B) := τ (A) → τ (B)
τ (∀xA)
:= ∀x τ (A)
τ (∃xA)
:= ∃x τ (A)
Recall that intuitionistic negation is defined as ¬A := A → ⊥, so we also have
τ (¬A) = ¬ τ (A). On this basis, the main result can be states as follows.
Theorem 8. For every L1 -formula A derivable in AX1 , τ (A) is derivable in AX2 .
Proof. Both systems are based on NI, so it will suffice to prove the result for formulas that are AX1-axioms. In fact, the translation of the three apartness axioms
(3)–(5) coincides with the corresponding axioms of AX2, (25)–(27), so we only
need consider the other four axioms of AX1, viz. (1), (2), (6), and (7). The case
for (1) is straightforward, since τ (¬x x) := ¬(x ≮ x ∧ x 6= x) is an immediate
consequence of (25). The case for (7) is even easier, since τ (x y → x 6= y) :=
x ≮ y ∧ x 6= y → x 6= y is a theorem of NI. The other two cases, however, involve
disjunctions and require more care, so we give detailed proofs.
y → x
z∨z
y) := x ≮ y ∧ x 6= y →
Proof of (2) – We have τ (x
(x ≮ z ∧ x 6= z) ∨ (z ≮ y ∧ z 6= y). Let A be the antecedent, x ≮ y ∧ x 6= y,
and let B and C be the two disjuncts in the consequent, x ≮ z ∧ x 6= z and
z ≮ y ∧ z 6= y. Consider the following derivation D of B ∨ C from A and x ≮ z:
A
x≮y
x 6= z ∨ z ≮ y
x≮z
(29)
2
[x 6= z]
B
B∨C
B∨C
A
x 6= y
x 6= z ∨ z 6= y
1
x≮z
(27)
[x 6= z]
B
B∨C
B∨C
2
[z ≮ y]
1
[z 6= y]
C
B∨C
1
2
In a perfectly similar fashion, we may construct a derivation E of B ∨ C from
A and z ≮ y. Using D and E , we obtain a derivation of A → B ∨ C as follows:
15
2
2
1
[x ≮ z]
.. ..
2
..
[A]
D.
..
x≮y
.
(24)
B∨C
x≮z∨z ≮y
B∨C
2
A→B∨C
1
[z ≮ y]
.. ..
..
E.
..
.
B∨C
[A]
[A]
1
Proof of (6) – Here we have τ (x 6= y → x y ∨ y x) := x 6= y → (x ≮ y
∧ x 6= y) ∨ (y ≮ x ∧ y 6= x). Recall once more the linearity property from Theorem 6(i), namely x ≮ y ∨ y ≮ x, and let A and B abbreviate x ≮ y ∧ x 6= y and
y ≮ x ∧ y 6= x, respectively. Then we have the following derivation:
2
1
[x ≮ y]
x≮y∨y ≮x
2
1
[x 6= y]
A
A∨B
A∨B
x 6= y → A ∨ B
[x 6= y]
y 6= x
[y ≮ x]
B
A∨B
(26)
1
2
Theorem 8 answers one half of our question. It tells us that AX1 can be fully
recovered within AX2, which means that the excess relation can effectively be
recast in terms of weak excess. Turning now to the other half, it would be nice
to have a parallel result—a way of defining in AX1 a binary relation of weak
excess that obeys the axioms characteristic of AX2. Unfortunately things are
not so easy.
The natural option would be to define x ≮ y as x y ∨ ¬ x 6= y, dualizing
the definition of in AX2. Such a definition, however, would not deliver the intended result. Particularly, it would not secure the derivability in AX1 of the
co-transitivity of ≮, which is axiom (24) of AX2. This is shown by the following
Kripke model, with two worlds α, β and three objects a, b, c in the domain of
α such that (i) β is accessible from α, (ii) a 6= c and c 6= b hold at β, and (iii)
a c does not hold at α.
a 6= c
c 6= b
α
β
In such a model, all axioms of AX1 are valid but ≮ as defined above is not cotransitive, i.e. x ≮ y → x ≮ z ∨ z ≮ y is not valid. For notice that ¬ a 6= b holds
at α, since a 6= b does not hold at any world accessible from α. Hence the disjunction a b ∨ ¬ a 6= b also holds at α. According to the definition, this means
that a ≮ b holds at α. On the other hand, since a 6= c holds at the accessible world β, ¬ a 6= c does not hold at α. Moreover a c doesn’t hold at α by
assumption. Hence the disjunction a
c ∨ ¬ a 6= c, which would amount to
16
a ≮ c, does not hold either. Similarly we can see that the definiens of c ≮ b
does not hold at α. It follows that according to the definition the disjunction
a ≮ c ∨ c ≮ b does not hold at α even though a ≮ b does, which means that we
have a counterexample to the validity of x ≮ y → x ≮ z ∨ z ≮ y.
Are there any alternatives? We do not have a general answer, but we conjecture that any attempt to define ≮ in terms of will suffer from similar defects,
short of adding (24) ad hoc. Weak excess appears to be intuitionistically irreducible to excess. It is nonetheless noteworthy that the above definition would
work if we allowed some classical reasoning in AX1. Consider the following translation function σ from L2 to L1 .
σ(x ≮ y) := x y ∨ ¬ x 6= y
σ(x 6= y) := x 6= y
σ(⊥)
:= ⊥
σ(A ∧ B) := σ(A) ∧ σ(B)
σ(A ∨ B) := ¬(¬ σ(A) ∧ ¬ σ(B))
σ(A → B) := σ(A) → σ(B)
σ(∀xA)
σ(∃xA)
:= ∀x σ(A)
:= ∃x σ(A)
Again, we have that σ(¬A) = ¬ σ(A). But notice that translating A ∨ B as
¬(¬ σ(A) ∧ ¬ σ(B)) betrays a classical, non-constructive understanding of disjunction. If this is accepted, then we have the following general result.
Theorem 9. For every L2 -formula A derivable in AX2 , σ(A) is derivable in AX1 .
Proof. As with Theorem 8, we may ignore the pure apartness axioms (25)–(27)
and focus on the remaining axioms of AX2, namely (23), (24), (28), and (29). Of
these, the first is obviously derivable in AX1, since σ(x ≮ x) := x x ∨ ¬ x 6= x
is an immediate consequence of the irreflexivity of apartness (3). The derivability
of the other three axioms calls for detailed proofs.
Consider (24). We have σ(x ≮ y → x ≮ z ∨ z ≮ y) := x y ∨ ¬ x 6= y →
¬(¬(x z ∨ ¬ x 6= z) ∧ ¬(z y ∨ ¬ z 6= y)). Since ¬(B ∨ C) and ¬B ∧ ¬C are
intuitionistically equivalent, the consequent of this conditional can be rewritten
as ¬(¬ x z ∧ ¬¬ x 6= z ∧ ¬ z y ∧ ¬¬ z 6= y). Now let A abbreviate ¬ x z ∧
¬¬ x 6= z ∧ ¬ z
y ∧ ¬¬ z 6= y and consider the following derivation D of ⊥
from A, ¬ x 6= y, and z x.
1
[y
z
z
1
x
y∨y
x
(2)
[z
y]
A
¬z y
⊥
x]
y 6= x
x 6= y
(7)
(4)
¬ x 6= y
⊥
1
⊥
Using D, we obtain the following derivation E of ⊥ from A and ¬ x 6= y.
17
1
A
2
z∨z
x
x
[x
(6)
x] ¬ x 6= y
.. ..
..
D.
..
.
⊥
A
¬x z
1
[x 6= z]
[z
z]
⊥
1
⊥
¬ x 6= z
A
¬¬ x 6= z
2
⊥
We now use E to construct a derivation of the desired conditional.
3
2
[x
4
[x
x
z∨z
[A]
1
y]
y
y ∨ ¬ x 6= y]
(2)
[x
3
¬x
z]
[A]
1
z
[z
y]
⊥
⊥
1
⊥
x
⊥
3
¬A
y ∨ ¬ x 6= y → ¬A
¬z
y
3
2
[A]
[¬ x 6= y]
.. ..
..
E.
.
.
.
⊥
2
4
This completes the proof for axiom (24). The proofs for axioms (28) and (29) are
similar and left to the reader.
6
Extensionality
We are now in a position to go back to our initial point concerning mereological
extensionality. Recall the three general principles mentioned in Section 1, which
may be stated formally as follows.
(EP)
(EO)
x6y∧y 6x→x=y
∀z(z ◦ x ↔ z ◦ y) → x = y
∃z z < x ∨ ∃z z < y → (∀z(z < x ↔ z < y) → x = y)
(EPP)
In classical mereology, with 6 as a primitive, EP (Extensionality of Parthood)
is typically assumed as an axiom whereas EO (Extensionality of Overlap) and
EPP (Extensionality of Proper Parthood) are derived from EP with the help
of Strong Supplementation.11 We said that such derivations require classical
reasoning that is not constructively admissible, and we can now be more precise.
Typically, the derivation of EO in classical mereology makes use of the following Overlap Principle:
(OP)
∀z(z ◦ x → z ◦ y) → x 6 y
11 See [47, §3.2]. A notable exception is Leonard and Goodman’s Calculus of Individuals [25],
which is based on a primitive of mereological disjointness and includes EO (or, rather, its
equivalent formulation in terms of disjointness) as an axiom. See also [31] for a systematic overview of classical mereology based on overlap, following Goodman [16].
18
This principle, in turn, is usually established through a proof by contradiction:
given the antecedent of OP, one further assumes ¬ x 6 y to derive by modus
ponens the consequent of Strong Supplementation, ∃z(z 6 x ∧ ¬ z ◦ y), whence
a contradiction quickly follows from ∀z(z ◦ x → z ◦ y).12 The last step is intuitionistically valid, but then one would need classical logic to drop the double
negation from ¬¬ x 6 y to obtain x 6 y.
Similarly, the derivation of EPP typically relies on the following theorem of
classical mereology, known from [40] as the Proper Parts Principle:
(PPP)
∃z z < x → (∀z(z < x → z < y) → x 6 y)
As above, to prove this theorem one begins by assuming ¬ x 6 y in order to obtain ∃z(z 6 x ∧ ¬ z ◦ y) from Strong Supplementation and then reach a contradiction from this formula together with the antecedents of PPP. Thus, again,
the proof eventually requires an application of ¬¬-elimination, which is not
intuitionistically available.
The following NI-model shows that it is indeed possible for 6 to obey the axioms of a strongly supplemented partial order while violating both OP and PPP.
c<a
c<b
d<a
d<b
a=b
α
β
In the model we have a and b with the same proper parts, c and d, with a = b
holding only at β. The partial order axioms for 6 are satisfied at both worlds,
as are the strict order properties of <. (Recall that such models satisfy the
inclusion requirement, so all mereological relations that hold at α continue to
hold at the accessible worlds, hence at β.) Moreover, a 6 b does not hold at α
even though ¬¬ a 6 b does, since a 6 b holds at β and hence ¬ a 6 b holds at
neither world (and similarly for b 6 a, ¬ b 6 a, and ¬¬ b 6 a). It is also easy
to check that Strong Supplementation is satisfied at both worlds (vacuously so
with regard to a and b). Yet OP and PPP fail to hold at α, since the relevant
antecedents ∀z(z ◦ a → z ◦ b), ∃z z < a, and ∀z(z < a → z < b) hold despite the
fact that the consequent a 6 b doesn’t (and similarly for b 6 a etc.). Clearly EO
and EPP fail as well.
All of this confirms that a purely logical intuitionistic counterpart of classical
mereology, i.e. an intuitionistic mereology obtained simply by revising the logic
while adopting the proper axioms of classical mereology verbatim, is not going
to be fully extensional. We can assume EP along with Strong Supplementation,
but EO and EPP will not follow.
There is, to be sure, a sense in which this verdict may be appealed. After all,
there are several ways of embedding classical logic into intuitionistic logic by
means of translation functions that preserve classical equivalence, and any such
12
Some authors actually identify Strong Supplementation with OP; see e.g. [21, p. 187].
19
function would yield a complete intuitionistic counterpart of any classical theory, including the whole of classical extensional mereology.13 A case in point is
the so-called Gödel-Gentzen negative translation (after [15] and [13]), which is
obtained by appending double negations to atomic formulas, disjunctions, and
existentially quantified formulas. Effectively, this amounts to associating each
formula A with a formula g(A) defined inductively as follows:
g(x 6 y) := ¬¬ x 6 y
g(x = y) := ¬¬ x = y
g(⊥)
:= ⊥
g(A ∧ B) := g(A) ∧ g(B)
g(A ∨ B) := ¬(¬g(A) ∧ ¬g(B))
g(A → B) := g(A) → g(B)
g(∀xA)
g(∃xA)
:= ∀x g(A)
:= ¬∀x¬ g(A)
This translation has the property that, for any set of formulas Σ in the language,
A follows from Σ classically iff g(A) follows from {g(B) : B ∈ Σ} intuitionistically. (For details, see e.g. [45, ch. 2].) Thus, when Σ comprises the axioms of
classical extensional mereology, the translation yields an intuitionistically acceptable counterpart of every theorem thereof, including EP along with EO and
EPP. This is telling, since these counterparts are classically equivalent to the
originals. However it is hardly what we wanted. The constructive extensionalist
isn’t just interested in asserting some logically sanitized rendering of classical
extensionality; she wants to assert extensionality tout court.
So the verdict stands: one way or the other, a purely logical revamping of
classical mereology on intuitionistic grounds is not going to be fully extensional.
Is the picture any different when it comes to intuitionistic mereologies that, like
IM1 and IM2, involve a substantive revision of the proper axioms of classical
mereology along with its logical axioms? We said the answer is in the affirmative.
Let us finally see why and to what extent.
To begin with, we know from Theorems 1(i) and 7(i) that 6 is antisymmetric
in IM1 as well as in IM2 (under the relevant definitions), so both theories will be
extensional with regard to their parthood relations. While EP is not assumed
as an axiom in either theory, it is derivable as a theorem in both.
Importantly, the same will be true of EO as soon as we assume Strong Supplementation. Theorems 1(v) and 7(v) tell us that in both theories 6 is a stable
relation. Thus, although ¬¬-elimination is not intuitionistically available as a
general rule of inference, in IM1 and IM2 the special case corresponding to the
inference from ¬¬ x 6 y to x 6 y is legitimate. Both theories may therefore rely
on the standard reductio argument mentioned above to obtain a derivation of
the Overlap Principle from Strong Supplementation, which can certainly be
added to the relevant set of proper axioms. Given OP, EO will then follow in
13
Here, again, we are indebted to a referee for bringing this point to our attention.
20
both cases by the antisymmetry of 6, i.e., effectively, by EP. Of course, strictly
speaking there is more than one way of understanding Strong Supplementation
intuitionistically, depending on the primitive we choose: in IM1 we have two
distinct formulations, corresponding to SS1 and SS2; in IM2 we can recast the
standard formulation SS. As it turns out, however, OP will follow equally from
each of these principles.
Theorem 10. (i) SS1 → OP and SS2 → OP are derivable in IM1 ; (ii) SS → OP
is derivable in IM2 .
Proof. With reference to (i), we know from Theorem 4(vii) that SS1 → SS2 is
derivable in IM1. Thus, SS1 → OP follows directly from SS2 → OP, for which we
have the following derivation. (We assume as given a derivation D of z ◦ x from
z 6 x, which is straightforward given the reflexivity of 6; see Theorem 1(i).)
1
[z 6 x ∧ ¬ z ◦ y]
z6
.. x
..
D.
..
.
z◦x
2
[x
y]
∃z(z 6 x ∧ ¬ z ◦ y)
(SS2)
⊥
2
¬x y
(8)
x6y
∀z(z ◦ x → z ◦ y) → x 6 y
3
[∀z(z ◦ x → z ◦ y)]
z◦x→z◦y
z◦y
⊥
1
[z 6 x ∧ ¬ z ◦ y]
¬z ◦ y
1
3
Note that the second-last step, from ¬ x y to x 6 y, is precisely the special
case of ¬¬-elimination licensed by definition (8). As for (ii), the proof is similar
(using definition (31)) and left to the reader.
With regard to EPP, IM1 and IM2 behave differently. The standard reductio
argument to obtain the Proper Parts Principle does not only rely on the inference
from ¬¬ x 6 y to x 6 y, which is admissible in both theories; it also makes use
of the classical conditional x 6 y → x < y ∨ x = y, which, as we saw, is available
only in IM2 (see our earlier discussion of (19) and (38)). This difference happens
to be crucial.
On the one hand, it turns out that IM1 is indeed too weak to derive PPP
from SS1 (and hence, a fortiori, from SS2). More precisely, there are four ways
of recasting PPP in IM1, one for each properi parthood predicate <i :
∃z z <i x → (∀z(z <i x → z <i y) → x 6 y)
(PPPi )
In view of Theorem 2, this really gives us two non-equivalent principles in the
language of IM1, and neither turns out to be derivable. Here, for instance, is a
model that illustrates the point for i = 4.
21
a
a
e
a
a
b
b
c
d
c
d
b
e
b
β
α
γ
b
b
e
a
e
a
Given definition (14), this is a model of IM1 where a has exactly two proper4
parts at world α, namely c and d, both of which are also proper4 parts of b.
Yet a is not, at α, part of b, since a b holds at β and therefore ¬ a b, which
amounts to a 6 b by definition (8), holds at neither world. It follows that α
violates the conditional ∃z z <4 a → (∀z(z <4 a → z <4 b)) → a 6 b), which is
an instance of PPP4 . (A perfectly parallel story applies to b 6 a.) However, it’s
easy to check that SS1 does hold at α. In particular, since ¬ a b holds at γ,
¬¬ a b does not hold at α and hence ¬ a 6 b does not hold either (again by
definition (8)), so the conditional ¬ a 6 b → ∃z(z 6 a ∧ ¬ z ◦ b) holds vacuously.
It is also easy to check that, thanks to e, SS1 may hold (non-vacuously) at β
and at γ as well, e.g. when e is atomic and disjoint from c and d. Then SS1 will
hold at every world, and since SS1 implies SS2 by Theorem 4(vii), the same is
true of SS2. The model will satisfy SS1 and SS2 but not PPP4 . Similar models
will establish the same result for the other PPPi ’s.
By contrast, IM2 turns out to be strong enough to warrant the derivation:
PPP follows directly from Strong Supplementation, as in classical mereology.
Theorem 11. SS → PPP is derivable in IM2 .
Proof. The derivation is a bit long, so we shall split it into four steps. Let A be
∃z z < x and let B be ∀z(z < x → z < y). We want to construct a derivation of
A → (B → x 6 y). We begin with a derivation D of u ◦ y from B and u < x.
1
u<x
[u ≮ y ∧ u 6= y]
u≮y
B
u<x→u<y
u<y
(30)
¬u ≮ y
⊥
¬(u ≮ y ∧ u 6= y)
u6y
u6u
u6u∧u6y
∃v(v 6 u ∧ v 6 y)
(33)
u◦y
1
(31)
(Here the left-most leaf, u 6 u, comes from the reflexivity of 6 established in
Theorem 7(i).) Next, consider a derivation E of z 6 y from B and z < x.
22
z<x
1
[z ≮ y ∧ z 6= y]
z≮y
B
z<x→z<y
z<y
(30)
¬z ≮ y
⊥
¬(z ≮ y ∧ z 6= y)
z6y
1
(31)
We now use E to construct a derivation F of u ◦ y from B, z < x, and u = x.
2
1
[z ≮ u ∧ z 6= u]
z≮u
z ≮ x ∨ x 6= u
z<x
[z ≮ x] ¬ z ≮ x
⊥
⊥
¬(z ≮ u ∧ z 6= u)
z6u
1
(28)
[x 6= u]
u 6= x
(30)
(26)
⊥
u=x
¬ u 6= x
(32)
1
2
z 6u∧z 6y
∃v(v 6 u ∧ v 6 y)
u◦y
z<x B
.. ..
..
E.
..
.
z6y
(33)
Finally, using D and F , and abbreviating u 6 x ∧ ¬ u ◦ y as C, we obtain the
following derivation of the entire Proper Parts Principle A → (B → x 6 y) from
the relevant instance of Strong Supplementation, ¬x 6 y → ∃uC.
1
2
4
[¬ x 6 y]
∃uC
(SS)
[C]
u6x
u<x∨u=x
⊥
A
⊥
4
¬¬ x 6 y
x6y
5
B→x6y
A → (B → x 6 y)
5
1
[u < x] [B]
.. ..
..
D.
2
..
[C]
.
u◦y
¬u ◦ y
⊥
⊥
[u = x]
3
[z < x]
.. ..
..
F
..
..
u◦y
5
[B]
2
[C]
¬u ◦ y
⊥
1
2
3
6
Notice that the inference step from u 6 x to u < x ∨ u = x in this final derivation requires an instance of the conditional x 6 y → x < y ∨ x = y. This is precisely the IM2-theorem (38) noted above, whose <i -counterparts do not hold in
IM1 for any i.
Given Theorem 11, the derivability of EPP in IM2 + SS is now an immediate
consequence of the antisymmetry of 6, i.e., effectively, of EP, exactly as with
the derivation of EO. This is how the three extensionality principles are related
also in classical mereology. Since IM1 + SS1 and IM1 + SS2 fail to derive PPP,
and EPP with it, we conclude that it is the theory of weak excess that provides
the best resources for a natural intuitionistic counterpart of classical extensional
23
mereology.14 In IM1 we would achieve full extensionality only by assuming PPP
explicitly, as a further axiom along with SS1 or SS2. While perfectly legitimate
in its own right, such a move would defy the classical way of understanding
extensionality—as a property of any parthood relation that is antisymmetric
and strongly supplemented.
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