[In Claudio Calosi and Pierluigi Graziani (eds.), Mereology and the Sciences, Berlin: Springer, 2014, pp. 359–370]

Appendix. Formal Theories of Parthood
Achille C. Varzi
This Appendix gives a brief overview of the main formal theories of parthood, or
mereologies, to be found in the literature.1 The focus is on classical theories, so thesurvey is not meant to be exhaustive. Moreover, it does not cover the many philosophical issues relating to the endorsement of the theories themselves, concerningwhich the reader is referred to the Selected Bibliography at the end of the volume. In
particular, we shall be working under the following simplifying assumptions:2— Absoluteness: Parthood is a two-place relation; it does not hold relative to time,
space, spacetime regions, sortals, worlds, or anything else.3— Monism: There is a single relation of parthood that applies to every entity independently of its ontological category.4— Precision: Parthood is not a source of vagueness: there is always a fact of the matter as to whether the parthood relation obtains between any given pair of things.5
For definiteness, all theories will be formulated in a standard first-order languagewith identity, supplied with a distinguished binary predicate constant, ‘P’, to be interpreted as the parthood relation. The underlying logic will be the classical predicate calculus.
1 Core Principles
As a minimal requirement on ‘P’, it is customary to assume that it stands for a partial
order—a reflexive, transitive, and antisymmetric relation:6
(P.1)
(P.2)
(P.3)

Pxx
(Pxy ∧ Pyz) → Pxz(Pxy ∧ Pyx) → x = y

Reflexivity
Transitivity
Antisymmetry

Together, these three axioms are meant to fix the intended meaning of the parthoodpredicate. They form the “core” of any standard mereological theory, and the theory
1

The exposition follows Varzi (2014). For a thorough survey, see Simons (1987).
The labels and formulations of these assumptions are from Sider (2007).
3
For the view that parthood should be a three-place relation relativized to time, see e.g. Thomson(1983). For the view that it should be a four-place relation, see Gilmore (2009).
4
For misgivings about Absoluteness and related worries, see e.g. Mellor (2006) and McDaniel (2009).
5
For mereologies that allow for indeterminate or “fuzzy” parthood relations, see e.g. Smith (2005)
and Polkowski (2011).
6
Unless otherwise specified, all formulas are to be understood as universally closed.
2

1that comprises just them is called Ground Mereology, or M for short.7 A number of additional mereological predicates may then be introduced by definition:
(1)
(2)
(3)
(4)
(5)

EQxy =df
PPxy =df
PExy =df
Oxy =df
Uxy =df

Pxy ∧ Pyx
Pxy ∧ ¬x = y
Pyx ∧ ¬x = y∃z (Pzx ∧ Pzy)∃z (Pxz ∧ Pyz)

equality
proper parthood8
proper extension
overlap
underlap

Given (P.1)–(P.3), it follows immediately that EQ is an equivalence relation. Moreover, PP and PE are irreflexive, asymmetric, and transitive whereas O and U are reflexive and symmetric, but not transtive. Since the following is a also theorem of M,(6) Pxy ↔ (PPxy ∨ x = y)‘PP’ could have been used as a primitive instead of ‘P’. Similarly for ‘PE’. Sometimes‘P’ is also defined in terms of ‘O’ via the biconditional(7) Pxy ↔ ∀z(Ozx → Ozy).
However, (7) is not provable in M and calls for stronger axioms (specifically, the axioms of theory EM defined below). Since those stronger axioms reflect substantive
philosophical theses, ‘P’ and ‘PP’ (or ‘PE’) are the best options to start with. Here we
stick to ‘P’.

Figure 1. Basic patterns of mereological relations. (Shaded cells indicate parthood).

2 Decomposition Principles
M is standardly viewed as embodying the common core of any mereological theory.
Yet not every partial order qualifies as parthood, and establishing what further re7

For a survey of the motivations that may lead to the development of non-standard mereologies inwhich P is weaker than a partial order, see Varzi (2014: §2.1).
8
In the literature, proper parthood is sometimes defined as asymmetric parthood: PPxy =df Pxy ∧ ¬Pyx.
Given Antisymmetry, this definition is equivalent to (2). Without Antisymmetry, however, the two definitions would come apart. (Similarly for ‘PE’.) See Cotnoir (2010).

2

quirements should be added to (P.1)–(P.3) is precisely the question a good mereological theory is meant to answer.
One way to extend M is by means of decomposition principles, i.e., principles concerning the part structure of a given whole. Here, one fundamental intuition is that nowhole can have a single proper part. There are several ways in which this intuition
can be captured, beginning with the following:(P.4a) PPxy → ∃z(PPzy ∧ ¬z = x)
(P.4b) PPxy → ∃z(PPzy ∧ ¬Pzx)(P.4) PPxy → ∃z(PPzy ∧ ¬Ozx)(Weak) Company
Strong Company(Weak) Supplementation9(P.4a) is the literal rendering of the idea in question, but it is too weak: it rules out
certain implausible finitary models (Fig. 2, left) but not, for example, models with
infinitely descending chains in which the additional parts do not leave any mereological “remainder” (Fig. 2, center). (P.4b) is stronger, but it still admits of models inwhich a whole can be decomposed into several proper parts all of which overlap one
another (Fig. 2, right). In such cases it is unclear what would be left of the whole
upon the removal of any of its proper parts (along with all proper parts thereof). It isonly (P.4) that appears to capture the full spirit of the above-mentioned intuition:
every proper part must be “supplemented” by another part—a proper part that iscompletely disjoint (i.e., does not overlap) the first. (P.4) entails both (P.4a) and(P.4b) and rules out each of the models in Fig. 2. The extension of M obtained byadding this principle to (P.1)–(P.3) is called Minimal Mereology, or MM.10

Figure 2. Three unsupplemented models. (Connecting lines going upwards indicate proper parthood.)

There is another, stronger way of expressing the supplementation intuition. It corresponds to the following axiom, which differs from (P.4) in the antecedent:(P.5)¬Pyx → ∃z(Pzy ∧ ¬Ozx)

Strong Supplementation

In M this principle entails (P.4). The converse, however, does not hold, as shown by
9

In the literature, this principle is sometimes formulated using ‘P’ in place of ‘PP’ in the consequent.
In M the two formulations are equivalent.
10
Strictly speaking, in MM (P.3) is redundant, as it follows from (P.4) along with (P.1) and (P.2).
For ease of reference, however, we shall continue to treat (P.3) as an axiom.

3the model in Fig. 3. The stronger mereological theory obtained by adding (P.5) tothe three core principles of M is called Extensional Mereology, EM.

Figure 3. A weakly supplemented model violating strong supplementation.

The extensional character of EM may not be manifest in (P.5) itself, but it becomes
clearer in view of the following theorem:(8)∃zPPzx → (∀z(PPzx → PPzy) → Pxy)from which it follows that sameness of mereological composition is both necessary
and sufficient for identity:(9)∃zPPzx → (x=y ↔ ∀z(PPzx ↔ PPzy)).

Thus, EM is “extensional” precisely insofar as it rules out any model of the sort depicted in Fig. 3, where distinct objects decompose into the same proper parts.
There is yet a further way of capturing the supplementation intuition. It corresponds
to the following axioms, which differs from (P.5) in the consequent:(P.6) ¬Pyx → ∃z∀w(Pwz ↔ (Pwy ∧ ¬Owx)).

Complementation11

Informally, (P.6) states that whenever an object fails to include another among its
parts, there is something that amounts exactly to the difference or relative complement
between the first object and the second. Once again, it is easily checked that in M
this principle entails (P.5)—thus, a fortiori, (P.4)—whereas the converse does nothold (Fig. 4).

Figure 4. A strongly supplemented model violating complementation.

It should be noted, however, that (P.6) goes beyond the original supplementation
intuition. For while it guarantees that a whole cannot have a single proper part, it
11

In the literature, (P.6) is also known as the Remainder Principle.

4also pronounces on the specific mereological makeup of the supplementary part. In
particular, it requires the relative complement to exist regardless of its internal
structure. If, for example, y is a wine glass and x the stem of the glass, (P.6) entails
the existence of something composed exactly of the base and the bowl—a spatially
disconnected entity. Whether there exist entities of this sort, and more generally
whether the remainder between a whole and any one of its proper parts adds up to abona fide entity of its own, is really a question about mereological composition, over
and above the conditions on decomposition set by (P.4) and (P.5).
Before turning to issues regarding composition, a different sort of decomposition
principles is worth mentioning. Let a mereological atom be any entity with no proper parts:(10)

Ax =df ¬∃yPPyx.

atom

Obviously, all the theories considered so far are compatible with the existence ofsuch things. But one may want to demand more than mere compatibility, just as one
may want to preclude it. Thus, one may want to require that everything is ultimately
composed of atoms, or else that everything is made up of “atomless gunk”12 that divides forever into smaller and smaller parts. These two options are usually formulated as follows:
(P.7)
(P.8)

∃y(Ay ∧ Pyx)
∃yPPyxAtomicity
Atomlessness

These postulates are mutually inconsistent, but taken in isolation they can consistently be added to any mereological theory mentioned so far to yield either an atomistic
variant or an atomless variant, respectively.
Atomistic mereologies admit significant semplifications in the axiomatics. For example, Atomistic EM can be simplified by merging Strong Supplementation (P.5) and Atomicity (P.7) into a single axiom:(P.5ʹ′) ¬Pxy → ∃z(Az ∧ Pzx ∧ ¬Pzy)

Atomistic Supplementationand the the extensionality thesis (9) can be put more perspicuously as follows:(9ʹ′)

x=y ↔ ∀z(Az → (Pzx ↔ Pzy))

This is especially significant if one considers that (P.7) does not quite say that everything is made up of atoms; it merely says that everything has atomic parts, which isconsistent with the possibility of infinitely descending chains of decomposition that
12

The phrase is from Lewis (1991: 20).

5

never bottom out (Fig. 5). Whether stronger versions of (P.7) can be formulated
that rule out such dubious patterns is, at the moment, a question that has not been
fully explored.13

Figure 5. An infinitely descending atomistic model.

Concerning atomless mereologies, one may similarly remark that (P.8) is by itself
rather weak. For one thing, the unsupplemented model in Fig. 2, middle, qualifies
as atomless. To the extent that such models run afoul of the intended notion of a
gunky world, this means that (P.8) calls for teories at least as strong as MM, inwhich case the relevant axiomatization may again be simplified by merging (P.4) and(P.8) into a single axiom:(P.4ʹ′) Pxy → ∃z(PPzy ∧ (Ozx → x = y))Atomless Supplementation

Moreover, infinite divisibility is loose talk. Given (P.8), gunk may have as few as denumerably many parts; but can it have more? Is there an upper bound on the cardinality on the number of pieces of gunk? Should it be allowed that for every cardinal
number there may be more than that many pieces of gunk? (P.8) is silent on thesequestions, yet these are certainly aspects of atomless mereology that deserve further
scrutiny.
3 Composition Principles
The other main way of extending M is via composition principles, i.e. principles governing the behavior of P in the bottom-up direction: from the parts to the wholesthat they compose. We have already seen that the Complementation axiom (P.6) is, ina way, a principle of this sort. Another such principle would be the dual of Atomlessness, to the effect that everything might be “worldless junk”14 that composes forever
into greater and greater wholes:(P.9)∃yPPxyAscentBoth (P.6) and (P.9) are consistent with any of the theories considered so far. They
are, however, fairly strong principles, which reflect specific views on the overall
13
14

See Cotnoir (2013) for some work in this direction.
The phrase is from Schaffer (2010: 64).

6

mereological structure of the universe. More generally, it is customary to consider
ways of extending M by means of composition principles that specify the conditions
under which one or more things qualify as parts of a larger whole.15
The most basic principles of this sort have the following form, to the effect that forany pair of suitably related entities, i.e., any two entities satisfying a given provision
ξ, there is something of which both are part—an underlapper:(P.10) ξxy → Uxz

ξ-BoundSuch principles are quite weak. For example, regardless of how exactly ξ is construed, (P.10) is trivially satisfied in any model that includes a universal entity ofwhich everything is part.
A stronger sort of requirement is that any pair of suitably related entities have a minimal underlapper, something composed of their parts and nothing else. There are atleast three ways of formulating such a requirement, corresponding to three different
ways of characterizing the relevant notion of a minimal underlapper, also known as a
mereological sum of the two entities in question:16
(11a)
(11b)
(11c)

Sazxy =df Pxz ∧ Pyz ∧ ∀w((Pxw ∧ Pyw) → Pzw)
Sbzxy =df Pxz ∧ Pyz ∧ ∀w(Pwz → (Oxw ∨ Oyw))
Sczxy =df ∀w(Owz ↔ (Oxw ∨ Oyw))

a-sum17
b-sum
c-sum

In M these three notions are pairwise distinct (Fig. 6), though they may coincide in
the presence of further axioms. For instance, given Strong Supplementation, (11b) and(11c) are equivalent (though stronger that (11a)), whereas in the presence of Complementation all three notions coincide so long as there is a universal entity: in that case,each sum of any two things is just the complement of the difference between thecomplement of one minus the other. (Such is the strength of (P.6)—a genuine crossbetween decomposition and composition principles.)Figure 6. An a-sum that is not a b- or c-sum, and a c-sum that is not an a- or b-sum.18
15

This is a version of the so-called “Special Composition Question”. See Van Inwagen (1990: Ch. 2).
The first notion may be found in Eberle (1967) and Bostock (1979), the second in Tarski (1935)
and Lewis (1991), the third in Simons (1987) and Casati and Varzi (1999).
17
Given Reflexivity and Transitivity, the definiens in (11a) is equivalent to ∀w(Pzw ↔ (Pxw ∧ Pyw)).
18
The non-extensional model of Fig. 3 also depicts a case in which x and y have a c-sum, in fact two csums (themselves), though no a- or b-sum. This runs contrary the intended meaning of ‘sum’, suggesting that (11c) is best suited to theories at least as strong as EM. See Hovda (2009) for discussion.
16

7

For each i ∈ {a, b, c}, we can then extend M by adding a corresponding axiom as follows, where again ξ specifies a suitable binary condition:(P.11i) ξxy → ∃zSizxy

ξ-SumiThe non-equivalence of these axioms is immediately verified by taking ξ to be satisfied by all pairs of objects and considering the models in Fig. 6. But the axioms
may also differ when ξ is more restrictive. For instance, with ξ expressing overlap, the model in Fig. 6, right, still satisfies (P.11c), but not (P.11a) or (P.11b),
whereas the model in Fig. 7 satisfies (P.11a), but not (P.11b) or (P.11c). In EM, however, (P.11b) or (P.11c) are equivalent, since the corresponding notions of sum coincide.

Figure 7. A model of ξ-suma violating ξ-sumb and ξ-sumc.

The intuitive force of each (P.11i) is in fact best appreciated in the context of EM,
for in that case the relevant sums must be unique. If we introduce a corresponding
binary operator (using ‘ι’ for the definite descriptor),(12i)

x +i y =df ιzFizxyi-sumthen is then easy to see that EM warrants all the “Boolean” properties one might expect. For instance, as long as the arguments satisfy the relevant condition ξ,19 each
operator is idempotent, commutative, and associative:
(13)
(14)
(15)

x = x +i x
x +i y = y +i x
x +i (y +i z) = (x +i y) +i z

and well-behaved with respect to parthood:
(16)
(17)
(18)
(19)

Px(x +i y)
Pxy → Px(y +i z)
P(x +i y)z → Pxz
Pxy ↔ x +i y = y

19

If the condition is not satisfied, the sum may not exist, in which case the standard treatment of descriptive terms implies that the corresponding instances of the theorems that follow are false. In classical logic, (13)–(19) should therefore be taken to hold conditionally on the assumption that the relevant variables range over ξ-related entities.

8Each (P.11i) is still fairly weak, for it governs only finitary composition. We get evenstronger composition principles by requiring a minimal underlapper to exist for any
set of objects satisfying a given condition, including infinite sets (whose sums—or
fusions—cannot be generated by means of the binary operators defined above).
There is, of course, a technical obstacle to formulating such principles in their fullgenerality without resorting to explicit quantification over sets, since a standard
first-order language does not have the resources to specify all sets, but only a denumerable number (in any given domain).20 However, one can achieve a sufficient degree of generality by relying on axiom schemas where the relevant sets are identified
through open formulas. Thus, let ‘φ’ be any formula in the language, and let ‘ψ’ expresses the condition in question. Infinitary variants of the three notions of sum in(11a)–(11c) can be defined as follows, respectively:21
(20a)
(20b)
(20c)

Fazφw =df ∀w(φw → Pwz) ∧ ∀v(∀w(φw → Pwv) → Pzv)
Fbzφw =df ∀w(φw → Pwz) ∧ ∀v(Pvz → ∃w(φw ∧ Owv))
Fczφw =df ∀v(Ovz ↔ ∃w(φw ∧ Owv))

a-fusionb-fusionc-fusion(‘Fizφw’ may be read as ‘z is an i-fusion of the φ-ers’.) For each such notion, we maythen introduce a corresponding principle of infinitary fusion through the following
axiom schema, which asserts the existence of an i-fusion (i ∈ {a, b, c}) for every nonempty set of objects satisfying ψ:(P.12i) (∃wφw ∧ ∀w(φw → ψw)) → ∃zFizφw

ψ-Fusioni

It can be checked that each (P.12i) includes the corresponding finitary principle(P.11i) as a special case, taking ‘φw’ to be the formula ‘w = x ∨ w = y’ and ‘ψw’ thecondition ‘(w = x → ξwy) ∧ (w = y → ξxw)’. Thus, again, these principles are pairwise distinct in M, though it turns out that in the presence of Strong Supplementation(P.12b) and (P.12c) are equivalent.
Finally, the strongest versions of all these composition principles are obtained by asserting them as axiom schemas holding for every condition ψ, i.e., effectively, byforegoing any reference to ψ altogether. Formally this amounts in each case to dropping the second conjunct of the antecedent of (P.12i), i.e., to asserting the schema
expressed by the relevant consequent for any non-empty set of objects specifiable in
the language:(P.13i) ∃wφw → ∃zFizφw

Unrestricted Compositioni

20

To overcome this limitation, some early theories such as those of Tarski (1929) and Leonard andGoodman (1940) resort to explicit quantification over sets. Others, such as Lewis (1991), resort tothe machinery of plural quantification.
21(20a)–(20c) are to be read on the assumption that the variables ‘z’ and ‘v’ do not occur free in φ.
Similar restrictions will apply below.

9

Once again, the relative strength of these principles varies for each i ∈ {a, b, c}. In
particular, it is noteworthy that adding (P.13b) to MM would suffice to warrant
the equivalence of Weak and Strong Supplementation, (P.4) and (P.5), whereas adding(P.13c) would not (Fig. 4 would still count as a counteremodel). Given (P.5), however, the two composition principles are equivalent, which means that the theory
obtained by adding every instance of (P.13b) to MM22 is the same theory obtained byadding every instance of (P.13c) to EM. This theory is known in the literature as General Extensional Mereology, or GEM. The same theory can be obtained by extending
MM with (P.13a), provided the following axiom is also added:23(P.14)(Fazφw ∧ Pxz) → ∃w(φw ∧ Owx)

Filtration

4 Classical Mereology
GEM is a powerful theory, and it was meant to be so by its nominalistic forerunners,
who were thinking of mereology as a fundamental alternative to set theory.24 Indeed,
GEM has such a distinguished pedigree that it has earned the title of Classical Mereology. It is also a decidable theory, whereas for example M, MM, EM, and many extensions thereof are not.25 To see just how powerful GEM is, consider the following
operator, where ‘F’ is any of the ‘Fi’s defined above (which GEM forces to coincide):(21)

σxφx =df ιzFzφx

general fusion

In terms of this operator—the fusion of all φ-ers—GEM can be further simplified,for example by merging (P.5) and (P.13c) into a single axiom schema:(P.13)∃xφx → ∃z(z = σxφx)

Unique Unrestricted Fusion

and we can introduce the following definitions:
(22)
(23)
(24)
(25)
(26)

x + y =df
x × y =df
x – y =df
~x =df
U =df

sum26
product
differencecomplement
universe

σz(Pzx ∨ Pzy)
σz(Pzx ∧ Pzy)
σz(Pzx ∧ ¬Ozy)
σz¬Ozx
σzPzz

The full strength of GEM can then be appreciated by considering that its models areclosed under each of these notions, subject to the satisfiability of the relevant condi-

22

Indeed, (P.2) and (P.4) would suffice.
From Hovda (2009).
24
See the classical works of Leśniewski (1927–1931) and Leonard and Goodman (1940).
25
For a comprehensive picture of decidability in mereology, see Tsai (2013).
26
In GEM, this definition is equivalent to (12i), for each i ∈ {a, b, c}.
23

10

tions. More exactly: the condition ‘¬OzU’ is unsatisfiable, so U cannot have a complement. Likewise products are defined only for overlappers and differences only forpairs that leave a remainder. In all other cases, however, (22)–(26) yield perfectly
well-behaved operators. Since such operators are the natural mereological analoguesof the familiar set-theoretic operators, with ‘σ’ in place of set abstraction, it follows
that the parthood relation axiomatized by GEM has essentially the same properties as
the inclusion relation in standard set theory, modulo the absence of a null entity corresponding to the empty set. Indeed, P is virtually isomorphic to the inclusion relation restricted to the set of all non-empty subsets of a given set, which is to say acomplete Boolean algebra with the zero element removed. We say ‘virtually’ because, strictly speaking, this is only true of stronger version of GEM in which infinitary sums are defined using explicit quantification over sets.27 For set-free formulations that, like those considered here, strictly adhere to a standard first-order language with a denumerable supply of open formulas, the isomorphism does not quite
hold. However, this is only a minor limitation, and we can still characterize the exact algebraic strength of GEM in as follows: any model of this theory is isomorphic
to a Boolean subalgebra of a complete Boolean algebra with the zero element removed (a subalgebra that is not necessarily complete if Zermelo-Frankel set theory
with the axiom of Choice is consistent).28
In this connection, two further points are worth stressing. First, the existence of a“null entity” which is part of everything—the analogue of the empty set—is not in
principle incompatible with GEM. However, it is easy to see that the only models of
GEM with such an entity are trivial one-element models, owing to Weak Supplementation. It is for this reasons that the principles of Unrestricted Composition in (P.13i) arestated as conditionals warranting the existence of a fusion for any given non-empty setof φ-ers. Dropping such a proviso would have disastrous effects, for then the existence of a null entity—the null entity—would be guaranteed by taking ‘φw’ to be thecondition ‘∀xPwx’. The only way around the disaster would be to revisit the nonbasic vocabulary by carefully distinguishing trivial cases of parthood and overlap (involving the ubiquitous null entity) and non-trivial, genuine ones, as in
(27)
(28)

GPxy =df Pxy ∧ ∃z¬Pxz
GOxy =df ∃z(GPzx ∧ GPzy)genuine parthoodgenuine overlap

and by reformulating all non-core axioms accordingly.29 In this way, one can actually
arrive at a variant of GEM that inherits all the strength of a complete Boolean algebra. Nonetheless, the philosophical import of such a theory would remain dubious.
27

As such, the result goes back to Tarski (1935: n. 4).
See Pontow and Schubert (2006), Theorem 34, for details and proof.
29
This strategy is not uncommon in the mathematically oriented literature; see again Pontow and
Schubert (2006) for a comprehensive treatment.
28

11

Second, note that GEM is fully committed to the existence of U, a “universal entity”
of which everything is part. This is not by itself a problem, barring any philosophicalconcerns about the gerrymendered nature of such an entity. It is, however, notwithout consequences. In particular, while GEM admits of models in which everything is composed of atoms as well as “gunky” models in which everything divides
forever, the necessary existence of U deprives GEM of any “junky” model in which
everything composes forever. Thus, while GEM admits of both atomistic and atomless extensions, adding the Ascent principle (P.9) would immediately result in an inconsistent theory.
5 Summary of GEM
For ease of reference, we conclude by summarizing the main axiomatizations of GEM
mentioned above, with some rewriting of bound variables and dropping all redundancies:30(I)(Pxy ∧ Pyz) → Pxz
PPxy → ∃z(PPzy ∧ ¬Ozx)∃xφx → ∃zFazφx(Fazφx ∧ Pyz) → ∃x(φx ∧ Oxy)Transitivity
Weak Supplementation
Unrestricted Compositiona
Filtration

(P.2)
(P.4)
(P.13a)(P.14)(II)(Pxy ∧ Pyz) → Pxz
PPxy → ∃z(PPzy ∧ ¬Ozx)∃xφx → ∃zFbzφx

Transitivity
Weak Supplementation
Unrestricted Compositionb

(P.2)
(P.4)
(P.13b)(III)

Pxx(Pxy ∧ Pyz) → Pxz
Pxy ∧ Pyx) → x = y¬Pyx → ∃z(Pzy ∧ ¬Ozx)∃xφx → ∃zFczφx

Reflexivity
Transitivity
Antisymmetry
Strong Supplementation
Unrestricted Compositionc

(P.1)
(P.2)
(P.3)
(P.5)
(P.13c)(IV)(Pxy ∧ Pyz) → Pxz∃xφx → ∃z(z = σxφx)

Transitivity
Unique Unrestricted Fusion(P.3)(P.13)

30
See also Simons (1987) and Hovda (2009) for additional axiom sets. The elegant axiomatization in(IV) is essentially due to Tarski (1929), though the axioms are explicitly given only in the 1956 English translation

12