Two notions of fusion and the landscape of extensionality

There are two main ways in which the notion of mereological fusion is usually defined in the current literature in mereology which have been labelled ‘Leśniewski fusion’ and ‘Goodman fusion’. It is well-known that, with Minimal Mereology as the background theory, every Leśniewski fusion also qualifies as a Goodman fusion. However, the converse does not hold unless stronger mereological principles are assumed. In this paper I will discuss how the gap between the two notions can be filled, focussing in particular on two specific sets of principles that appear to be of particular philosophical interest. The first way to make the two notions equivalent can be used to shed some interesting light on the kind of intuition both notions seem to articulate. The second shows the importance of a little-known mereological principle which I will call ‘Mild Supplementation’. As I will show, the mereology obtained by adding Mild Supplementation to Minimal Mereology occupies an interesting position in the landscape of theories that are stronger than Minimal Mereology but weaker than what Achille Varzi and Roberto Casati have labelled ‘Extensional Mereology’.


Introduction
The notion of mereological fusion is central to many debates in contemporary metaphysics. Two main definitions of this notion have been proposed in the literature which, following Cotnoir (2018), we can call 'Leśniewski' and & Roberto Loss robertoloss@gmail.com 1 Philosophisches Seminar, University of Hamburg, Ü berseering 35, 22297 Hamburg, Germany 'Goodman' fusion,respectively. ,12 Let '\' stand for the relation of proper parthood (which I will be assuming to be a primitive notion), and let us stipulate that x is part of y just in case x is either a proper part of y or identical to y, and that x and y overlap just in case they have a common part: By adopting the following abbreviations for ease of exposition (where 'yy' is a plural variable and '0' is the one-many 'one of' relation) ('xOyy' and 'xx y' can be read as 'x overlaps the yy', and 'all the xx are part of y', respectively) the notions of Leśniewski fusion and Goodman fusion can be defined as follows: The mereological theory known in the literature as 'Minimal Mereology' (henceforth 'MM') can be axiomatized by means of the following two principlestransitivity of proper parthood and Weak Supplementation: ð\-TransitivityÞ 8x8y8z x\y^y\z ð Þ!x\z ð Þ WSP ð Þ 8x8y x\y ! 9z z y^$ Ozx ð Þ ð Þ (notice that MM entails that proper parthood is a strict partial order, and so it is not only transitive but also irreflexive and asymmetric). As is well-known (see, for instance , Pietruszczak 2005: 216), in the presence of (\-Transitivity) every Leśniewski-fusion (or 'L-fusion' for short) qualifies as a Goodman-fusion (or 'G-fusion' for short): 3 1 The notion of Leśniewski fusion corresponds to the definition given by Leśniewski (1916). Notice that although for the ease of exposition I choose here to follow Cotnoir (2018) and use the label 'Goodman fusion', the second notion of fusion was also originally formulated by Leśniewski (1931) (in terms of the relations 'exterior to' and 'discrete from', respectively; many thanks to an anonymous referee of this Journal). The notion of Goodman fusion is also used by Leonard and Goodman (1940) and Goodman (1951). Hovda (2009) calls Goodman fusions 'type 1 fusions', and Leśniewski fusions 'type 2 fusions'. Varzi (2019) calls Goodman and Leśniewski fusions 'General Sums 2', and 'General Sums 3', respectively. 2 A third definition of fusion that can be found in the literature is the notion of 'Algebraic' fusion which consists in taking the fusion of a plurality of entities to be their least upper bound with respect to parthood (see Hovda 2009: 61 and below: Sect. 2). 3 Proof. Suppose that x L-fuses the yy. If some entity z overlaps x, then z and x have at least a part w in common. By (L-def), w overlaps the yy and so has a part j in common with some of the yy. By (\-Transitivity) it follows that j is also part of z, so that also z overlaps the yy. By generalization, it follows that if something overlaps x then it overlaps the yy. Suppose, instead, that some entity z overlaps the yy. Then, z has at least a part w in common with some of the yy. By (L-def) the yy are all parts of x. Since w is L-to-G ð Þ 8x8yy xF L yy ! xF G yy À Á However, even under the assumption of MM, we don't have that every G-fusion qualifies as an L-fusion, as one can easily appreciate from the model depicted in Fig. 1. In fact, as noted by Pietruszczak (2018: 147), Hovda (2009: 64-65) and Varzi (2009: 602), in Fig. 1 many pluralities of entities have a G-fusion without having an L-fusion. For instance, the plurality ½a; b; c; d (that is, the plurality of entities that are identical to either a, b, c, or d) 4 is G-fused by both a and b, and yet there is no entity that is their L-fusion, given that no entity has all of them as parts (as required by the first conjunct of (L-def)). It follows, therefore, that the following principle-stating that something is an L-fusion if and only if it is a G-fusion-is not a theorem of MM: This naturally raises the question as to which pluralities of pairwise logically independent principles jointly entail (EqLG)-with MM in the background-and are entailed by it. In other words: let say that a plurality S 1 ; . . .; S n of sentences (taken together) are a way of expressing (or, more simply, express) the difference between L-fusions and G-fusions just in case the following is true: The question raised by the considerations just made is, thus, the following: which pluralities of sentences comply with (Diff) and, thus, 'express the difference' between L-fusions and G-fusions? The aim of this paper is not to list all the possible ways to express the difference between L-fusions and G-fusions. Rather, I will only discuss two ways to do so that appear to be of particular interest. In the literature, it is a well-known fact that, assuming (\-Transitivity), (EqLG) is equivalent to Strong Supplementation (and, thus, that (SSP) complies with (Diff)): SSP ð Þ 8x8y xy ! 9z z x^$ Ozy ð Þ (Pietruszczak 2005(Pietruszczak : 218, 2018Gruszczyński 2013: 140;Varzi 2016: section 4.3). As I will argue in Sect. 2, however, there is a way to reach this result which shows how (L-def) and (G-def) can be seen as articulating the same, highly plausible intuition concerning the notion of mereological fusion. In Sect. 3, I will argue that there is a second way of articulating the difference between L-fusions and G-fusions on the background of MM which reveals the importance of a little-known mereological principle that I will label 'Mild Supplementation'. As I will show, Footnote 3 continued part of one of the yy, we have, by (\-Transitivity), that w is part of x, so that z has a part in common with x. By generalization it follows that if something overlaps the yy, then it overlaps x. Therefore, something overlaps x if and only if it overlaps the yy, which means that x G-fuses the yy. j 4 Letting 'ixx:/xx' stand for the plural definite description 'the xx that /', '½x 1 ; . . .; x n ' is short for Two notions of fusion and the landscape of extensionality 3445 Mild Supplementation will allow us to unearth a theory which appears to occupy an interesting position in the logical space lying between MM and the theory that Casati and Varzi (1999) call 'Extensional Mereology'.

L-fusions, G-fusions and strong supplementation
According to the definition of L-fusion, an entity x is an L-fusion of a plurality yy of entities just in case all the yy are part of x and every part of x overlaps at least one of the yy. Instead, according to the definition of G-fusion, an entity x is a G-fusion of a plurality yy of entities just in case something overlaps x if and only if it overlaps one of the yy: Consider, then, the second conjunct of (L-def).
(1) 8z z x ! zOyy ð Þ Notice that, under the assumption of MM, (1) is equivalent to 5 : (2) 8z Ozx ! zOyy ð Þ But (2) is just the left-to-right direction of (G-def). Consider, then, the right-toleft direction of (G-def): (3) 8z zOyy ! Ozx ð Þ (3) can be unpacked as: In turn, (4) is logically equivalent to: (1) entails (2). Suppose z overlaps x. Then, some entity w is part of both z and x. It follows, thus by (1) that w overlaps at least some of the yy. By the transitivity of parthood, every part of w is also a part of z, so that also z overlaps the yy.
(2) entails (1). Suppose z is part of x. Then, z overlaps x (since by the reflexivity of parthood they have at least a part in common, that is, z itself). By (2), x also overlaps the yy. j Let, then, an entity x be covered by an entity y ('xY y') if and only if everything that overlaps x also overlaps y: The right-to-left direction of (G-def) can, then, be rewritten as follows: Putting everything together, we have that, under the assumption of MM, (L-def) and (G-def) are equivalent to the following two definitions: Consider the following principle employing pre-theoretic notions of 'containment' and 'covering' (and in which 'fusion' can be taken to be a dummy term for an object that is made from a plurality of entities): (Fusion) A an entity x is a fusion of a plurality yy of entities just in case (i) x 'contains' all the yy, and (ii) the yy 'cover' all of x.
(Fusion) strikes me as a highly intuitive principle. If x didn't 'contain' z and z was one of the yy, how could one say that x a fusion of them? At the same time, if some part of x wasn't completely 'covered' by the yy-if some part of x was free of the yy, so to speak-how could a fusion of the yy be identical to x? (L-def?) and (G-def?) allow one to appreciate at a glance how close the two definitions are. Both (L-def?) and (G-def?) consist, in fact, in a conjunction which can be seen as formulating the two conditions expressed by (Fusion). The 'covering condition' is expressed in the second conjunct of (L-def?) and (G-def?) in the same way: x is covered by the yy in the sense that every part of x overlaps at least one of the yy, so that no part of x is disjoint from-and, in this sense, 'free of'-all of them. We can call this sense of covering plural covering (or many-one covering). The difference between (L-def?) and (G-def?) concerns the way in which the 'containment condition' is expressed in their first conjunct. According to (L-def?), to contain an entity is to have it as a part, so that x can contain all the yy only if it has all of them as parts. Instead, according to (G-def?), to contain an entity is to cover it, in the sense of overlapping everything it overlaps (we can call this sense of covering singular covering, or oneone covering, for disambiguation). Therefore, (L-def?) and (G-def?) can be seen as disagreeing on how the pre-theoretical notion of (mereological) 'containment' should be properly expressed. 6 Before moving further, notice that a third definition of the notion of fusion can be found in the literature, which following Cotnoir (2018) we can call 'Algebraic' fusion: (A-def) gets the 'containment part' of (Fusion) right (or, at least, in a way that is at least prima facie legitimate). The problem with (A-def) is the way it gets the 'covering part' of (Fusion). Consider, for instance, the model depicted in Fig. 2. In Fig. 2 a counts as an A-fusion of ½b; d. Yet there is clearly more to a than just b and d taken together, namely, c. b and d taken together clearly fail to cover all of a. Therefore, as long as we follow (Fusion) as our guiding intuition, the notion of A-fusion appears to fall short of being a live option to define the notion of mereological fusion. 7 Also the notion of G-fusion may be subjected to some criticism. Consider, in fact, the model depicted in Fig. 1. a is a fusion of ½a; b; c; d and yet it doesn't have b as a part. However, as Varzi (2009) claims, it may seem that […] no matter how exactly one defines the word, one should always expect a fusion to include, among its parts, all the things it fuses. (Varzi 2009: 602) Even worse, a is a fusion of [b]-the 'improper' plurality of the things that are identical to b-without having it a part. Yet, it may seem plausible to object that surely a fusion is supposed to include, among its parts, at least some of the things it fuses. (Varzi 2009: 603) Notice, first, that G-mereologists endorsing a notion of 'containment-ascovering' agree that a fusion must contain all the entities it fuses. What they deny is that, in order to be contained by their fusion, the entities fused must be part of the fusion. Therefore, simply replying that a fusion must contain all the entities it fuses as parts seems to be dialectically ineffective in this case. Second, G-mereologists appear to have an independent way to argue that in Fig. 1 a and b do contain each other, namely, by appealing to the idea that a fusion is 'nothing over and above' the plurality of its proper parts. This argument can be presented as follows: (A1) For every x and y and zz, if x contains the zz and y is nothing over and above the zz, then x also contains y (A2) In Fig. 1, a contains [c,d] (A3) b is nothing over and above [c,d] 7 Cotnoir and Bacon (2012: 196) argue that, if parthood is not assumed to be antisymmetric, then the notion of A-fusion may be adequate if paired with the following Complementation axiom which (in the presence of MM) is strictly stronger than both (WSP) and (SSP)): Complementation (A2) is something that also containment-as-parthood theorists like Varzi (2009) accept. Therefore, whether or not this argument is successful depends on whether G-theorists can appeal to a notion of nothing-over-and-aboveness that is both compatible with anti-extensionalism and such as to make (A1) and (A3) true.
Many philosophers share the intuition that a whole is nothing over and above the parts it fuses 8 : if I buy separately the four parts composing a certain lot of land, I don't need to also buy the lot they compose. Similarly, if I buy six cans of beer, I don't need to spend extra money to also buy the six-pack (Baxter 1988). Several ways to cash out the slippery notion of nothing-over-and-aboveness have been proposed in the literature. 9 However, many of them appear to be incompatible with anti-extensionalist models like the one depicted in Fig. 1. For instance, G-theorists cannot take a whole to be nothing over and above its parts in the sense that it is literally identical to its parts taken together (a thesis commonly known as 'Strong Composition as Identity'). 10 In fact, if both a and b in Fig. 1 were identical to the plurality of their proper parts then (by symmetry and transitivity of identity) they themselves would be identical. Similarly, G-mereologists cannot take a composite entity to be nothing over and above its parts in the sense that it is the only object that is composed of those parts (Smid 2017: 2; 12, fn 2), as that would also be straightforwardly inconsistent with Fig. 1. More generally, the idea that antiextensionalists cannot claim that entities sharing the same proper parts are nothing over and above them may indeed seem to have the ring of intuitiveness to it: if some Xs compose two things, then wholes could not be ''nothing over and above their parts.'' How could distinct things each be nothing over and above the same parts? (Sider 2007: 70; my italics) Gilmore (2010: 181) formulates the principle that appears to be behind Sider's (2007) intuition as follows: 'P1 For any Xs, any y, and any z, if y is nothing over and above the Xs, and z is nothing over and above the Xs, then y = z' (Gilmore 2010: 181) c d a b Fig. 2 a is an A-fusion of [b,d] 8 See, among many others, Lewis (1991), Varzi (2014), and for additional references Smid (2017). 9 Smid (2017), for instance, recently distinguishes between five readings of 'x is nothing over and above the yy': (i) x is not an additional commitment with respect to the yy, (ii) the existence of the yy is sufficient for the existence of x, (iii) x is the only object that is composed of the yy, (iv) x has no properties that are not reducible to the yy, (v) x is identical to the yy. 10 See Cotnoir (2014) for an introduction to Composition as Identity.
As Gilmore argues, as intuitive as P1 may initially sound, there seem to be good reasons to doubt it. Consider, in fact, the following principle P1? which extends P1 to pluralities of entities: 'P1? For any Xs, any Ys, and any Zs, if [the Ys are nothing over and above the Xs, the Zs are nothing over and above the Xs, and there are exactly as many of the Ys as there are of the Zs], then the Ys = the Zs.' (Gilmore 2010: 181) If being nothing over and above is 'identity-like' as P1 suggests, it seems that also P1? should be accepted as true. However, P1? is intuitively false: both the rows and the columns in my chessboard (which are equal in number) are nothing over and above the squares of my chessboard, and yet they are not identical (the rows run horizontally, the columns vertically). It seems, therefore, that if we are 'free to endorse the nonidentity of pluralities […] each of which [are] nothing over the same things' we should also be 'free to endorse the nonidentity of single individuals [that are] nothing over and above the same things' (Gilmore 2010: 182). Furthermore, there seems to be at least one possible way to account for the notion of 'being nothing over and above' that is both consistent with anti-extensionalist models like the one of Fig. 1 and with premises (A1) and (A3). The general idea behind this account can be briefly sketched as follows. Intuitively, certain entities seem to depend for their existence on other entities. In particular, composite entities are often claimed to depend for their existence on their proper parts. This notion of existential dependence can be understood by means of the notion of metaphysical grounding, in at least two ways. 11 The first one, proposed by Correia (2005) and Schnieder (2006), is that of taking an entity x to depend existentially on a plurality yy of entities if and only if the fact that x exists is grounded in some facts concerning the yy. Alternatively, one could follow Schaffer (2009) and identify this notion of dependence with a primitive, cross-categorical relation of grounding. 12, 13 In both cases, there seems to be at least some intuitive appeal to the idea that, if x's existence is grounded in the yy, then x owes its existence to the yy, so that x and the yy are really the same 'portion of reality', or the same 'amount of being', so to speak. 14 This suggests the possibility of the following kind of grounding-based account of nothing-over-and-aboveness: (NOA) The xx are nothing over and above the yy if and only if either (i) the xx depend for their existence on the yy, (ii) the yy depend for their existence on 11 For a general introduction on the notion of grounding see Correia and Schnieder (2012). On the idea that a whole is grounded in its parts see, among others, Cameron (2014) and Loss (2016). 12 See Schnieder (2020) for some discussion of these two grounding-based approaches to ontological dependence. 13 Schaffer (2010) himself defends the idea that the cosmos is prior to its parts. However, this choice is clearly independent from the choice of identifying existential dependence with a primitive grounding relation. 14 On the idea that, in general, the grounded is the same portion of reality of its grounds see Schaffer (2016). On the general idea that grounding entails nothing-over-and-aboveness see Fine (2001: 15-16;2012: 39) and Schaffer (2009: 353). the xx, or (iii) for some zz, both the xx and the yy depend for their existence on the zz.
According to (NOA), in order for the xx to really 'add something' to the yy in the required sense, the portion of reality in which the xx consist must not be completely contained in the portion of reality in which the yy consist. Therefore, in order to be something over and above the yy the xx cannot stand in a grounding relation to the yy or share a common ground with them. Given (NOA), (A3) follows directly from the assumption that b is grounded in [c,d], while (A1) appears to have at least the ring of plausibility to it. Suppose, in fact, that an entity x contains the zz and that the existence of a certain other entity y is grounded in the zz. In this case, it seems plausible to say that-in virtue of containing the 'ontological root' of y (as we may call it)-x also contains y. 15 Both this kind of account of the notion of being nothing over and above in terms of grounding and the general idea that anti-extensionalism is compatible with nothing-over-and-aboveness may, of course, be challenged. Be that as it may, however, the foregoing considerations seem to show at least that the case against the notion of G-fusion is not as tight as it may appear at first sight and certainly not as straightforward as the case against the notion of A-fusion. Therefore, we seem to have at least some prima facie reason to conclude that, pace Sider (2007) and Varzi (2009), both the notion of L-fusion and the notion of G-fusion can be taken to be admissible notions of fusion.
Let us now return to the issue concerning the difference between L-fusions and G-fusions. It is a theorem of MM that parthood entails covering 16 : It is, thus, straightforward to observe from the first conjuncts of (L-def?) and (Gdef?) that (assuming MM) if something is an L-fusion (of a certain plurality of entities) it is also a G-fusion (of those entities): (9) 8x8yy xF L yy ! xF G yy À Á 15 Skiles (2015: 739-741) has argued that the idea that grounding entails nothing-over-and-aboveness is compatible with 'grounding contingentism' (namely, the idea that grounds do not necessitate what they ground). Notice, however, that G-mereologists don't appear to be forced to endorse grounding contingentism. In fact, at least under the assumption of Correia's and Schnieder's account of ontological dependence, entities that share the same proper parts can existentially depend on them by being grounded in different facts about them. For instance, the fact that a exists may be grounded simply in the fact that c exists and the fact that d exist, taken together, while the fact that b exists may be grounded in the fact that c and d are arranged in a certain way R. Therefore, when c and d cease to be R-arranged the fact that only b ceases to exist (while a keeps existing) doesn't appear to be problematic for 'grounding necessitarians'. 16 Proof: Suppose that x is a part of y and assume that z overlaps x. There is, therefore, some entity w that is part of both z and x. So w is a part of x and x is a part of y. By the transitivity of parthood, w is a part of y. Therefore, z also overlaps y. By generalization, everything that overlaps x also overlaps y or, in other words, x is covered by y. j In the same way, it is also straightforward to see what principle can fill by itself the gap between the two notions. In fact, if an entity x is a G-fusion of some entities yy, then although the yy cover all of x (as required by the second conjunct of (Ldef?)) we only have that if something is one of the yy, it is only guaranteed to be covered by x and not to be part of it. It is, thus, sufficient to add to the mix the principle according to which if x is covered by y, then x is part of y-(10) 8x8y xY y ! x y ð Þ -in order to guarantee that every G-fusion is also an L-fusion: (11) 8x8yy xF G yy ! xF L yy À Á Notice that if we unpack (10)  Proof Suppose that x is covered by y. Clearly, y covers itself. Therefore, each of the ½x; y (that is, the plurality of x and y taken together) is covered by y. On the other hand, every part of y overlaps at least some of the ½x; y, since y clearly overlaps itself. Therefore y is a G-fusion of ½x; y. From (EqLG) it follows that y is also an Lfusion of the ½x; y, so that x is a part of y. We have, thus, proved (10), and namely that, for every x and every y, if x is covered by y, then x is a part of y. As we have just seen above, given MM (10) is equivalent to (SSP). j We can, thus, conclude that (SSP) is a way to express the difference between L-fusions and G-fusions on the background of MM: we have done in this section is to unpack both definitions of mereological fusion so as to make a 'containment' part (requiring the fusion to 'contain' in some sense all the things it fuses) and a 'covering' part (requiring the things fused to completely 'cover' the fusion) explicit. This allowed us to interpret the difference between the two notions of fusion as a different way of articulating the notion of containment at play in the definition of fusion: containment-as-parthood and containment-as-covering. At that point, the 'gap' between the two notions emerged very naturally as the requirement that covering entail parthood, which is just a different way to express (SSP). Therefore, in this section we haven't just proved that (SSP) is a way to express the difference between L-fusions and G-fusions. We have also provided a seemingly intuitive explanation as to why that is the case, namely because (SSP) functions as a bridge principle between the two different notions of containment in play.

Mild Supplementation and Extensionality
The model of Fig. 1 was used to show that, if MM is the only mereological assumption in the background, then not every G-fusion is an L-fusion. Figure 1 is a counterexample to Extensionality of Proper Parthood: This may lead one to suppose that (EPP) is another way to express the difference between the two notion of fusions. However, the model depicted in Fig. 3 clearly shows that this is not the case. Consider, in fact, the plurality [b 1 ,b 2 ,b 3 ] (the 'bs'). a is clearly a G-fusion of the bs, as it covers all of them and it is covered by them (taken together). However, no entity in Fig. 3 is an L-fusion of the bs, as no entity in Fig. 3-not even a-has all the bs as parts. Figure 3 is a model of (EPP). Therefore, (EPP) is not enough to fill the gap between L-fusions and G-fusions.
In Fig. 3 a would qualify as a fusion of the bs if it had all of the bs as parts. In that case each of the bs would be a proper part of a. This is, thus, what may appear to be odd about Fig. 3 19 : a covers each of the bs, none of the bs covers a, and yet Fig. 3 A model of (EPP) that is not a model of (EqLG) 19 Pietruszczak (2018: 145) uses this model to show that MM, what we may call 'G-Universalism' (that is, the principle according to which every non-empty set of entities has a G-fusion-where in this case the notion of G-fusion is defined in terms of sets, instead of pluralities) and the principle Extensionality of Overlap (see Sect. 4 below) do not entail either (SSP) or 'L-Universalism'. Hovda (2009: 71) uses this none of the bs is a proper part of a. Therefore, it is sufficient to add the following principle to MM (which I will label 'Mild Supplementation') to ensure that in Fig. 3 a is also an L-fusion of the bs 20 : which is equivalent to the following formulation: Notice that, given MM, (SSP) is equivalent to the following principle: Therefore, for every x and y such that x is not a proper part of y and yet it is covered by y we have that, while (SSP2) demands that x be identical to y (thus excluding both models like the one depicted in Figs. 1 and in 3), (MSPc2) requires only that x cover y (thus excluding only models like the one in Fig. 3 and leaving the door open to failure of extensionality).
(MSP) and (EPP) are independent principles. Figure 1 is a model of (MSP) but not of (EPP), while Fig. 3 is a model of (EPP) but not of (MSP). Furthermore, as MM; ðSSPÞ ' ðEPPÞ c.

MM; ðMSPÞ; ðEPPÞ ' ðSSPÞ
The proofs of both (15a) and (15b) are straightforward. 21 The proof of (15c) can be presented as follows. First, we prove that MM and (MSP) jointly entail the principle '(OPP)' according to which entities overlapping the same entities have the same proper parts Footnote 19 continued model to argue against the notion of G-fusion. His argument is the following. Classical Mereology can be axiomatized by just two axioms: (i) the transitivity of parthood and (ii) the existence of a unique L-fusion of any plurality of entities. Instead, it is not sufficient to assume that parthood is transitive and that every plurality of entities has a unique G-fusion to get Classical Mereology, as Fig. 3 shows. 20 Notice that (MSP) is the right-to-left reading of Goodman's (1951: 49) definition of proper part when expressed solely in terms of overlap. 21 Proof of (15a): Suppose x is covered by y. By (SSP), x is part of y. Suppose, furthermore, that y is not covered by x. It follows from Leibniz's Law that x is different from y. Therefore, x is a proper part of y. We can, thus, conclude that if x is covered by y and y isn't covered by x, then x is a proper part of y. j Proof of (15b): Suppose x and y are two composite entities with the same proper parts. Therefore, every part of x overlaps y, and every of y overlaps x. It follows from (SSP) that x is part of y and y is part of x. By the anti-symmetry of parthood it follows that x is identical to y. j OPP ð Þ 8z Ozx $ Ozy ð Þ!8 z z\y $ z\x ð Þ

(16) MM; ðMSPÞ ' ðOPPÞ
Proof Suppose that x and y overlap the same entities. Suppose, for reductio, that some entity w is a proper part of only one of them (say, of x). w is not a proper part of y. It follows from (MSP) that either w is not covered by y or y is covered by w. Since w is a proper part of x, everything that overlaps w also overlaps x (by the transitivity of parthood). But we are assuming that everything that overlaps x also overlaps y. Therefore, everything that overlaps w overlaps y so that w is covered by y. Therefore, y is covered by w, so that everything that overlaps y also overlaps w. w is a proper part of x. It follows by (WSP) that some entity k is part of x and doesn't overlap w. Therefore, k doesn't overlap y. But we are assuming that x and y overlap the same entities, so that k doesn't overlap x either. Yet, k is a proper part of x and so it overlaps x. Contradiction! Therefore, for every x and y, if x and y overlap the same entities they have the same proper parts. j Proof Suppose that x is not part of y and that yet every part of x overlaps y. It follows by the transitivity of parthood that everything that overlaps x also overlaps y or, in other words, that x is covered by y. Since x is not a part of y, x is not a proper part of y. By (MSP) we have that either x is not covered by y or y is covered by x. y is, thus, covered by x. x and y cover each other. By (OPP) they have the same proper parts. Suppose that x is an atom (the reasoning being similar in the case of y).
x is, thus, a part of y that is different from y. Therefore, x is a proper part of y. But x and y have the same proper parts, so that x is a proper part of itself, contra the irreflexivity of proper parthood. Therefore, x and y are two different composite entities with the same proper parts. By (EPP), they are identical. Contradiction! Therefore, some part of x doesn't overlap y. By generalization it follows that, for every x and y, if x is not a part of y, then some part of x doesn't overlap y. j From (15) and (Difference-1) it follows, thus, that (MSP) and (EPP) taken together are a further way to express the difference between L-fusions and G-fusions: 4 Between minimal mereology and extensional mereology Casati and Varzi (1999) call 'Extensional Mereology' ('EM') a theory that is equivalent to MM?(SSP) (and, thus, to MM?(EPP)?(MSP)). They justify the label 'extensional' with the fact that EM rules out countermodels to (EPP) (Casati and Varzi 1999: 40). Other extensionality principles that can be found in the literature are the principles 'Extensionality of Overlap', 'Uniqueness of G-Fusion', and 'Uniqueness of L-Fusion': In addition we also have the principle that  calls 'Extensionality of Composition': (EC) 'If x and y are composed of the same things, then x = y' (Varzi 2008: 109).
where (C) 'x is composed of the zs = df x is a [fusion] of the zs and the zs are pairwise disjoint (i.e., no two of them have any parts in common)' (Varzi 2008: 109; see also van Inwagen 1990: 29) Letting 'Dxx' stand for 'the xx are pairwise disjoint' we can define two notions of composition in this sense, namely, Goodman-composition (or G-composition) and Leśniewski-composition (or L-composition): In turn, (GC) and (LC) allow us to state two different versions of Varzi's principle (EC): One elegant way to prove this is due to Pietruszczak (2018: 85-86, 144) and can be briefly reformulated within this framework as follows. 24 First, it is proved from (\-Transitivity) that if every part of x overlaps y, then x is an L-fusion of the plurality of entities that are parts of both x and y: 25 26 Fig. 4 A model of (EPP) that is not a model of either (EO), (UGF), (ULF), (EGC), or (ELC) 23 Proof. Suppose that x and y are composite entities having the same proper parts. Suppose, furthermore, that some z overlaps, say, x but not y (the reasoning being identical if we assume that something overlaps y but not x). Then, there is some w that is a common part of z and x but is disjoint from y. x has all of its proper parts in common with y, and so it clearly overlaps y. Therefore, z must be a proper part of x. But we are supposing that every proper part of x is also a proper part of y. Therefore, z is also a proper part of y and, thus, overlaps y. Contradiction! Therefore, everything that overlaps x also overlaps y. By (EO) it follows that x is identical to y. Therefore, if x and y are composite entities having the same proper parts, they are identical. j See also Pietruszczak (2018: 174;2020: 42). 24 As Pietruszczak (2018: 85-86, 144) shows, it is sufficient to assume that parthood is transitive in order to prove that ( , where in this context POS can be taken to be the mereology axiomatised by (\-Transitivity) and (\-Irreflexivity) and entailing thus that proper parthood is a strict partial order (see on this Pietruszczak 2018: 86, 144;2020: 46, 58). 25 Recall that 'ixx:/xx' stands for the plural definite description 'the xx that /' (see footnote 4). 26 Proof (see Pietruszczak 2018: 85). Suppose that every part of x overlaps y and let the zz be the entities that are parts of both x and y. The zz are all part of x, so that they comply with the first conjunct of (L-def).
Consider an arbitrary part k of x. k overlaps y and so there is some j such that j is part of both k and y. By (\-transitivity) j is part of both x and y and so it is one of the zz. Therefore, k overlaps one of the zz. By generalization, every part of x overlaps one of the zz, so that they comply also with the second conjunct of (L-def). It follows that x is an L-fusion of the entities that are parts of both x and y. j . Assume (EO) and suppose that both x and y L-fuse the zz and that w overlaps x. Therefore, w has a part k in common with x. By (L-def), k overlaps the zz. k has thus a part j in common with some of the zz. Each of the zz is part of y. By (\-transitivity), j is part of both w and y. Therefore, w overlaps y. It follows that everything that overlaps x also overlaps y. By symmetry of reasoning we also have that everything that overlaps y overlaps x. It follows from (EO) that x is identical to y. (20b). Assume (ULG) and suppose that x and y overlap the same entities. It follows that every part of x overlaps y and every part of y overlaps x. By (19) we have that both x and y are an L-fusion of the same plurality of entities, namely, the entities that are part of both x and y. By (ULG) it follows that x and y are identical. j 28 Proof (see Pietruszczak 2018: 144). Left-to-right. Assume (UGF) and suppose that both x and y are an L-fusion of the zz. By (L-to-G) (Sect. 1) it follows that they are a G-fusion of the zz so that, by (UGF), they are identical. Right-to-left. Assume (ULF) and suppose that both x and y are a G-fusion of the zz. By (G-def) it follows that if something overlaps either x or y, it overlaps the zz, and that if something overlaps the zz, then it overlaps both x and y. Therefore, x and y overlap the same entities. It follows, thus, from (EO)-which, by (20), is equivalent to (ULF)-that x and y are identical. j 29 Proof. Suppose that x and y overlap the same entities and yet they are different. By the anti-symmetry of parthood, either x is not a part of y or y is not a part of x. Suppose that x is not a part of y. By (SSP) there is a part of x that doesn't overlap y. But every part of x clearly overlaps x, so that this means that something overlaps x without overlapping y. Contradiction! The same kind of reasoning applies if we suppose that y is not a part of x. Therefore, x and y are identical. j See, also, Pietruszczak (2018: 92-93) for a proof that EM entails (ULG).
(24) MM; ðULFÞ ' ðELCÞ and MM; ðUGFÞ ' ðEGCÞ Also, it follows directly from the fact that MM entails (L-to-G) (see Sect. 1) that (EGC) and MM jointly entail (ELC): Interestingly, given the axiom of choice it can also be proved that (ELC) and MM jointly entail (UGF): Proof Part I. Suppose a and b are both G-fusions of the zz (which may or may not be pairwise disjoint). Let W be the non-empty set of parts that a and b have in common, 30 R be a well-order on W, 31 e 1 be the least element of W under R, and S be a subset of W that is defined as follows: Þ (an item in W is in S whenever every R-smaller item in S is disjoint from it); (iii) nothing else is a member of S. 32 30 We can prove that W is non-empty as follows. It follows from the definition of G-fusion that a and b overlap the same entities. Since overlap is reflexive, a and b overlap and have, thus, some part in common. j 31 The existence of R is guaranteed by the Well-Ordering Theorem, which is equivalent to the axiom of choice (see e.g. Moschovakis 2006: 112). 32 The fact that S is well-defined can be shown as follows. Let f be a function such that, for every part x of b, f x ð Þ ¼ 1 if it can be decided, given the definition of S, whether x is a member of S (f x ð Þ ¼ 0 otherwise). We have that: Proof. (a) From (i) we have that e 1 2 S. Therefore, f e 1 ð Þ ¼ 1. (b) Assume that for every y, such that yRx, f y ð Þ ¼ 1. It follows that for every y, such that yRx, it can be decided, given the definition of S, whether x is a member of S or not. From (ii) and (iii) in the definition of S we have, thus, that, if for every y, such that yRx and y 2 S, x is disjoint from y, then x 2 S, otherwise x 6 2 S. Therefore, f x ð Þ ¼ 1]. By the Transfinite Induction Theorem (see e.g. Moschovakis 2006: 94) it follows from (a) and (b) that, for every part x of b, f x ð Þ ¼ 1. Therefore, for every part x of b, the definition of S allows us to decide whether x is a member of S or not, so that S is well-defined. j We have, thus, the following: (a) e 1 is disjoint from any other member of S 33 ; (b) any two other members c and d of S are disjoint. 34 Therefore, the members of S are pairwise disjoint. 35 Part II. Suppose that z is a part of a. If z is a member of S, then it clearly overlaps some member of S (by -reflexivity). Suppose that z is not a member of S. Since a and b are G-fusions of the same plurality of entities it follows from (G-def) that they overlap the same entities. By the reflexivity of parthood, z overlaps a. Therefore, z also overlaps b and has, thus, a part v in common with b. By the transitivity of parthood v is a member of W (namely, the set of entities that are parts of both a and b; see Part I).
If v is also a member of S, then z clearly overlaps some member of S (by -reflexivity). If v is not a member of S, it follows from the definition of S that some member w of S is such that wRv and v overlaps w. Since v is part of z, the part that v has in common with w is also part of z (by -transitivity) so that z overlaps w. Therefore, in each of these cases z overlaps some member of S. By generalization, we have that every part of a overlaps some member of S. By symmetry of reasoning, we can also conclude that every part of b overlaps some member of S.
Part III. Let the cc be the plurality of entities that are in S. Each of the cc is part of both a and b. As we just proved in Part II, every part of either a or b overlaps the cc. Therefore, both a and b are an L-fusion of the cc. The cc are pairwise disjoint. It follows, thus, from (ELC) that a and b are identical. 36 j It follows from (21) For instance, the model depicted in Fig. 3 is a model of MM?(EO) but not of MM?(MSP). Therefore, it can be concluded that that MM?(MSP) appears to occupy an interesting place among the theories under consideration (see Fig. 5). 38 In fact, while MM?(MSP) is a non-extensional mereology that is stronger than MM, 39 the only extensional mereology (among those under consideration) that is stronger than (PPP) 8x8y 9z z\x ð Þ^8zðz\x ! z\y ð Þ ! x y ð Þ we obtain a system that is stronger than MM?(EO) but weaker than EM (see Pietruszczak 2020: 42) (many thanks to an anonymous referee of this Journal). 39 MM?(MSP) is not the strongest mereology containing MM but not (EPP). For instance, MM?(MSP) could be extended by the addition of the following 'artificially weaker' version of (EPP): (EPP3) 8x8y 9v9w9z v\x^w\x^z\x^v 6 ¼ w^w 6 ¼ z^v 6 ¼ z ð Þ ! 8 z z\x $ z\y ð Þ ! x ¼ y ð Þ ð Þ (many thanks to an anonymous referee for this Journal).
Two notions of fusion and the landscape of extensionality 3461 In this paper I have addressed the question concerning the 'difference'-with MM in the background-between the notion of Leśniewski fusion and the notion of Goodman fusion. Although it is well known that the Strong Supplementation principle is sufficient to fill this gap, I have argued that this fact can be proved in a way that sheds some interesting light on the relationship between the two notions of fusion. I have also shown how the difference between Leśniewski fusions and Goodman fusions can be broken down into two logically independent components, namely, the well-known principle Extensionality of Proper Parthood and the lesserknown Mild Supplementation. Finally, the theory combining Minimal Mereology and Mild Supplementation has also emerged as a non-extensional theory that occupies an interesting position in the logical space of theories that are stronger than Minimal Mereology but weaker than Casati and Varzi's (1999) Extensional Mereology. 40 Funding For the work on this paper I received funding from the DFG project The Structure of Fundamentality (project number 448954791). Open Access funding enabled and organized by Projekt DEAL.
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