An Explication of Extensions, Sets and Membership. The following is a formal system in which "object extension of a predicate", "set" and "set membership" are defined concepts, it is hoped that the set concept finds some explication in it. It is a blend of Frege's extension of predicates method and Mereology especially that of David Lewis. EXPOSITION Language: Second Order Logic where predicate symbols are only allowed to take object symbols as arguments. Extra-logical primitives: A Two-place relation symbol "P" denoting part-hood. A two place object-to-predicate partial function symbol "<,>" denoting fulfillment instance of .. in .. so y= is read as y is the fulfillment instance of Q by x, or equivalently the fulfillment instance of x in Q. Axioms: [1] Mereology: All first order axioms of General Extensional Atomic Mereology "GEAM" without bottom. [2] Fulfillment: = <=> [x=y ^ Q<=>R]. [3] Instances: y= => y is an atom. Define "+": +(x) <=> Exist Q,y (x= ^ Q(y)). Define "-" : -(x) <=> not (+(x)). "+(x)" is read as "x is positive". "-(x)" is read as "x is negative". [4] Rejection: [not Q(x)] => Exist y (y= ^ -(y)). Define "eQ": x=eQ <=> for all y((y is an atom ^ y P x) <=> Exist z(y=)). eQ is read as the "object extension of Q". Define "Set": Set(x) <=> Exist Q (x=eQ ^ for all y (Q(y) =>Exist z(z=))). Define "E": y E x <=> Exist Q,z (x=eQ ^ z= ^ Q(y)). "E" stands for membership. [5] Acceptance: If phi is a formula with set parameters that only uses E and = as predicates, then: ([for all x(Q(x) <=> phi)] ^ y is a set ^ Q(y) => Exist z (z=)) is an axiom. [6] Subsets: [for all m E y (m E x)] ^ x is a set => y is a set. / Theory definition finished. A SUMMARY OF THE THEORY To General Extensional Atomic Mereology we add a new primitive partial function named as "fulfillment instance". When an object x fulfills a predicate Q, i.e. Q(x) is true, we say that there "might" exist an object that indicates that fulfillment which is considered here to be a positive kind of fulfillment, also when an object x is rejected by a predicate Q, i.e. we have ~Q(x) is true, then also we say that there "may" exist an object that indicates that rejection, which is seen here as a kind of negative fulfillment. so fulfillment instance is a partial two place function symbol between an object and a predicate denoted by "<,>" symbol, so y= is read as y is the fulfillment instance of object x in predicate Q. This theory grants the existence of a fulfillment instance for every rejection of an object in a predicate, i.e. for any object x and predicate Q if we have ~Q(x) then there would exist an object y such that y=, those fulfillment instances are termed as rejection instances, however this theory doesn't grant the existence of a fulfillment instance for every acceptance of an object in a predicate (acceptance instance). Now a fulfillment instance is stipulated to be an atom and also to be unique per its arguments, and EXTENSIONS are defined as mereological totalities of all fulfillment instances of a predicate by all objects in the universe of discourse. Membership in an extension of a predicate Q is defined as an object fulfilling Q provided that there exists a fulfillment instance of it in Q, otherwise even if it fulfills Q it won't be an element of the extension of Q. A set is defined as a complete extension of a predicate, i.e. one in which for every object fulfilling that predicate there is a fulfillment instance of it in that predicate. So membership in sets would exactly mirror fulfillment of the predicates they extend. While with the case of partial extensions membership in them would lag behind fulfillment of the predicates they extend, i.e. there will be objects SATISFYING those predicates yet not being elements in their extensions. Pathological predicates like 'not x E x', 'x is a ordinal', 'x is a singleton that is not in its sole element' etc.. would have incomplete extensions and thus won't have set extensions, this coupled with the limitation on the language in which those predicates are defined would prevent going into paradoxes.The last axiom states that a subextension of a set is a set. I think this is enough to interpret the reflection scheme of Ackermann's and thus enough to interpret ZFC. While Z is clearly interpreted by this method. For a more intuitive account and explanation of the axioms please see: http://zaljohar.tripod.com/extensions.txt MINOR CONUNDRUMS Some concern might be drawn about the mereological nature of sets as presented here, for example it would be expected from an extension of a predicate to have those objects fulfilling it being among its parts, it looks odd not to have them and only have their fulfillment instances in the extension. The answer is that this would complicate the structure of extensions, we can easily add the totality of all objects fulfilled by the predicate to the totality of all fulfillment instances to serve as the extension, but all processing would still pass through the fulfillment instances, for example the definition of set membership would not change at all, the 'functional' parts are the fulfillment instances, the added objects would only serve as a backlog that do not have a clear usage. Lewis in his explanation of sets also used a mereological collection of labels of objects to stand for the set rather than the objects themselves. On the other hand the thought that we can formulize sets in mereology by being just about the fulfilling objects without adding extra material, i.e. a set is its members, this thought would only manage to formalize a set theory with one tier of membership, i.e. all non singleton sets would be proper classes; and singletons would be identical to their sole elements, and existence of empty sets would be shunned, now to go up the hierarchy, i.e. to have set of sets of set...etc, it is almost impossible to find a mereological interpretation of such sets without adding extra material, so Ontological distinction is forced between a set and its members, this is well known [see Varzi at: http://www.academia.edu/5001160/Basic_Problems_of_Mereotopology], and it is a usual point mereologists hold against set theory, namely it is not Ontologically innocent, you need extra ontology, the material of the members is just not enough, or at least not easily figured out mereologically, not only that! adding it doesn't seem to do something and there are other considerations that might justify getting rid of them in the extension like for example Lewis's aesthetic principle of subclasses being identical to parts of classes [Lewis: Parts of Classes] and this can be reached in some versions of this method (see below) . However in this account the main answer is that this addition won't be that functional and it only complicates the mereological structure of extensions. Another conundrum is about the existence of indefinable sets, an uncountable set of sets would have more elements than the available descriptions "formulas" of the language, so can those indefinable sets be extensions of predicates, also a choice set has no written rule in the language of set theory. All those are answered by the fact that predicates are not formulas, and we may have a non expressible predicate, definability of a predicate is not the same as its existence, even choice is a predicate but not definable in terms of E and =, the philosophical debate about what predicates are? is very old traced back to the problem of universals discussed by Plato. VERSIONS There is a sense that sets according to this formulation have excessive mereological structure, mainly that of granting a tail of rejection instances that plays no role in deciding membership in extensions, a minimalistic approach would be to abandon Axiom 2, and define sets as complete positive extensions, i.e. a mereological collection of just positive fulfillment instances (i.e. acceptance instances) of objects fulfilling the predicate, or if there do not exist any object fulfilling the predicate then the set would be composed of a single rejection instance, now completeness of extension would be defined in the same manner above, so is set membership, also to aim for minimalism we may state that each object would have a unique acceptance instance should it exist, formally this is: ((Q(x) ^ R(y) ^ z= ^ u=) =>[z=u <=> x=y]). This would collapse into having the same structure David Lewis desired about 'subclasses being identical to parts of classes' for mereological interpretation of classes, although here representing the empty set would be nicer than that of Lewis. All of this won't have much technical impact, I prefer axiom 2 because it renders all sets mereologically separate, i.e. disjoint, and this adds more distinctiveness of structure into sets and helps ease definition of transitive closures of sets and the alike, albeit this can be achieved by stipulating that every acceptance instance is unique per its arguments and shunning multiplicity of rejection instances as above, however to me this doesn't give a nice sense of completeness as does the original formulation do (even though it does it at the expense of having a huge unnecessary ontology) Of course as with Lewis's approach Ur-elements (non extensions), Proper Classes, non well founded membership are all definable here. From the intuitive standpoint the minimalistic approach is preferred but it doesn't seem to have a big impact on the technical side. WITH LEWIS: This method depends highly on Lewis's "Parts of Classes", so some understanding of the main ideas of Lewis is important to understand some of its detail. Lewis's Parts of Classes is a mereological interpretation of classes and sets. In that book Lewis poses an important principle for such interpretation that of: To understand sets we need to Represent Multiplicity with an individual, so there is an interchange between Individuals and Multiplicities: Multiplicities are represented by individuals and those collect to form multiplicities and so on.. A nice mereological depiction of Multiplicities is that of a mereological body of "nice parts", to do that we add another primitive to mereology, that of 'connection', and we say a nice object is one that doesn't have external connection (contact) with another object and is not the fusion of two proper parts that are not connected to each other. Then because of the aesthetic principle of parts of a class being identical its subclasses, those nice parts would be atoms, so we have multiplicities of atoms represented by atoms. Of course not all multiplicities are representable by an atom, only some are, now Lewis considers a representative atom of a multiplicity to be a LABEL of it, and then goes to pose a size sensitive kind of labeling, that in order for a multiplicity to have a label then it should have 'few' members, then he goes and validate axioms of ZF by stating that the infinitude of the naturals is 'few', and that the subsets of a set are 'few', and so on regarding union. To me I don't see Labeling as size sensitive, The universe has a label (the Cosmos) but a lot of tiny objects doesn't receive any, Labeling might better be linked to the concept of definability of sets rather than being a size sensitive axiom, so to me I don't see labeling as constituting a substantial explanation for the axioms of set theory, to me it is just a kind of re-speaking them in other terms. However I do agree with the general structural framework he gives to explain sets, especially the idea of an atom representing a multiplicity of atoms. My explanation is that this atom (which I would term as Lewis's atom) is a fulfillment instance of a multiplicity in a predicate, and I think it is a more intimate understanding of sets, since it immediately engages it with comprehension principles which are the most important kind of axioms for sets. REVISITING FULFILLMENT INSTANCE. Here we take a closer look at this concept to understand truth about it. Fulfillment of a predicate can be depicted as the sum of fulfillments of objects that fulfill it, this sum is definitely a complex entity and we are to break it down to its constituents, we come down to the individual fulfillment of an object in a predicate, now this fulfillment itself is a simple concept, so if x fulfills Q, then no matter how many atoms x is composed of, the whole of x would function as a single unit to fulfill Q, we do not usually see this fulfillment as a complex process composed of stages, it seems to appear as a process having no proper parts, an all at once phenomenon, thus there is some intuitive justification into considering such fulfillment to constitute a mereological atom, same rationale applies to simple rejection. Fulfillment doesn't usually come into attention, it is a product of an object and a predicate, so it is overlooked since both objects and predicates have some clearer meaning than fulfillment, or at least come in mind prior to fulfillment. Now one might criticize this method for it not showing the fulfillment of some objects in predicates even though those objects do fulfill the predicate, so a fulfillment is there, so why they don't have a fulfillment instance? The answer to that is that those kinds of fulfillment are not those of simple acceptance nor of simple rejection, they seem to be a complex kind of fulfillment that requires multi-valued logic, something that classical logic cannot depict and thus looks as absence from its standpoint, for example the fulfillment of the Russell extension (i.e. the extension of the predicate defined after "not x E x") now the Russell extension fulfillment of the predicate "not x E x" seems to be wondering, ambivalent, it doesn't seem to be a simple straightforward concept, it cannot be understood in classical logic, it definitely doesn't give the impression of it being atomic, so it is not an atomic fulfillment and thus it would be absent from atomic fulfillment, and since it is atomic fulfillment that we are meant about here, then its absence is justified. Of course to compare it to Lewis the fulfillment instance has a similar role to Labels, there are some differences about existence of those relative to defining predicates here, but still the main function of both in interpreting classes and deciding upon class membership, and deciding which classes are sets, etc.. is very similar. I do concede that intuitively speaking a fulfillment instance is still a foggy concept compared to labels, and definitely looks foggy compared to its arguments, however I still think that it can be clearly and rigorously understood as this method strides to achieve. AVOIDING KNOWN PARADOXES: This method seemingly avoids paradoxes by not posing a general rule of acceptance for every predicate, in this manner pathological predicates like those defined after formulas "is a set" , "not x E x" , "is an ordinal", "is well founded", "is a singleton that is not an element of its sole element", would have extensions but those would be INCOMPLETE, i.e. they have an object fulfilling them yet not being a member of their extensions, now acceptance axioms are restricted to the language of set theory with set parameters, and to fulfillment by sets, so only predicates so restricted if they are only fulfilled by sets, can have set extensions, this restriction avoids going into the known paradoxes because there is no proof that every object in the universe of discourse fulfilling pathological predicates formulated within that restricted language would be a set. Another protection comes from the formula 'x is a set' not abiding those restrictions so no formula containing it would abide by the above restrictions, and of course it is a theorem of this theory that no formula abiding the above restrictions can be equivalent to the formula 'x is a set'. On the other hand pathological predicates defined after formulas not abiding by the above restriction that only hold of sets, like the formula 'x is a set', won't have all sets fulfilling them necessarily being members of their extension, since there is no rule stipulated here for the general acceptance of sets in predicates that are only fulfilled by sets, so those would have incomplete extensions. In nutshell the known paradoxes of Cantor, Russell, Burali-Forti, and Lesniewski are discharged as having incomplete extensions of the predicates involved in those paradoxes. MAIN ROLE Now the main question this method is to answer is which predicates described in the language of set theory (i.e. only using membership and identity as predicates) can have set extensions. To do that one should have some presumption about what causes incomplete extensions? One school of thought is that the reason lies in some kind of circularity, this can be expressed as a predicate D such that there would be an object x that fulfills D and yet has the extension of D as a subextension of the transitive closure of it, this is clearly the case with the extensions of the Russell predicate and the Burali Forti predicate, both extensions of those would be not in themselves and the later would be an ordinal. Axioms for sets must be stipulated carefully so that it won't result in such circularity, actually it should give the impression of them being far away from such altitudes. There are two ways to answer to that: [1] The controlled situation approach, which is a piecemeal approach that deals with specific sets and specific rules defined on them that enforces non circular predication, as an example is a theory that have axioms of Extensionality for non empty objects, Existence of an infinite set of all empty objects, Power-{0} operator (i.e. for each set x a set of all non empty subsets of x exists), and Separation scheme. Now this theory is equi-interpretable with Simple Type Theory with Ur-elements + Infinity "TSTU+Infinity" [Holmes: private messaging with me]. All sets would have all of their members having the same rank, and it is eligible to assume that no circular predication would arise from those axioms. Z and ZFC belong also to this kind of controlled situation approach and the structure of the sets being well founded is thought to force predication in a non circular manner thus might avoid the circularity resulting in paradoxes. On the other hand stratified comprehension has been provably reduced to Acyclic comprehension [a published result of mine] and thus shares the same general outline of avoiding circularity but this time not on the basis of set structure being non cyclic but rather on the basis of avoiding cyclicity within defining the predicates themselves, so a restriction to acyclic formulas is imposed and this eludes pathological formulation of predicates; this is also a kind of limited controlled situation approach. All those kinds of limited approaches can be done with the usual abstract notion of sets based on a primitive membership concept, and it doesn't beg any of the mereological understanding of extensions present here, even the concept of non circularity depicted above is well known (see Forester' article on Quine's new foundation at http://plato.stanford.edu/entries/quine-nf/). [2] The principled approach which looks at the root of pathological predication and give a more general notion of how to escape it. For example even the structure non circularity concept spoken about above can be summed up collectively in this method as restricting the axiom scheme of acceptance to hereditarily acyclic sets such that if P is a predicate defined in the language of set theory with hereditarily acyclic set parameters, and if x is a hereditarily acyclic set (a set in which every element of its transitive closure is not in its own transitive closure, and all being sets), and if P(x) is true then there exists a fulfillment instance of x in P, and thus this leads to: every predicate P with above restrictions that is only fulfilled by hereditarily acyclic sets would have a set extension! Also instead of the axiom of subsets we replace it with the axiom that any subextension of a hereditarily acyclic set is complete!. I think these non circular predication principles are equivalent to ZFC. However the approach here is a more principled one more akin to that of Ackermann's and actually explains set theorems of the later! Now we come to an answer to the paradoxes that is somewhat different from the above and yet in some ways depends on it. That is about "offshoots" of this methodology, an offshoot is a statement that is instigated by contemplating some discipline, of course not every offshoot statement is intuitive nor it would necessarily be consistent, but some of those offshoots might turn to be so, those statements are readily guessed and thought of when working with the method, now those statements must be scrutinized after the above controlled situation method, and some might help to instigate further thought about extending the controlled situation method itself. For example the axiom of acceptance written in the exposition of this theory, this axiom would definitely come into the mind of the investigator of this method! Because the most important predicates are those defined in the language of set theory with set parameters, and the objects of the main concern of this method are the sets no doubt and the main question of this method is about which fulfillment leads to existence of fulfillment instances, so the first thing that comes to the mind is the straightforward answer, that of the general statement of axiom of acceptance given above, This is what I call an offshoot statement, it is definitely not that intuitive, it is technical in nature, but it is something that would be thought of easily and pops up to the mind when investigating this method. Now if this can be proved consistent relative to ZFC, then this would add to our experience about guessing which offshoots are the consistent ones, and this might result in discovering some stronger principles that can extend the known standard axioms of set theory. It is a kind of investigative logicism having some heuristic value for discovering set axioms that I'm purporting here. CONCLUSION: A blend between Mereology, especially that of David Lewis, and Frege's method of extending predicates, would yield a body of knowledge that seems to have the potential of aiding understanding of classes and sets on the intuitive level as well as on the technical side, and so might prove useful to motivate further extensions of set theory, investigating useful alternatives, and motivating overall research in this field. Zuhair Al-Johar 23/6/2015 Update: Jul.9.2015 The above theory has an inconsistency that I've mentioned to FOM, see: http://www.cs.nyu.edu/pipermail/fom/2015-July/018802.html I already gave the name of "Mereo-Logicism" to describe the above methodology, it is very rich, it can have many versions. The following is one of the purest versions and most aesthetic one. Language: second order logic with predication limited to be over objects only. Primitives: Part-hood symbolized by P fulfillment instance symbolized by <,> (an object, predicate two place partial function symbol), so y= is read as y is the fulfillment instance of x in Q, or alternatively the fulfillment instance of Q by x. Axioms: [1] Mereology: General extensional atomic mereology without bottom [2] Fulfillment: R(x) ^ Q(y) ^ a= ^ b= -> [a=b <-> x=y] [3] Instances: y= -> y is an atom. Define: x=eQ <-> for all y ((y is an atom ^ y P x) <-> Exist z(y=)) "eQ" is read as: the extension of Q. Define: x is a set <-> Exist Q (x=eQ ^ for all y (Q(y) -> Exist z (z=))) Define: y E x <-> Exist Q (x=eQ ^ Exist z (z=) ^ Q(y)) "E" is read as "is a member of" [4] Extensionality: [for all x (x E eQ <-> x E eR)] -> [Q<->R] [5] Rejection: [Not exist x (Q(x))] <-> Exist y,z (y= & not Q(z)) [6] Acceptance: if phi is a formula only using E and = as predicates, then [(for all x (Q(x) <-> phi)) ^ y is a set ^ Q(y) -> Exist z (z= y E eR) -> [Q<->R] A nice feature of this theory is that it is compatible with the following aesthetic principle. "Every part of an extension of a predicate is an extension of a predicate". This will render the empty set a single atom, and prove Lewis's aesthetic principle of parts of classes being identical to subclasses, since all classes after Lewis are non empty. Here this account, unlike Lewis's, enrolls the empty set among classes and thus views it as a pure object, thus would prove here that: every part of a non empty extension is identical to a non empty subextension of it, and every part of an empty set is identical to that empty set itself, which is the most possible aesthetic result. Now this theory does interpret Z and I'd think ZF also over the realm of hereditarily set extensions of it. And those are definable here as: Define: x is hereditarily a set <-> x is a set ^ for all y (Exist Q,z(z= ^ z P TC(x)) -> y is a set) (TC standing for transitive closure is definable using the unrestricted composition principle of Mereology, this is the mereological intersection of all mereological collections having x as a part of them and for each positive fulfillment instance (i.e. acceptance instance) that is part of them every atom of the first argument of that fulfillment instance is a part of them also) ALL the above axioms are emanations of Mereologicism, none of them emanates from intuitive contemplation of sets on the abstract level (i.e. taking set membership as a primitive concept), so none of them is a set theory rule. This means that Set theory is explained in pure Mereologicism! Mereologicism seems to be more 'concrete' than the abstract Set Theory, so one can say that the later is founded in the former. So it seems that Mathematics would ultimately be founded in Mereo-Logicism. Of course the aesthetic principle mentioned above would open the door for increasing the language coverage of acceptance, so one would contemplate adding some formulas using part-hood relation in axioms of acceptance in a careful manner, this might result in stronger extensions of this method, on the other hand one might need to be more cautious and imposed restrictions on the language of formulas of acceptance or the type of objects accepted, for example a plausible restriction is to restrict the objects accepted to be hereditarily sets rather than just the general case of sets stipulated above. I do generally think that this method is positive in envisioning what sets are and in directing us to more powerful set axioms, not only that perhaps it can also interpret other primitives directly and thus be of more foundational significance since it does have the general outlook of LOGIC, and thus would have a wider coverage of concepts. Zuhair Al-Johar Update July 28, 2015: For the above mentioned system there is no real need for Extensionality (axiom 4) despite the very aesthetic nature of it. Versions of Mereo-Logicism Lets' call the above version as theory A. Now we need to justify the consistency of theory A by another theory that is also Mereo-logical. I'll call the following theory as theory B1. In this theory the axiom of "instances" would be modified to just require fulfillment instances to be disjoint, i.e.; not necessarily being atoms. The axiom of fulfillment is as that in theory A but we add the requirement of both fulfillment instances being atoms on the left side. Now we modify the axiom schema of acceptance in such a manner that for every predicate Q definable in the language of set theory (only using E or = as predicates) if fulfilled by object x then exists, and would be an atom if x is Fregean acyclic. Here we'll be having two types of extensions of predicates, the complete one and the partial one. The complete extension of predicate Q, denoted as e`Q, would be the mereological collection of fulfillment instances of Q whether atoms or not (I usually call the non atom fulfillment instances as Rocks) The partial extension of predicate Q, denoted as eQ, would be the mereological collection of fulfillment instances of Q that are atoms, so no rock is a part of eQ, eQ is only made of atoms. Now eQ is a set if and only if eQ=e'Q, in other words eQ is a set iff e'Q has no rocks as parts of it. Now the membership relation E shall be defined as: x E y <=> Exist Q (y=eQ ^ Exist z (z= ^ z Part of y)) Now the Fregean membership relation E* would be defined as: x E* y <=> Exist Q (y =e'Q ^ Exist z (z=)) which is equivalent to Exist Q (y=e'Q ^ Q(x)) which is Frege's original membership definition as long as Q is definable in the language of set theory. Now the transitive closure of an extension x of a predicate Q, whether x=eQ or x=e'Q, denoted as TC(x), would be defined as the mereological intersection of all objects having e'Q as a part of and such that for every positive fulfillment instance , whether an atom or not, that is a part of them, every atom of y is a part of them also. Now y is said to be 'within' TC(x) if there exists a part of TC(x) that is a positive fulfillment instance (whether atom or rock) and y is the first argument of that instance. Now x is called Fregean acyclic if and only if for every y within TC(x) we have not y within TC(y). Of course x is Fregean cyclic iff x is not Fregean acyclic. Now adding the further restriction on parameters of the formulae defining the predicates in acceptance to also be Fregean acyclic [just add the phrase "and all parameters of Q are Fregean Acyclic" to the end of the statement of modified scheme of acceptance written above], and if we add the axiom that every extension having a rocky part must be Fregean cyclic. Then I'd think we can have a theory that also might be equi-interpretable with ZF, we'll call this system as B2 which seems to be the safest. However it seems that the restriction on parameters would allow us to have extensions for predicates definable in languages other than that of set theory, and so we may introduce fulfillment instances as total functions over the domain of discourse and the resulting theory would be called B3. B1 have a clear strategy to avoid paradoxes, actually an explicit rigorous formal one, that of avoiding Fregean cyclicity; and theory A seems to be interpretable in it. And so B1 motivates the consistency of Theory A. Theory B1 unravel the nature of absences of fulfillment instances from eQ of objects fulfilling Q, those absences in theory A are the rocks in theory B1, and theory B1 adds the condition that the rocky backlog of an incomplete extension would harbour cyclicity that is captured formally as Fregean cyclicity depicted above, so extensions like those of all extensions that are not E members of themselves, or of those of ordinals, and the alike all would be seen as Fregean cyclical ones. This nicely captures the Acyclicity of structure school for avoiding paradoxes. And theory A seems to be directly interpretable in it. That those theories prove separation is obvious, so is the proof of power; infinity is proved in a coercive way as the extension all "finite Von Neumann ordinals for which a power set exist and for which an intersectional set K of all sets containing its predecessor that are closed on predecessor-ship should exist and for which every element m of K there exist K-{m}, and such that if K is non empty then not every non empty element of K has a non empty predecessor in K" is clearly acyclic and thus must be a set. In this way Z is clearly interpretable in those systems. Moreover those systems can be formulated in multi-sorted first order logic as well. Another mereo-logical theories that might be interesting are those working in three valued logic, so a positive fulfillment instance would be one of acceptance, a negative one would be that of rejection and an ambivalent fulfillment instance would portray that of neither rejection nor acceptance. A set is an extension of a predicate, that do not have any ambivalent fulfillment instance that is a part of it. Now if we take all theorems about sets of such a theory and modify it to suit classical logic, then we might get what I designate as theories C, and I'd think those might be nearest to theories with a universal set in them, like NF and the alike. In nutshell theory A is the most neat one and the simplest in terms of Mereo-Logicism. Theory B1 serves to motivate the consistency of theory A in mereological terms through avoidance of paradoxes by evading cyclicity. Theory B2 is more conservative and it might be the safest one. Theory B3 is a possible strengthening of theory B2 and might enable extensions of it that speak about languages wider than that of set theory. Theories C might be nearer to stratified theories (acyclicity of comprehension method). Such systems mean that an argument for consistency of Mereological systems can come from within Mereo-Logicism itself and thus doesn't essentially beg interpretability in ZF and the alike abstract set theoretical systems. IF this can be achieved nicely then Mereo-Logism would stand on its own as a foundation of mathematics. Now if Mereo-Logicism is interpretable in a fragment of ZF, and if it proves to be useful in guidance of development of extensions of ZF, or of establishing equi-consistency of alternative set theories with fragments of ZF (like for example of NF with a fragment of ZF), and especially if those alternative theories prove to be more useful in interpreting some mathematics than ZF itself, then this would make Mereo-Logicim of more foundational importance than ZF, and by then it would turn to be the ultimate foundation of mathematics. Moreover if bi-interpretability with ZF was proved and if it was derived by ideas guided by Mereo-Logicsm itself rather than by those of ZF, then also Mereo-Logicsm would have the upper lead over ZF. I'd conjecture really that Mereo-Logicism would be the ultimate Foundational theory of Mathematics and Set theory. Zuhair Al-Johar