Extensional Mereology is a formal calculus for dealing with the part-whole relationships encountered in the study of Mereology. Through its formalism we can reason about how entities are composed of others much in the way that set theory allows us to manipulate classes of mathematical objects. The prime assumption of extensional mereology is that the whole is equal and identical to the sum of its parts. In the same way that 6 is nothing more than 4+2 or 2x3, (meaning no additional elements are necessary to make it 6), so are the objects of the world.

Historical Background

At the end of the 19th century and the beginning of the 20th, much progress was being made in understanding and creating formal logical systems. At the time, Gottlob Frege was developing a set-theoretic logical system that was aimed at supplying the foundation on which the whole of mathematics would be grounded. During the course of its development, however, Bertrand Russell discovered a gaping hole in its formulation, namely that paradoxes could result if one allowed sets to include themselves. This paradox threatened the whole endeavor for how could mathematics be based on a system that very easily resulted in paradoxes? This was no foundation!

Russell's response was to mandate a strict hierarchy of sets when he formulated his own system in the Principia Mathematica. At the same time, Stanislaw Lesniewski, a Polish logician, was working on an alternative logical system that was based on the principles of Mereology. His system avoided the paradox because it conceived of a "class" as a mereological sum, and since every sum was a part of itself due to the concept of an improper part, no paradox was entailed.1 (Mereological sums and improper parts will be explained shortly) Because classes (sets) were unable to account for part-whole relations, Russel's system came to be used for mathematics while Lesniewski's unorthodox axiomatic system gradually became the means to deal with ontology. Over the course of time, Whitehead, Tarski, and Leonard and Goodman all formulated their own mereological systems, but it was Lesniewski's that is still studied today.

Fundamental Assumptions

• The whole is identical to the sum of its parts. If a hammer is made up of the metal head and the wooden handle, then the wooden handle and metal head equal the hammer.
• Two objects are identical iff their parts are identical
• A sum of individuals is always a part of itself (an "improper part")
• Any two objects have a sum, meaning that for any two objects, there is another object which is made up of the two. (See mereological sum, below)
The Calculus

x << y

x is a proper part of y. The proper part relation is how we normally conceive of something being a part of another.

1. Existence entailing: if one of the two exists, the other does too
2. Asymmetrical: if one thing is a proper part of another, the second is not part of first.
3. Transitive: a part of a part of a whole is itself part of the whole.
4. Irreflexivity: Nothing is a proper part of itself.
5. Supplementative: If an object has a part, it has another part disjoint from the first. (This is necessary, for if the whole is made up of only that part, then it is not a part but the whole thing)

x < y

x is a part of y. x can either be a proper part or an improper part of y. (An improper part of a whole is the whole itself. Whereas a chocolate chip would be a proper part of a cookie, only the entire cookie is an improper part of itself)

x º y

x overlaps y. If two elements have parts in common, then they are considered to be overlapping mereologically. Overlapping is symmetric and reflexive, but not transitive. It is somewhat difficult to conceive of two real-world objects that overlap. One example of overlap occuring in humans is when siamese twins share a heart or other organ. (Gritchka has pointed out to me that one could consider a wall to be the intersection of two rooms since it is a part of both)

x | y

x is disjoint from y. This is the opposite case from overlap; here x and y have no elements in common.

x • y

Binary product of x and y. The individual which results is such that any common part of both x and y is part of the product. This operation results in the overlapping object, mentioned above.

x + y

Binary sum of x and y. The sum is simply the individual that includes both x and y. A broom is roughly speaking the sum of its handle and head. One of the main theses of Extensional Mereology is that any two items have a mereological sum.2

x - y

Difference of x and y. The largest part of x that has no part in common with y.

U

The universe. The universe is the unique individual that is formed by taking the mereological sum of all object. All objects are a part of the universe individual. It is important to note that the universe is an actual object rather than a container, so that it cannot exist if no objects exist.

U - x

Complement. The individual comprising the rest of the universe outside of x.

At x

Atom. x is an atom which means that there are no objects which are a proper part of x.

These simple mereologial operations are combined with elements from formal logic such as quantifiers and equality operators and are able to express many more complicated facts about the objects' ontologies. We can build up rules for relations, restrict membership, demonstrate dependence, and so on.

Examples

Extensional Mereology (like most other logics) very quickly gets extremely complicated and tedious as specialized symbols and functions are included in order to prove things. In order to give some examples of how it works that are simple and brief, I will restrict my examples to sentences describing essential parts.3 x is an essential part of y if y cannot exist unless it has y (A man may lose his hands and still exist, but one cannot lose one's head and continue to exist). We can show this by the following statement:

(E!a → Fa) - a is essentially F (a is essentially F iff a cannot exist unless a is F)
(E!a → x<<a) - x is an essential part of a (a cannot exist unless x is a (proper) part of a)

Note that I'm using '' to symbolize the box necessity operator of modal logic. It's supposed to look like a box. If it doesn't, I'm sorry =). In these examples, All(x) means for all x, Exists(x) means that there exists an x, 'E!h' means that h exists, and 'Exth' means that h exists at time t.

Now let's extend this to talk about bicycles. If we wanted to say that any possible object has a wheel as a part of it if it is a bicycle, we could express this like so:

All(x)(Bx → Exists(y)(Wy & y << x))

But this is a very weak formulation. If we take time into account, denoted by subscript t, then we can express the same claim in at least 3 different ways:

()(Bx → All(t)(Btx → Exists(y)(Wty & y <<t x)) - Whenever something is a bicycle, it has a wheel then, but this need not be the same wheel at all times
()(Bx → Exists(y)All(t)(Btx → Wty & y <<t x)) - There is something which is a wheel and part of the bicycle at all times at which it is a bicycle (Rules out changing the wheels)
()(Bx → Exists(y)Exists(t)(Bt & Wty & y <<t x)) - Every bicycle must have some wheel at some time.

Just one more quick example. A helium atom is composed of two protons and electrons. Now the electrons may come and go, but when we designate a particular atom (say h), we mean that it is made up of two specific protons. If one proton (p) were to change, it would be a different helium atom. So each proton is an essential part of that helium atom.

(E!h → All(t)(Exth → p <<t h)) - The proton is an essential permanent part of the helium atom
()(Hx → Exists(y)(Py & (E!x → All(t)(Extx → y <<t x)))) - General case

Objections

The following are some of the objections that have been raised to Extensional Mereology.

• Proper Parts Objection There are senses of 'part' where a whole does not count as one of its parts. This argument is against the inclusion of the improper part relation which allows say the cookie itself to be included as one of the parts of the cookie. This notion runs against common sense and in many respects is simply non-sensical.

Response: All mereologies have the proper part relation where the whole cannot be a proper part of itself. We still allow the normal sense of 'part', so common notions shouldn't be threatened.

• Nontransitivity Parts aren't necessarily transitive. A handle is a part of a door, and a door is part of a house yet a handle is not part of a house. Similarly, a platoon is part of a company and an army is made up of companies, but a platoon is not a part of an army.

Response: If a handle is not part of a house, then where is it? Is it somehow detached from the house? For the most part, notions of 'part' are considered to be of a spatio-temporal sense. If you narrow the definition to a direct functional contribution then it's true that things may not be transitive, but there is no reason to believe this means that a more basic part-whole theory can't exist and be used by extensional mereology.

• Too Many Sums The mereological sum operation allows us to create a sum (individual) from any two other individuals in the world, even though they may have nothing in common to each other, even physical location. This gives rise to individuals which we would not consider as such.

Response: The notion of an individual in a metaphysical sense may be more abstract than our common sense notion, but this doesn't mean that it can't be considered part of the mereology. If we don't consider the possibility that we can unite any two individuals then we are seriously limiting the expressive power of our ontology. Apart from this, I am not aware of a real objection to this argument. It seems to me that it simply is a decision one has to make when creating the ontology; either you allow these types of sums or you don't.

• Rescher's Objection Mereology looks at content but not organization. Structure is responsible for the identity of individuals rather than the content. Nicholas Rescher describes a figure where the words 'Cardinals' and 'Multiply' are arranged in a circle so it is ambiguous whether it is 'Cardinals Multiply' or 'Multiply Cardinals'. In each case the content is the same but the meaning is different, and this is due to the structure. A similar problem can be found in ordered pairs (<1,2> has the same content as <2,1> but are certainly not identical), committees (even though the Acquisitions Comittee may have the exact same members as the PR Committee, they are vastly different entities) and states of affairs ('John loves Mary' has the elements: 'John', 'Mary' and 'x loves y' but it is different from 'Mary loves John').

Response: None that I am aware of, but we should note that this objection is not raised against Extensional Mereology per se, but against the larger view that the whole is equal to the sum of its parts. Extensional Mereology is just a tool or a branch of investigation under that assumption, so this objection really is on a different level from others.

• Four-Dimensionality The ontology forced onto us by the acceptance of the principles of extensional part-whole theory is a materialistic ontology of four dimensional objects.

Note: This is a very complicated and involved argument that is dealt with in the node Flux Argument.

Some of these arguments and responses are more effective than others. There is a debate as to whether a formal ontology must adhere to our common notions of existence or whether it should be allowed to delve into strange new ideas. Extensional Mereology, because it assumes that the whole is identical to its parts is obviously subject to the larger debate of the nature of parts and wholes. If we are to assume that it is the case that a whole is identical to its parts then extensional mereology is the way to go. It is important to realize that Extensional Mereology is still a very new area and the full ramifications of the logic are still being explored. It's important to develop a strong understanding of the semantics so that we can fully develop and realize the reasoning power of the system.

Of course, I personally feel that the entire system is flawed due to the fact that I believe wholes are not merely the sum of their parts. If we reject this fundamental premise, the entire system falls apart. I do appreciate, however, the fact that it gives us a clear and unambiguous way to describe the relations between objects. It just ain't on the right track.

1We can see how Extensional Mereology was immune to Russell's paradox by introducing the concept of the improper part. Since every sum contains itself, there can be no classes of classes that do not include themselves.

2This thesis is rather implausible... how is it possible that I and a star in some remote part of the universe have a sum? What does it mean? This is roughly equivalent, though, to how one could have a set that contains only edibleplastic and the star: {edibleplastic, remote star}.

3The study of essential parts is called Mereological Essentialism. Are some parts essential while others are not? What happens to the whole if an essential part is lost?

Simons, Peter Parts: A Study in Ontology Clarendon Press 1987.