Is a wall or a brick house a whole of its parts? Obviously -- that's a pre-analytic datum. But is it a sum of its parts? I have been arguing, with no particular originality, in the negative. I have been arguing that it is a big mistake to assume that, just because y is a whole of the xs, that y is a sum of the xs. But it depends on what exactly is meant by 'sum.' My point is well-taken if 'sum' is elliptical for 'classical mereological sum.' But what does that mean? Since 'classical mereological sum' is a technical term, it has all and only the meaning conferred upon it by the definitions and axioms of classical mereology. I will now present what I take to be the essentials of classical mereology. I will use 'sum' as short for 'classical mereological sum.' Later we will look at neoclassical variants that result from tampering with the classical definitions and axioms.
If anything in what follows is original, it is probably a mistake on my part. Feel free to correct me -- but only if you know the subject matter.
I will take proper parthood and identity as primitives. To simplify the exposition I will drop universal quantifiers. They are there in spirit if not in letter.
D1. x is a PART of y =df x is a proper part of y or x = y.
D2. x OVERLAPS y =df there is a z such that z is part of x and z is part of y.
D3. x is DISJOINT from y =df it is not the case that x overlaps y.
D4. y is a SUM of the xs =df z overlaps y iff z overlaps one of the xs.
A1. Asymmetry of Proper Parthood. If x is a proper part of y, then y is not a proper part of x.
A2. Transitivity of Proper Parthood. If x is a proper part of y, and y is a proper part of z, then x is a proper part of z.
A3. Supplementation of Proper Parthood. If x is a proper part of y, then there is a z such that z is a proper part of y and z is disjoint from x.
A4. Uniqueness of Summation. If u is a sum of the xs and v is a sum of the xs, then u = v.
A5. Unrestricted Summation. For any xs, there is a y such that y is a sum of the xs.
When I used the word 'sum' in previous posts, I intended that its meaning be not merely the meaning assigned to it by (D4), but the meaning assigned to it by (D4) in conjunction with the rest of the definitions and the axioms (not to mention the theorems that follow as logical consequences of the definitions and axioms).
Extensionality is a feature of classical mereology. I leave it as an exercise for the reader to derive Extensionality of Parthood -- if x and y are sums with the same proper parts, then x = y -- as a theorem from the above.