Part-Whole Relations
Part-Whole Relations
This Demonstration introduces the basic notions of Leśniewski's mereology (a theory of part-whole relations). A 2×2 square is divided into four 1×1 squares , , , and . The same object also consists of two 2×1 rectangles and , or alternatively, two 1×2 rectangles and . Then, for example, is a part of , is a part of , and is a part of . In symbols, ∈pt(S), ∈pt(), and ∈pt(S). Let be the general name for a 1×1 square; then is the class of these squares. In symbols, . If is the general (unshared) name of 2×1 rectangles, then . Also, is not part of itself. Squares are shown in yellow, while rectangles are colored light gray.
S
S
1
S
2
S
3
S
4
S
R
1
R
2
R
3
R
4
S
1
S
S
1
R
1
R
1
S
S
1
S
1
R
1
R
1
S
1×1
S
S∈Kl()
S
1×1
R
2×1
S∈Kl()
R
2×1
S
The notion of ingredient (element) is defined by —that is, is an ingredient of object if and only if is the same object as or is a part of . So .
A∈el(B)≡A=BorAϵpt(B)
A
B
A
B
A
B
S∈el(S)
It is possible to define class in terms of ingredient.
P
a
1. is an object;
P
2. every is an ingredient of the object ;
a
P
3. for any , if is an ingredient of the object , then some ingredient of the object is an ingredient of some .
Q
Q
P
Q
a
You can use the diagrams to illustrate the following truth relations: , , ∈pt(), ….
S∈Kl()
S
1×1
S∈Kl()
R
2×1
S
1
R
1