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Mereotopology Cheat Sheet by

topology     logic     mereology     mereotopology

Ground Mereology Axioms

axiom
meaning
defn.
𝗠
Ground Mereology
Pxy
x is a part of y
Reflexivity
x is a part of itself
Pxx
Antisymmetry
x and y can't be parts of each other, unless they are actually the same thing
Pxy ∧ Pyx → x=y
Transitivity
if x is a part of y, and y is a part of z, then x is a part of z
Pxy ∧ Pyz → Pxy

Ground Mereology Defini­tions

sym.
meaning
defn.
PP
Proper Part
PPxy := Pxy ∧ ¬Pyx
O
Overlap
Oxy := ∃z (Pzx ∧ Pzy)
U
Underlap
Uxy := ∃z (Pxz ∧ Pyz)
OX
Over-C­rossing
OXxy := Oxy ∧ ¬Pxy
UX
Under-­Cro­ssing
UXxy := Uxy ∧ ¬Pyx
PO
Proper Overlap
POxy := OXxy ∧ OXyx
PU
Proper Underlap
PUxy := UXxy ∧ UXyx

Derived Statements

Overla­pping is Reflexive
Oxx
Overla­pping is Transitive
Oxy → Oyx
Proper Parts are not Reflexive
¬PPxx

Extens­ional Mereology

𝗘𝗠
Extens­ional Mereology
Supplementation Axiom
¬Pxy → ∃z(Pzx ∧ ¬Ozy)
Weak Supple­men­tation
𝗘𝗠 ⊢ PPxy → ∃z(PPzy ∧ ¬Ozx)
If all the proper parts of X are proper parts of Y, X is part of Y
If two objects have the exact same proper parts, they are the same object

Closed (Exten­sional) Mereology

𝗖𝗘𝗠
Closed Extens­ional Mereology
descri­ption operator
℩x is "the unique x such that"
x+y
sum (or fusion)
Oxy→∃x∀w(Pwz↔(Pwx∧Pwy))
defined as:
℩z∀w(Owz↔(Owx∨Owy))
x×y
product
Uxy→∃z∀w(Owz↔(Owx∨Owy))
defined as:
℩z∀w(Pwz↔(Pwx∧Pwy))
x-y
difference
∃z(Pzx∧¬Ozy)→∃z∀w(Pwz↔(Pwx∧¬Owy))
defined as:
℩z∀w(Pwz↔(Pwx∧¬Owy))
𝑈
universe
∃z∀x(Pxz)
defined as:
℩z∀x(Pxz)
∼x
compliment
U-x

General (Exten­sional) Mereology

𝗚𝗘𝗠
General Extens­ional Mereology
Fusion Axiom
∃xΦ → ∃z∀y(Oyz ↔ ∃x(Φ∧Oyx))

Ground Topology Axioms

𝗧
Ground Topology
Cxy
x is connect to y
Reflex­ivity
x is connected to itself
Cxx
Symmetry
 
Cxy → C yx
Transitivity
 
Pxy → ∀z(Czx → Czy)

Ground Topology Defini­tions

EC
External Connection
TP
Tangential Part
TPP
Tangential Proper Part
IP
Internal Part
IPP
Internal Proper Part
E
Enclosure
IE
Internal Enclosure
TE
Tangential Enclosure
S
Superp­osition
PS
Proper Superp­osition
I
Coinci­dence
A
Abutting
 

Predicate Logic

¬
not
and
or
for every
there exists
implies
:=
definition
iff
provable
entails
tautology
contra­diction

Basic Patterns in Mereology

Credit: Varzi 1996, used without permis­sion. The relations
in parent­hesis hold if there is a larger z including both x and y.

Basic Patterns in Mereot­opology

Credit: Varzi 1996, used without permis­­sion. Seven basic patterns of the connection relati­onship.

Examples

Part
Your finger is part of your hand
Reflex­ivity
Your finger is part of your finger
Antisymmetry
Your finger is part of your hand, but your hand is not part of your finger
Transitivity
Your finger is part of your hand, and your hand is part of your body, so your finger is part of your body
Proper Part
A tail is a proper part of a cat
Overla­pping
Two roads overlap at their inters­ection
Underlapping
Your finger and thumb are underl­apping parts of your hand
Supplementation
Road A is not part of Road B, because there is at least some of Road A that doesn't overlap Road B
Weak Supplementation
Road A is not a proper part of Road B, because at least some of Road A is outside Road B

Alternate Notations

symbol
meaning
from
is a proper part of
Simon 1987
is an improper part of
Simon 1987
overlaps
Simon 1987
is disjoint from
Simon 1987
Pxx
is a part of
Smith

Mereol­ogical Operations

binary product
x⋅y
+
binary sum
x+y
-
difference
x-y
σx⌜Fx⌝
fusion
𝜋x⌜Fx⌝
nucleus

Smith (1996) Mereology Defini­tions

sym.
meaning
ex.
defn.
P
is a part of
xPy
O
overlaps
xOy
∃z(zPx ∧ zPy)
D
discrete
xDy
¬xOy
Pt()
is a point
Pt(x)
∀y(yPx­→y=x)

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