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axiom
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meaning
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defn.
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𝗠
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Ground Mereology
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Pxy
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x is a part of y
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Reflexivity
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x is a part of itself
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Pxx
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Antisymmetry
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x and y can't be parts of each other, unless they are actually the same thing
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Pxy ∧ Pyx → x=y
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Transitivity
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if x is a part of y, and y is a part of z, then x is a part of z
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Pxy ∧ Pyz → Pxy
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sym.
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meaning
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defn.
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PP
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Proper Part
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PPxy := Pxy ∧ ¬Pyx
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O
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Overlap
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Oxy := ∃z (Pzx ∧ Pzy)
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U
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Underlap
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Uxy := ∃z (Pxz ∧ Pyz)
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OX
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Over-Crossing
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OXxy := Oxy ∧ ¬Pxy
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UX
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Under-Crossing
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UXxy := Uxy ∧ ¬Pyx
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PO
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Proper Overlap
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POxy := OXxy ∧ OXyx
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PU
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Proper Underlap
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PUxy := UXxy ∧ UXyx
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Overlapping is Reflexive
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Oxx
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Overlapping is Transitive
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Oxy → Oyx
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Proper Parts are not Reflexive
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¬PPxx
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𝗘𝗠
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Extensional Mereology
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Supplementation Axiom
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¬Pxy → ∃z(Pzx ∧ ¬Ozy)
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Weak Supplementation
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𝗘𝗠 ⊢ PPxy → ∃z(PPzy ∧ ¬Ozx)
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If all the proper parts of X are proper parts of Y, X is part of Y
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If two objects have the exact same proper parts, they are the same object
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𝗖𝗘𝗠
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Closed Extensional Mereology
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℩
|
description operator
℩x is "the unique x such that" |
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x+y
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sum (or fusion)
Oxy→∃x∀w(Pwz↔(Pwx∧Pwy)) defined as: ℩z∀w(Owz↔(Owx∨Owy)) |
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x×y
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product
Uxy→∃z∀w(Owz↔(Owx∨Owy)) defined as: ℩z∀w(Pwz↔(Pwx∧Pwy)) |
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x-y
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difference
∃z(Pzx∧¬Ozy)→∃z∀w(Pwz↔(Pwx∧¬Owy)) defined as: ℩z∀w(Pwz↔(Pwx∧¬Owy)) |
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𝑈
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universe
∃z∀x(Pxz) defined as: ℩z∀x(Pxz) |
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∼x
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compliment
U-x |
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𝗚𝗘𝗠
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General Extensional Mereology
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Fusion Axiom
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∃xΦ → ∃z∀y(Oyz ↔ ∃x(Φ∧Oyx))
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𝗧
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Ground Topology
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Cxy
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x is connect to y
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Reflexivity
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x is connected to itself
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Cxx
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Symmetry
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Cxy → C yx
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Transitivity
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Pxy → ∀z(Czx → Czy)
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EC
|
External Connection
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TP
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Tangential Part
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TPP
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Tangential Proper Part
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IP
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Internal Part
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IPP
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Internal Proper Part
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E
|
Enclosure
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IE
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Internal Enclosure
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TE
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Tangential Enclosure
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S
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Superposition
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PS
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Proper Superposition
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I
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Coincidence
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A
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Abutting
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¬
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not
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∧
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and
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∨
|
or
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∀
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for every
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∃
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there exists
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→
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implies
|
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:=
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definition
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↔
|
iff
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⊢
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provable
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⊨
|
entails
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⊤
|
tautology
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⊥
|
contradiction
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Part
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Your finger is part of your hand
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Reflexivity
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Your finger is part of your finger
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Antisymmetry
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Your finger is part of your hand, but your hand is not part of your finger
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Transitivity
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Your finger is part of your hand, and your hand is part of your body, so your finger is part of your body
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Proper Part
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A tail is a proper part of a cat
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Overlapping
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Two roads overlap at their intersection
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Underlapping
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Your finger and thumb are underlapping parts of your hand
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Supplementation
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Road A is not part of Road B, because there is at least some of Road A that doesn't overlap Road B
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Weak Supplementation
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Road A is not a proper part of Road B, because at least some of Road A is outside Road B
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symbol
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meaning
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from
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≪
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is a proper part of
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Simon 1987
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≺
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is an improper part of
|
Simon 1987
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○
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overlaps
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Simon 1987
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⎱
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is disjoint from
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Simon 1987
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Pxx
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is a part of
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Smith
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⋅
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binary product
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x⋅y
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+
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binary sum
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x+y
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-
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difference
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x-y
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σx⌜Fx⌝
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fusion
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𝜋x⌜Fx⌝
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nucleus
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sym.
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meaning
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ex.
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defn.
|
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P
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is a part of
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xPy
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O
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overlaps
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xOy
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∃z(zPx ∧ zPy)
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D
|
discrete
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xDy
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¬xOy
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Pt()
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is a point
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Pt(x)
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∀y(yPx→y=x)
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