Show Menu

# Mereotopology Cheat Sheet by apowers313

### 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 productUxy→∃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 complimentU-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)

## Download the Mereotopology Cheat Sheet

3 Pages
//media.cheatography.com/storage/thumb/apowers313_mereotopology.750.jpg

PDF (recommended)

Alternative Downloads

## Comments

No comments yet. Add yours below!

## Add a Comment

Your Comment

Please enter your name.

Please enter your email address

Please enter your Comment.

## Related Cheat Sheets

Minecraft Cheat Sheet
Rhetoric Cheat Sheet
redes Cheat Sheet

## More Cheat Sheets by apowers313

Bikeshed Cheat Sheet