Cheatography

# 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.

### 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)