Reorganize ontology/RDF/XML.

notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json

@@ -967,7 +967,7 @@
     "_journal/2024-11/2024-11-20.md": "951b6034d60a40dbd8201c50abf0dbb9",
     "_journal/2024-11/2024-11-19.md": "d879f57154cb27cb168eb1f1f430e312",
     "set/cardinality.md": "270b48a3559c055cb15216f70101ecc2",
-    "geometry/area.md": "dc362cb5be314b2b9e4ecc53f2289af0",
+    "geometry/area.md": "7f947bb5ac782495a1fb4a63bb2463e7",
     "_journal/2024-11-23.md": "911f82ab8aede5ecdb96493aef64b0b9",
     "_journal/2024-11/2024-11-22.md": "51117030e2364dbce3a8d507dead86ae",
     "_journal/2024-11-24.md": "225661114ad1b605132873e83acdf777",

notes/_journal/2024-07/2024-07-28.md

@@ -8,5 +8,5 @@ title: "2024-07-28"
 - [ ] Sheet Music (10 min.)
 - [ ] Korean (Read 1 Story)
 
-* Initial notes on [[ontology/rdf/index|RDF]].
+* Initial notes on [[rdf|RDF]].
 * Notes and flashcards on [[uri|URIs]].

notes/_journal/2024-12-12.md

@@ -7,3 +7,6 @@ title: "2024-12-12"
 - [x] OGS
 - [ ] Sheet Music (10 min.)
 - [ ] Korean (Read 1 Story)
+
+* Notes on monotone property of area.
+* Initial proofs on some properties of integrals of step functions.

notes/encoding/rdf.md

@@ -1,9 +1,8 @@
 ---
 title: RDF
 TARGET DECK: Obsidian::STEM
-FILE TAGS: ontology::rdf
+FILE TAGS: rdf
 tags:
-  - ontology
   - rdf
 ---
 

notes/encoding/uri.md

@@ -3,7 +3,6 @@ title: Uniform Resource Identifiers
 TARGET DECK: Obsidian::STEM
 FILE TAGS: uri
 tags:
-  - rdf
   - uri
 ---
 

notes/geometry/area.md

@@ -189,6 +189,98 @@ Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Int
 <!--ID: 1732381333365-->
 END%%
 
+This property lets us prove $\varnothing$ is measurable with the expected area: $$a(\varnothing) = a(T - T) = a(T) - a(T) = 0$$
+
+%%ANKI
+Basic
+Is the empty set considered measurable?
+Back: Yes.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1734013716384-->
+END%%
+
+%%ANKI
+Basic
+Which axiom of area is used to prove $\varnothing$ is measurable?
+Back: The difference property.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1734013716403-->
+END%%
+
+%%ANKI
+Basic
+What does $a(\varnothing)$ evaluate to?
+Back: $0$
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1734013716406-->
+END%%
+
+%%ANKI
+Basic
+What chain of equalities is used to prove that $a(\varnothing) = 0$?
+Back: For any $T \in \mathscr{M}$, $a(\varnothing) = a(T - T) = a(T) - a(T) = 0$.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1734013716410-->
+END%%
+
+This property also lets us state the **monotone property**: $$\forall S, T \in \mathscr{M}, S \subseteq T \Rightarrow a(S) \leq a(T)$$
+
+%%ANKI
+Basic
+What does the monotone property of area state in FOL?
+Back: $\forall S, T \in \mathscr{M}, S \subseteq T \Rightarrow a(S) \leq a(T)$
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1734013716414-->
+END%%
+
+%%ANKI
+Basic
+What name is given to the following property? $$\forall S, T \in \mathscr{M}, S \subseteq T \Rightarrow a(S) \leq a(T)$$
+Back: The monotone property.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1734013767208-->
+END%%
+
+%%ANKI
+Basic
+Which axiom(s) of area are used to prove the monotone property?
+Back: The nonnegative and difference properties.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1734013716417-->
+END%%
+
+%%ANKI
+Basic
+Let $S, T \in \mathscr{M}$ such that $S \subseteq T$. Why is the difference property needed to show $a(T - S) \geq 0$?
+Back: It states $T - S$ is measurable.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1734013716421-->
+END%%
+
+%%ANKI
+Basic
+Let $S, T \in \mathscr{M}$ such that $S \subseteq T$. Why is the difference property needed to show $a(S - T) \geq 0$?
+Back: N/A. The difference property says nothing about $S - T$.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1734013716426-->
+END%%
+
+%%ANKI
+Basic
+Let $S, T \in \mathscr{M}$ such that $T \subseteq S$. Why is the difference property needed to show $a(S - T) \geq 0$?
+Back: It states $S - T$ is measurable.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1734013716430-->
+END%%
+
+%%ANKI
+Basic
+Let $S, T \in \mathscr{M}$ such that $S \subseteq T$. Why is the nonnegative property needed to show $a(T - S) \geq 0$?
+Back: It states the area of any measurable set is $\geq 0$.
+Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
+<!--ID: 1734013716434-->
+END%%
+
 ### Invariance Under Congruence
 
 If $S \in \mathscr{M}$ and $T$ is congruent to $S$, then $T \in \mathscr{M}$ and $a(S) = a(T)$.

notes/ontology/index.md

@@ -1,3 +1,124 @@
 ---
 title: Ontology
+TARGET DECK: Obsidian::H&SS
+FILE TAGS: ontology
+tags:
+  - ontology
 ---
+
+## Overview
+
+Ontology is the philosophical study of being. Generally *things* are split into two broad categories: **abstract** and **concrete** things. These words are "terms of art" and their definition is not standardized in any way.
+
+%%ANKI
+Basic
+What did Quine declare as *the* ontological question?
+Back: "What is there?"
+Reference: Simon Hewitt, “A Cardinal Worry for Permissive Metaontology,” _Metaphysica_ 16, no. 2 (September 18, 2015): 159–65, [https://doi.org/10.1515/mp-2015-0009](https://doi.org/10.1515/mp-2015-0009).
+<!--ID: 1720912238054-->
+END%%
+
+%%ANKI
+Basic
+Who is attributed *the* ontological question?
+Back: Willard Van Orman Quine.
+Reference: Simon Hewitt, “A Cardinal Worry for Permissive Metaontology,” _Metaphysica_ 16, no. 2 (September 18, 2015): 159–65, [https://doi.org/10.1515/mp-2015-0009](https://doi.org/10.1515/mp-2015-0009).
+<!--ID: 1720912259767-->
+END%%
+
+%%ANKI
+Cloze
+{Ontology} is the {philosophical study of being}.
+Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013).
+<!--ID: 1720912238058-->
+END%%
+
+%%ANKI
+Cloze
+{Epistemology} is the {philosophical study of knowledge}.
+Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013).
+<!--ID: 1720912238062-->
+END%%
+
+%%ANKI
+Cloze
+{Taxonomy} is the {branch of science concerned with categorization}.
+Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013).
+<!--ID: 1720912238066-->
+END%%
+
+%%ANKI
+Cloze
+{Mereology} is the {philosophical study of part-whole relationships}.
+Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013).
+<!--ID: 1720998380912-->
+END%%
+
+%%ANKI
+Basic
+What does Effingham mean when saying "concreta" and "abstracta" are terms of art?
+Back: They are terms defined freely by a person to mean whatever one wants.
+Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013).
+<!--ID: 1720894782942-->
+END%%
+
+%%ANKI
+Cloze
+In general, ontologists often categorize things as either {concreta} or {abstracta}.
+Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013).
+<!--ID: 1720894782951-->
+END%%
+
+%%ANKI
+Basic
+Generally speaking, what does someone *probably* mean by "concrete" things?
+Back: Things that exists in space and/or time.
+Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013).
+<!--ID: 1720894782957-->
+END%%
+
+%%ANKI
+Basic
+Generally speaking, what does someone *probably* mean by "abstract" things?
+Back: Things that exist in neither space nor time.
+Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013).
+<!--ID: 1720894782965-->
+END%%
+
+%%ANKI
+Basic
+Is a material object considered concreta?
+Back: Usually.
+Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013).
+<!--ID: 1720894782971-->
+END%%
+
+%%ANKI
+Basic
+Is an immaterial object considered concreta?
+Back: Possibly.
+Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013).
+<!--ID: 1720894782978-->
+END%%
+
+%%ANKI
+Basic
+Is a material object considered abstracta?
+Back: Not usually.
+Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013).
+<!--ID: 1720894782984-->
+END%%
+
+%%ANKI
+Basic
+Is an immaterial object considered abstracta?
+Back: Possibly.
+Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013).
+<!--ID: 1720894782989-->
+END%%
+
+## Bibliography
+
+* Francesco Orilia and Michele Paolini Paoletti, “Properties,” in _The Stanford Encyclopedia of Philosophy_, ed. Edward N. Zalta, Spring 2022 (Metaphysics Research Lab, Stanford University, 2022), [https://plato.stanford.edu/archives/spr2022/entries/properties/](https://plato.stanford.edu/archives/spr2022/entries/properties/).
+* Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013).
+* Simon Hewitt, “A Cardinal Worry for Permissive Metaontology,” _Metaphysica_ 16, no. 2 (September 18, 2015): 159–65, [https://doi.org/10.1515/mp-2015-0009](https://doi.org/10.1515/mp-2015-0009).

notes/ontology/philosophy/index.md

@@ -1,124 +0,0 @@
----
-title: Ontology
-TARGET DECK: Obsidian::H&SS
-FILE TAGS: ontology::philosophy
-tags:
-  - ontology
----
-
-## Overview
-
-Ontology is the philosophical study of being. Generally *things* are split into two broad categories: **abstract** and **concrete** things. These words are "terms of art" and their definition is not standardized in any way.
-
-%%ANKI
-Basic
-What did Quine declare as *the* ontological question?
-Back: "What is there?"
-Reference: Simon Hewitt, “A Cardinal Worry for Permissive Metaontology,” _Metaphysica_ 16, no. 2 (September 18, 2015): 159–65, [https://doi.org/10.1515/mp-2015-0009](https://doi.org/10.1515/mp-2015-0009).
-<!--ID: 1720912238054-->
-END%%
-
-%%ANKI
-Basic
-Who is attributed *the* ontological question?
-Back: Willard Van Orman Quine.
-Reference: Simon Hewitt, “A Cardinal Worry for Permissive Metaontology,” _Metaphysica_ 16, no. 2 (September 18, 2015): 159–65, [https://doi.org/10.1515/mp-2015-0009](https://doi.org/10.1515/mp-2015-0009).
-<!--ID: 1720912259767-->
-END%%
-
-%%ANKI
-Cloze
-{Ontology} is the {philosophical study of being}.
-Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013).
-<!--ID: 1720912238058-->
-END%%
-
-%%ANKI
-Cloze
-{Epistemology} is the {philosophical study of knowledge}.
-Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013).
-<!--ID: 1720912238062-->
-END%%
-
-%%ANKI
-Cloze
-{Taxonomy} is the {branch of science concerned with categorization}.
-Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013).
-<!--ID: 1720912238066-->
-END%%
-
-%%ANKI
-Cloze
-{Mereology} is the {philosophical study of part-whole relationships}.
-Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013).
-<!--ID: 1720998380912-->
-END%%
-
-%%ANKI
-Basic
-What does Effingham mean when saying "concreta" and "abstracta" are terms of art?
-Back: They are terms defined freely by a person to mean whatever one wants.
-Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013).
-<!--ID: 1720894782942-->
-END%%
-
-%%ANKI
-Cloze
-In general, ontologists often categorize things as either {concreta} or {abstracta}.
-Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013).
-<!--ID: 1720894782951-->
-END%%
-
-%%ANKI
-Basic
-Generally speaking, what does someone *probably* mean by "concrete" things?
-Back: Things that exists in space and/or time.
-Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013).
-<!--ID: 1720894782957-->
-END%%
-
-%%ANKI
-Basic
-Generally speaking, what does someone *probably* mean by "abstract" things?
-Back: Things that exist in neither space nor time.
-Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013).
-<!--ID: 1720894782965-->
-END%%
-
-%%ANKI
-Basic
-Is a material object considered concreta?
-Back: Usually.
-Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013).
-<!--ID: 1720894782971-->
-END%%
-
-%%ANKI
-Basic
-Is an immaterial object considered concreta?
-Back: Possibly.
-Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013).
-<!--ID: 1720894782978-->
-END%%
-
-%%ANKI
-Basic
-Is a material object considered abstracta?
-Back: Not usually.
-Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013).
-<!--ID: 1720894782984-->
-END%%
-
-%%ANKI
-Basic
-Is an immaterial object considered abstracta?
-Back: Possibly.
-Reference: Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013).
-<!--ID: 1720894782989-->
-END%%
-
-## Bibliography
-
-* Francesco Orilia and Michele Paolini Paoletti, “Properties,” in _The Stanford Encyclopedia of Philosophy_, ed. Edward N. Zalta, Spring 2022 (Metaphysics Research Lab, Stanford University, 2022), [https://plato.stanford.edu/archives/spr2022/entries/properties/](https://plato.stanford.edu/archives/spr2022/entries/properties/).
-* Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013).
-* Simon Hewitt, “A Cardinal Worry for Permissive Metaontology,” _Metaphysica_ 16, no. 2 (September 18, 2015): 159–65, [https://doi.org/10.1515/mp-2015-0009](https://doi.org/10.1515/mp-2015-0009).

notes/ontology/properties.md

@@ -1,7 +1,7 @@
 ---
 title: Properties
 TARGET DECK: Obsidian::H&SS
-FILE TAGS: ontology::philosophy
+FILE TAGS: ontology
 tags:
   - ontology
 ---