Difference between revisions of "User:DavidWhitten/mso"

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= Minimum Standard Ontology =
 
= Minimum Standard Ontology =
 +
==Basic logic symbols==
 +
{| class="wikitable"
 +
|- bgcolor=#a0e0a0
 +
! scope="col" |Symbol</div>
 +
!Name
 +
!Read as
 +
!Category
 +
! scope="col" |Explanation
 +
! scope="col" |Examples
 +
! scope="col" |Unicode<br />value<br />(hexadecimal)
 +
! scope="col" |HTML<br />value<br />(decimal)
 +
! scope="col" |HTML<br />entity<br />(named)
 +
! scope="col" |[[LaTeX]]<br />symbol
 +
|-
 +
| bgcolor="#d0f0d0" align="center" |<div style="font-size:200%;">⇒<br />→<br />⊃</div>
 +
||[[material conditional|material implication]]
 +
|implies; if ... then
 +
|[[propositional logic]], [[Heyting algebra]]
 +
|<math>A \Rightarrow B</math> is false when <math>A</math> is true and <math>B</math> is false but true otherwise.<ref>{{Cite web | url=https://en.wikipedia.org/wiki/Material_conditional |title = Material conditional}}</ref>{{Circular reference|date=May 2020}}<br /><br /><math>\rightarrow</math> may mean the same as <math>\Rightarrow</math> (the symbol may also indicate the domain and codomain of a [[function (mathematics)|function]]; see [[table of mathematical symbols]]).<br /><br /><math>\supset</math> may mean the same as <math>\Rightarrow</math> (the symbol may also mean [[superset]]).
 +
|<math>x = 2 \Rightarrow x^2 = 4</math> is true, but <math>x^2 = 4 \Rightarrow x = 2</math> is in general false (since <math>x</math> could be −2).
 +
| style="text-align:left;font-family:monospace" |U+21D2<br /><br />U+2192<br /><br />U+2283
 +
| style="text-align:left;font-family:monospace" |&amp;#8658;<br /><br />&amp;#8594;<br /><br />&amp;#8835;
 +
| style="text-align:left;font-family:monospace" |&amp;rArr;<br /><br />&amp;rarr;<br /><br />&amp;sup;
 +
| style="text-align:left;font-family:monospace" |<div><math>\Rightarrow</math>\Rightarrow<br /> <math>\to</math>\to or \rightarrow<br /><math>\supset</math>\supset<br /><math>\implies</math>\implies</div>
 +
|
 +
|-
 +
| bgcolor="#d0f0d0" align="center" |<div style="font-size:200%;">⇔<br />≡<br />↔</div>
 +
||[[material equivalence]]
 +
|if and only if; iff; means the same as
 +
|[[propositional logic]]
 +
|<math>A \Leftrightarrow B</math> is true only if both <math>A</math> and <math>B</math> are false, or both <math>A</math> and <math>B</math> are true.
 +
|<math>x + 5 = y + 2 \Leftrightarrow x + 3 = y</math>
 +
| style="text-align:left;font-family:monospace" |U+21D4<br /><br />U+2261<br /><br />U+2194
 +
| style="text-align:left;font-family:monospace" |&amp;#8660;<br /><br />&amp;#8801;<br /><br />&amp;#8596;
 +
| style="text-align:left;font-family:monospace" |&amp;hArr;<br /><br />&amp;equiv;<br /><br />&amp;harr;
 +
| style="text-align:left;font-family:monospace" |<math>\Leftrightarrow</math>\Leftrightarrow<br /><math>\equiv</math>\equiv<br /><math>\leftrightarrow</math>\leftrightarrow<br /><math>\iff</math>\iff
 +
|-
 +
| bgcolor="#d0f0d0" align="center" |<div style="font-size:200%;">¬<br />˜<br />!</div>
 +
||[[negation]]
 +
|not
 +
|[[propositional logic]]
 +
|The statement <math>\lnot A</math> is true if and only if <math>A</math> is false.<br /><br />A slash placed through another operator is the same as <math>\neg</math> placed in front.
 +
|<math>\neg (\neg A) \Leftrightarrow A</math><br /> <math>x \neq y \Leftrightarrow \neg (x = y)</math>
 +
| style="text-align:left;font-family:monospace" |U+00AC<br /><br />U+02DC<br /><br />U+0021
 +
| style="text-align:left;font-family:monospace" |&amp;#172;<br /><br />&amp;#732;<br /><br />&amp;#33;
 +
| style="text-align:left;font-family:monospace" |&amp;not;<br /><br />&amp;tilde;<br /><br />&amp;excl;
 +
| style="text-align:left;font-family:monospace" |<div><math>\neg</math>\lnot or \neg
 +
 +
<br /><math>\sim</math>\sim
 +
 +
 +
 +
</div>
 +
|-
 +
| bgcolor="#d0f0d0" align="center" |<div style="font-size:200%;">𝔻
 +
||[[Domain of discourse]]
 +
|Domain of predicate
 +
|[[Predicate (mathematical logic)]]
 +
|
 +
| <math>\mathbb D\mathbb :\mathbb R</math>
 +
| style="text-align:left;font-family:monospace" |U+1D53B
 +
| style="text-align:left;font-family:monospace" |&amp;#120123;
 +
| style="text-align:left;font-family:monospace" |&amp;Dopf;
 +
| style="text-align:left;font-family:monospace" |\mathbb{D}
 +
|-
 +
| bgcolor="#d0f0d0" align="center" |<div style="font-size:200%;">∧ <br/>·<br/>&</div>
 +
||[[logical conjunction]]
 +
|and
 +
|[[propositional logic]], [[Boolean algebra (logic)|Boolean algebra]]
 +
|The statement ''A'' ∧ ''B'' is true if ''A'' and ''B'' are both true; otherwise, it is false.
 +
|''n''&nbsp;< 4&nbsp;&nbsp;∧&nbsp; ''n''&nbsp;>2&nbsp;&nbsp;⇔&nbsp; ''n''&nbsp;= 3 when ''n'' is a [[natural number]].
 +
| style="text-align:left;font-family:monospace" |U+2227<br /><br />U+00B7<br /><br />U+0026
 +
| style="text-align:left;font-family:monospace" |&amp;#8743;<br /><br />&amp;#183;<br /><br />&amp;#38;<br />
 +
| style="text-align:left;font-family:monospace" |&amp;and;<br /><br />&amp;middot;<br /><br />&amp;amp;
 +
| style="text-align:left;font-family:monospace" |<div><math>\wedge</math>\wedge or \land<br /><math>\cdot</math>\cdot
 +
<math>\&</math>\&<ref>Although this character is available in LaTeX, the [[MediaWiki]] TeX system does not support it.</ref></div>
 +
|-
 +
| bgcolor="#d0f0d0" align="center" |<div style="font-size:200%;">∨<br />+<br />∥</div>
 +
||[[logical disjunction|logical (inclusive) disjunction]]
 +
|or
 +
|[[propositional logic]], [[Boolean algebra (logic)|Boolean algebra]]
 +
|The statement ''A'' ∨ ''B'' is true if ''A'' or ''B'' (or both) are true; if both are false, the statement is false.
 +
|''n''&nbsp;≥ 4&nbsp;&nbsp;∨&nbsp; ''n''&nbsp;≤ 2&nbsp;&nbsp;⇔ ''n''&nbsp;≠ 3 when ''n'' is a [[natural number]].
 +
| style="text-align:left;font-family:monospace" |U+2228<br /><br />U+002B<br /><br />U+2225
 +
| style="text-align:left;font-family:monospace" |&amp;#8744;<br /><br />&amp;#43;<br /><br />&amp;#8741;
 +
| style="text-align:left;font-family:monospace" |&amp;or;
 +
<br />&amp;plus;
 +
 +
 +
&amp;parallel;
 +
| style="text-align:left;font-family:monospace" |<math>\lor</math>\lor or \vee
 +
 +
 +
 +
<br /><math>\parallel</math>\parallel
 +
|-
 +
| bgcolor="#d0f0d0" align="center" |<br /><div style="font-size:200%;">⊕<br />⊻<br />≢</div>||[[exclusive or|exclusive disjunction]]
 +
|xor; either ... or
 +
|[[propositional logic]], [[Boolean algebra (logic)|Boolean algebra]]
 +
| The statement ''A'' ⊕ ''B'' is true when either A or B, but not both, are true. ''A'' ⊻ ''B'' means the same.
 +
| (¬''A'')  ⊕ ''A'' is always true, and ''A'' ⊕ ''A'' always false, if [[vacuous truth]] is excluded.
 +
| style="text-align:left;font-family:monospace" |U+2295<br /><br />U+22BB
 +
 +
 +
[[Mathematical Operators|U+]]2262
 +
| style="text-align:left;font-family:monospace" |&amp;#8853;<br /><br />&amp;#8891;
 +
 +
 +
&amp;#8802;
 +
| style="text-align:left;font-family:monospace" |&amp;oplus;
 +
<br />&amp;veebar;<br /><br />&amp;nequiv;
 +
| style="text-align:left;font-family:monospace" |<math>\oplus</math>\oplus
 +
 +
 +
<math>\veebar</math>\veebar
 +
 +
 +
<math>\not\equiv</math>\not\equiv
 +
|-
 +
| bgcolor="#d0f0d0" align="center" |<br /><div style="font-size:200%;">⊤<br />T<br />1</div>||[[Tautology (logic)|Tautology]]
 +
|top, truth
 +
|[[propositional logic]], [[Boolean algebra (logic)|Boolean algebra]]
 +
| The statement ''⊤'' is unconditionally true.
 +
|''A'' ⇒ ⊤ is always true.
 +
| style="text-align:left;font-family:monospace" |U+22A4<br /><br /><br /><br />
 +
| style="text-align:left;font-family:monospace" |&amp;#8868;<br /><br /><br />
 +
| style="text-align:left;font-family:monospace" |&amp;top;
 +
 +
 +
| style="text-align:left;font-family:monospace" |<math>\top</math>\top
 +
|-
 +
|-
 +
| bgcolor="#d0f0d0" align="center" |<br/><div style="font-size:200%;">⊥<br/>F<br/>0</div> ||[[Contradiction]]
 +
|bottom, falsum, falsity
 +
|[[propositional logic]], [[Boolean algebra (logic)|Boolean algebra]]
 +
| The statement ⊥ is unconditionally false. (The symbol ⊥ may also refer to [[perpendicular]] lines.)
 +
| ⊥ ⇒ ''A'' is always true.
 +
| style="text-align:left;font-family:monospace" |U+22A5<br /><br /><br /><br />
 +
| style="text-align:left;font-family:monospace" |&amp;#8869;<br /><br /><br /><br />
 +
| style="text-align:left;font-family:monospace" |&amp;perp;<br /><br /><br /><br />
 +
| style="text-align:left;font-family:monospace" |<math>\bot</math>\bot
 +
|-
 +
| bgcolor="#d0f0d0" align="center" |<div style="font-size:200%;">∀<br />()</div>
 +
||[[universal quantification]]
 +
|for all; for any; for each
 +
|[[first-order logic]]
 +
|∀&nbsp;''x'':&nbsp;''P''(''x'') or (''x'')&nbsp;''P''(''x'') means ''P''(''x'') is true for all ''x''.
 +
|∀&nbsp;''n''&nbsp;∈ ℕ: ''n''<sup>2</sup>&nbsp;≥ ''n''.
 +
| style="text-align:left;font-family:monospace" |U+2200<br /><br />
 +
| style="text-align:left;font-family:monospace" |&amp;#8704;<br /><br />
 +
| style="text-align:left;font-family:monospace" |&amp;forall;<br /><br />
 +
| style="text-align:left;font-family:monospace" |<math>\forall</math>\forall
 +
|-
 +
| bgcolor="#d0f0d0" align="center" |<div style="font-size:200%;">∃</div>
 +
||[[existential quantification]]
 +
|there exists
 +
|[[first-order logic]]
 +
|∃&nbsp;''x'': ''P''(''x'') means there is at least one ''x'' such that ''P''(''x'') is true.
 +
|∃&nbsp;''n''&nbsp;∈ ℕ: ''n'' is even.
 +
| style="text-align:left;font-family:monospace" |U+2203
 +
| style="text-align:left;font-family:monospace" |&amp;#8707;
 +
| style="text-align:left;font-family:monospace" |&amp;exist;
 +
| style="text-align:left;font-family:monospace" |<math>\exists</math>\exists
 +
|-
 +
| bgcolor="#d0f0d0" align="center" |<div style="font-size:200%;">∃!</div>
 +
||[[uniqueness quantification]]
 +
|there exists exactly one
 +
|[[first-order logic]]
 +
|∃!&nbsp;''x'': ''P''(''x'') means there is exactly one ''x'' such that ''P''(''x'') is true.
 +
|∃!&nbsp;''n''&nbsp;∈ ℕ: ''n''&nbsp;+ 5&nbsp;= 2''n''.
 +
| style="text-align:left;font-family:monospace" |U+2203&nbsp;U+0021
 +
| style="text-align:left;font-family:monospace" |&amp;#8707; &amp;#33;
 +
| style="text-align:left;font-family:monospace" |&amp;exist;!
 +
| style="text-align:left;font-family:monospace" |<math>\exists !</math>\exists !
 +
|-
 +
|-
 +
| bgcolor="#d0f0d0" align="center" |<div style="font-size:200%;">≔<br/>≡<br/>:⇔</div>
 +
||[[definition]]
 +
|is defined as
 +
|everywhere
 +
|''x''&nbsp;≔ ''y'' or ''x''&nbsp;≡ ''y'' means ''x'' is defined to be another name for ''y'' (but note that ≡ can also mean other things, such as [[congruence relation|congruence]]).<br /><br />''P''&nbsp;:⇔ ''Q'' means ''P'' is defined to be [[Logical equivalence|logically equivalent]] to ''Q''.
 +
|<math>\cosh x := \frac {e^x + e^{-x}} {2}</math><br /><br />''A''&nbsp;XOR&nbsp;''B'' :⇔ (''A''&nbsp;∨&nbsp;''B'')&nbsp;∧&nbsp;¬(''A''&nbsp;∧&nbsp;''B'')
 +
| style="text-align:left;font-family:monospace" |U+2254 (U+003A&nbsp;U+003D)<br /><br />U+2261<br /><br />U+003A&nbsp;U+229C
 +
| style="text-align:left;font-family:monospace" |&amp;#8788; (&amp;#58; &amp;#61;)
 +
<br />&amp;#8801;<br /><br />&amp;#8860;
 +
| style="text-align:left;font-family:monospace" |&amp;coloneq;
 +
<br />&amp;equiv;<br /><br />&amp;hArr;
 +
| style="text-align:left;font-family:monospace" |<div><math>:=</math>:=
 +
 +
 +
<math>\equiv</math>\equiv<br />
 +
 +
<math>:\Leftrightarrow</math>:\Leftrightarrow
 +
</div>
 +
|-
 +
| bgcolor="#d0f0d0" align="center" |<div style="font-size:200%;">( )</div>
 +
|[[precedence grouping]]
 +
|parentheses; brackets
 +
|everywhere
 +
| Perform the operations inside the parentheses first.
 +
|(8 ÷ 4) ÷ 2&nbsp;= 2 ÷ 2&nbsp;= 1, but 8 ÷ (4 ÷ 2)&nbsp;= 8 ÷ 2&nbsp;= 4.
 +
| style="text-align:left;font-family:monospace" | U+0028&nbsp;U+0029
 +
| style="text-align:left;font-family:monospace" |&amp;#40; &amp;#41;
 +
| style="text-align:left;font-family:monospace" |&amp;lpar;
 +
 +
&amp;rpar;
 +
| style="text-align:left;font-family:monospace" |<math>(~)</math> ( )
 +
|-
 +
| bgcolor="#d0f0d0" align="center" |<div style="font-size:200%;">⊢</div>
 +
||[[Turnstile (symbol)|turnstile]]
 +
|[[Logical consequence|proves]]
 +
|[[propositional logic]], [[first-order logic]]
 +
|''x'' ⊢ ''y'' means ''x'' proves (syntactically entails) ''y''
 +
| (''A'' → ''B'') ⊢ (¬''B'' → ¬''A'')
 +
| style="text-align:left;font-family:monospace" |U+22A2
 +
| style="text-align:left;font-family:monospace" |&amp;#8866;
 +
| style="text-align:left;font-family:monospace" |&amp;vdash;
 +
| style="text-align:left;font-family:monospace" |<math>\vdash</math>\vdash
 +
|-
 +
| bgcolor="#d0f0d0" align="center" | <div style="font-size:200%;">⊨</div>
 +
||[[double turnstile]]
 +
|[[Logical consequence|models]]
 +
|[[propositional logic]], [[first-order logic]]
 +
|''x'' ⊨ ''y'' means ''x'' models (semantically entails) ''y''
 +
| (''A'' → ''B'') ⊨ (¬''B'' → ¬''A'')
 +
| style="text-align:left;font-family:monospace" |U+22A8
 +
| style="text-align:left;font-family:monospace" |&amp;#8872;
 +
| style="text-align:left;font-family:monospace" |&amp;vDash;
 +
| style="text-align:left;font-family:monospace" |<math>\vDash</math>\vDash, \models
 +
|-
 +
|}
  
 
   
 
   
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| connective  
 
| connective  
 
|+  
 
|+  
−
| 16 LOGICAL OPERATORS<br>
+
| 16 LOGICAL OPERATORS<br>Class/Category/Set
 
|+  
 
|+  
 
|  
 
|  
 
|}
 
|}
 +
 +
forAll, thereExists, lambda, apply, mu,
 +
 +
aka, ako, isa,

Latest revision as of 15:46, 1 February 2021

Minimum Standard Ontology

Basic logic symbols

Symbol</div> Name Read as Category Explanation Examples Unicode
value
(hexadecimal) 		HTML
value
(decimal) 		HTML
entity
(named) 		LaTeX
symbol
⇒
→
⊃
 material implication implies; if ... then propositional logic, Heyting algebra <math>A \Rightarrow B</math> is false when <math>A</math> is true and <math>B</math> is false but true otherwise.<ref>Template:Cite web</ref>Template:Circular reference

<math>\rightarrow</math> may mean the same as <math>\Rightarrow</math> (the symbol may also indicate the domain and codomain of a function; see table of mathematical symbols).

<math>\supset</math> may mean the same as <math>\Rightarrow</math> (the symbol may also mean superset). 		<math>x = 2 \Rightarrow x^2 = 4</math> is true, but <math>x^2 = 4 \Rightarrow x = 2</math> is in general false (since <math>x</math> could be −2). U+21D2

U+2192

U+2283 		&#8658;

&#8594;

&#8835; 		&rArr;

&rarr;

&sup;
<math>\Rightarrow</math>\Rightarrow
<math>\to</math>\to or \rightarrow
<math>\supset</math>\supset
<math>\implies</math>\implies
⇔
≡
↔
 material equivalence if and only if; iff; means the same as propositional logic <math>A \Leftrightarrow B</math> is true only if both <math>A</math> and <math>B</math> are false, or both <math>A</math> and <math>B</math> are true. <math>x + 5 = y + 2 \Leftrightarrow x + 3 = y</math> U+21D4

U+2261

U+2194 		&#8660;

&#8801;

&#8596; 		&hArr;

&equiv;

&harr; 		<math>\Leftrightarrow</math>\Leftrightarrow
<math>\equiv</math>\equiv
<math>\leftrightarrow</math>\leftrightarrow
<math>\iff</math>\iff
¬
˜
!
 negation not propositional logic The statement <math>\lnot A</math> is true if and only if <math>A</math> is false.

A slash placed through another operator is the same as <math>\neg</math> placed in front. 		<math>\neg (\neg A) \Leftrightarrow A</math>
<math>x \neq y \Leftrightarrow \neg (x = y)</math> 		U+00AC

U+02DC

U+0021 		&#172;

&#732;

&#33; 		&not;

&tilde;

&excl;
<math>\neg</math>\lnot or \neg


<math>\sim</math>\sim


𝔻
Domain of discourse Domain of predicate Predicate (mathematical logic) <math>\mathbb D\mathbb :\mathbb R</math> U+1D53B &#120123; &Dopf; \mathbb{D}
∧
·
&
 logical conjunction and propositional logic, Boolean algebra The statement A ∧ B is true if A and B are both true; otherwise, it is false. n < 4  ∧  n >2  ⇔  n = 3 when n is a natural number. U+2227

U+00B7

U+0026 		&#8743;

&#183;

&#38;
 &and;

&middot;

&amp;
<math>\wedge</math>\wedge or \land
<math>\cdot</math>\cdot <math>\&</math>\&<ref>Although this character is available in LaTeX, the MediaWiki TeX system does not support it.</ref>
∨
+
∥
 logical (inclusive) disjunction or propositional logic, Boolean algebra The statement A ∨ B is true if A or B (or both) are true; if both are false, the statement is false. n ≥ 4  ∨  n ≤ 2  ⇔ n ≠ 3 when n is a natural number. U+2228

U+002B

U+2225 		&#8744;

&#43;

&#8741; 		&or;


&plus;


&parallel;

<math>\lor</math>\lor or \vee



<math>\parallel</math>\parallel


⊕
⊻
≢
 exclusive disjunction xor; either ... or propositional logic, Boolean algebra The statement A ⊕ B is true when either A or B, but not both, are true. A ⊻ B means the same. (¬A) ⊕ A is always true, and A ⊕ A always false, if vacuous truth is excluded. U+2295

U+22BB


U+2262

&#8853;

&#8891;


&#8802;

&oplus;


&veebar;

&nequiv;

<math>\oplus</math>\oplus


<math>\veebar</math>\veebar


<math>\not\equiv</math>\not\equiv


⊤
T
1
 Tautology top, truth propositional logic, Boolean algebra The statement ⊤ is unconditionally true. A ⇒ ⊤ is always true. U+22A4



 &#8868;


 &top;


<math>\top</math>\top

⊥
F
0
 Contradiction bottom, falsum, falsity propositional logic, Boolean algebra The statement ⊥ is unconditionally false. (The symbol ⊥ may also refer to perpendicular lines.) ⊥ ⇒ A is always true. U+22A5



 &#8869;



 &perp;



 <math>\bot</math>\bot
∀
()
 universal quantification for all; for any; for each first-order logic ∀ x: P(x) or (x) P(x) means P(x) is true for all x. ∀ n ∈ ℕ: n2 ≥ n. U+2200

 &#8704;

 &forall;

 <math>\forall</math>\forall
∃
 existential quantification there exists first-order logic ∃ x: P(x) means there is at least one x such that P(x) is true. ∃ n ∈ ℕ: n is even. U+2203 &#8707; &exist; <math>\exists</math>\exists
∃!
 uniqueness quantification there exists exactly one first-order logic ∃! x: P(x) means there is exactly one x such that P(x) is true. ∃! n ∈ ℕ: n + 5 = 2n. U+2203 U+0021 &#8707; &#33; &exist;! <math>\exists !</math>\exists !
≔
≡
:⇔
 definition is defined as everywhere x ≔ y or x ≡ y means x is defined to be another name for y (but note that ≡ can also mean other things, such as congruence).

P :⇔ Q means P is defined to be logically equivalent to Q. 		<math>\cosh x := \frac {e^x + e^{-x}} {2}</math>

A XOR B :⇔ (A ∨ B) ∧ ¬(A ∧ B) 		U+2254 (U+003A U+003D)

U+2261

U+003A U+229C 		&#8788; (&#58; &#61;)


&#8801;

&#8860;

&coloneq;


&equiv;

&hArr;

<math>:=</math>:=


<math>\equiv</math>\equiv

<math>:\Leftrightarrow</math>:\Leftrightarrow

( )
 precedence grouping parentheses; brackets everywhere Perform the operations inside the parentheses first. (8 ÷ 4) ÷ 2 = 2 ÷ 2 = 1, but 8 ÷ (4 ÷ 2) = 8 ÷ 2 = 4. U+0028 U+0029 &#40; &#41; &lpar;

&rpar;

<math>(~)</math> ( )
⊢
 turnstile proves propositional logic, first-order logic x ⊢ y means x proves (syntactically entails) y (A → B) ⊢ (¬B → ¬A) U+22A2 &#8866; &vdash; <math>\vdash</math>\vdash
⊨
 double turnstile models propositional logic, first-order logic x ⊨ y means x models (semantically entails) y (A → B) ⊨ (¬B → ¬A) U+22A8 &#8872; &vDash; <math>\vDash</math>\vDash, \models


connective
16 LOGICAL OPERATORS
Class/Category/Set

forAll, thereExists, lambda, apply, mu,

aka, ako, isa,
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