Difference between revisions of "User:DavidWhitten/mso"
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| style="text-align:left;font-family:monospace" |&#8658;<br /><br />&#8594;<br /><br />&#8835; | | style="text-align:left;font-family:monospace" |&#8658;<br /><br />&#8594;<br /><br />&#8835; | ||
| style="text-align:left;font-family:monospace" |&rArr;<br /><br />&rarr;<br /><br />&sup; | | style="text-align:left;font-family:monospace" |&rArr;<br /><br />&rarr;<br /><br />&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> | + | | 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> | | bgcolor="#d0f0d0" align="center" |<div style="font-size:200%;">⇔<br />≡<br />↔</div> | ||
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 |
⇒ → ⊃ |
⇒ → ⊃ |
<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 |
⇔ ≡ ↔ |
⇔ ≡ ↔ |
<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 |
¬ ˜ ! |
¬ ˜ ! |
<math>\neg</math>\lnot or \neg
| |
𝔻
|
Domain of discourse | Domain of predicate | Predicate (mathematical logic) | <math>\mathbb D\mathbb :\mathbb R</math> | U+1D53B | 𝔻 | 𝔻 | \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 |
∧ · & |
∧ · & |
<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 |
∨ + ∥ |
∨
|
<math>\lor</math>\lor or \vee
| |
⊕ ⊻ ≢ |
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
|
⊕ ⊻
|
⊕
|
<math>\oplus</math>\oplus
| |
⊤ T 1 |
Tautology | top, truth | propositional logic, Boolean algebra | The statement ⊤ is unconditionally true. | A ⇒ ⊤ is always true. | U+22A4 |
⊤ |
⊤
|
<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 |
⊥ |
⊥ |
<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 |
∀ |
∀ |
<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 | ∃ | ∃ | <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 | ∃ ! | ∃! | <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 |
≔ (: =)
|
≔
|
<math>:=</math>:=
<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 | ( ) | (
) |
<math>(~)</math> ( ) | |
⊢
|
turnstile | proves | propositional logic, first-order logic | x ⊢ y means x proves (syntactically entails) y | (A → B) ⊢ (¬B → ¬A) | U+22A2 | ⊢ | ⊢ | <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 | ⊨ | ⊨ | <math>\vDash</math>\vDash, \models |
| connective | 16 LOGICAL OPERATORS Class/Category/Set |
forAll, thereExists, lambda, apply, mu,
aka, ako, isa,
