List of logic symbols: Difference between revisions
m Add some to cause some "multiple words" to stay together |
m A few more   |
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|implies; if ... then |
|implies; if ... then |
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|[[propositional logic|propositional logic]], [[Heyting algebra | Heyting algebra]] |
|[[propositional logic|propositional logic]], [[Heyting algebra | Heyting algebra]] |
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|<math>A \Rightarrow B</math> is |
|<math>A \Rightarrow B</math> is false when {{mvar|A}} is true and {{mvar|B}} is false but true otherwise.<br /><br /><math>\rightarrow</math> may mean the same as <math>\Rightarrow</math><br>(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]]). |
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|<math>x = 2 \Rightarrow x^2 = 4</math> is true, but <math>x^2 = 4 \Rightarrow x = 2</math> is in general |
|<math>x = 2 \Rightarrow x^2 = 4</math> is true, but <math>x^2 = 4 \Rightarrow x = 2</math> is in general false<br> (since {{mvar|x}} could be −2). |
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|- |
|- |
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! scope"row" align="center" |<div style="font-size:200%;">⇔<br />≡<br />↔</div> |
! scope"row" align="center" |<div style="font-size:200%;">⇔<br />≡<br />↔</div> |
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|if and only if; iff; means the same as |
|if and only if; iff; means the same as |
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|[[propositional logic|propositional logic]] |
|[[propositional logic|propositional logic]] |
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|<math>A \Leftrightarrow B</math> is true only if both |
|<math>A \Leftrightarrow B</math> is true only if both {{mvar|A}} and {{mvar|B}} are false, or both {{mvar|A}} and {{mvar|B}} are true. |
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|<math>x + 5 = y + 2 \Leftrightarrow x + 3 = y</math> |
|<math>x + 5 = y + 2 \Leftrightarrow x + 3 = y</math> |
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|not |
|not |
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|[[propositional logic|propositional logic]] |
|[[propositional logic|propositional logic]] |
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|The statement <math>\lnot A</math> is true if |
|The statement <math>\lnot A</math> is true if and only if {{mvar|A}} is false.<br /><br />A slash placed through another operator is the same as <math>\neg</math> placed in front. |
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|<math>\neg (\neg A) \Leftrightarrow A</math><br /> <math>x \neq y \Leftrightarrow \neg (x = y)</math> |
|<math>\neg (\neg A) \Leftrightarrow A</math><br /> <math>x \neq y \Leftrightarrow \neg (x = y)</math> |
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|and |
|and |
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|[[propositional logic]], [[Boolean algebra (logic)|Boolean algebra]] |
|[[propositional logic]], [[Boolean algebra (logic)|Boolean algebra]] |
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|The statement ''A'' |
|The statement ''A'' ∧ ''B'' is true if ''A'' and ''B'' are both true; otherwise, it is false. |
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|{{math|''n'' < 4 ∧ }} {{math|''n'' >2 ⇔ }} {{math|1=''n'' = 3}} when ''n'' is a [[natural number]]. |
|{{math|''n'' < 4 ∧ }} {{math|''n'' >2 ⇔ }} {{math|1=''n'' = 3}} when ''n'' is a [[natural number]]. |
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|- |
|- |
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|or |
|or |
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|[[propositional logic]], [[Boolean algebra (logic)|Boolean algebra]] |
|[[propositional logic]], [[Boolean algebra (logic)|Boolean algebra]] |
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|The statement ''A'' |
|The statement ''A'' ∨ ''B'' is true if ''A'' or ''B'' (or both) are true; if both are false, the statement is false. |
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|{{math|''n'' ≥ 4 ∨ ''n'' ≤ 2 ⇔ ''n'' ≠ 3}} when ''n'' is a [[natural number]]. |
|{{math|''n'' ≥ 4 ∨ ''n'' ≤ 2 ⇔ ''n'' ≠ 3}} when ''n'' is a [[natural number]]. |
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|- |
|- |
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|xor; either ... or |
|xor; either ... or |
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|[[propositional logic]], [[Boolean algebra (logic)|Boolean algebra]] |
|[[propositional logic]], [[Boolean algebra (logic)|Boolean algebra]] |
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| The statement ''A'' |
| The statement ''A'' ↮ ''B'' is true when either A or B, but not both, are true.<br> ''A'' ⊻ ''B'' means the same. |
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| (¬''A'') ↮ ''A'' is always true, and ''A'' ↮ ''A'' always false, if [[vacuous truth]] is excluded. |
| (¬''A'') ↮ ''A'' is always true, and ''A'' ↮ ''A'' always false, if [[vacuous truth]] is excluded. |
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|- |
|- |
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|top, truth, full clause |
|top, truth, full clause |
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|[[propositional logic]], [[Boolean algebra (logic)|Boolean algebra]], [[first-order logic]] |
|[[propositional logic]], [[Boolean algebra (logic)|Boolean algebra]], [[first-order logic]] |
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| |
| {{math|⊤}} is unconditionally true. |
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|⊤(''A'') ⇒ ''A'' is always true. |
|⊤(''A'') ⇒ ''A'' is always true. |
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|- |
|- |
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|bottom, falsum, falsity, empty clause |
|bottom, falsum, falsity, empty clause |
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|[[propositional logic]], [[Boolean algebra (logic)|Boolean algebra]], [[first-order logic]] |
|[[propositional logic]], [[Boolean algebra (logic)|Boolean algebra]], [[first-order logic]] |
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| |
| ⊥ is unconditionally false. <br>(The symbol ⊥ may also refer to [[perpendicular]] lines.) |
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| ⊥(''A'') ⇒ ''A'' is always false. |
| ⊥(''A'') ⇒ ''A'' is always false. |
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|- |
|- |
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|for all; for any; for each |
|for all; for any; for each |
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|[[first-order logic]] |
|[[first-order logic]] |
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|{{math|∀ ''x'': ''P''(''x'')}} or {{math|(''x'') ''P''(''x'')}} means ''P''(''x'') is true for all |
|{{math|∀ ''x'': ''P''(''x'')}} or {{math|(''x'') ''P''(''x'')}} means ''P''(''x'') is true for '''all''' ''x''. |
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|<math>\forall n \isin \mathbb{N}: n^2 \geq n.</math> |
|<math>\forall n \isin \mathbb{N}: n^2 \geq n.</math> |
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|- |
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|there exists |
|there exists |
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|[[first-order logic]] |
|[[first-order logic]] |
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|{{math|∃ ''x'': ''P''(''x'')}} means there |
|{{math|∃ ''x'': ''P''(''x'')}} means there exists at '''least one''' ''x'' such that ''P''(''x'') is true. |
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|<math>\exists n \isin \mathbb{N}:</math> ''n'' is even. |
|<math>\exists n \isin \mathbb{N}:</math> ''n'' is even. |
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| style="text-align:left;font-family:monospace" |<math>\exists !</math>\exists ! |
| style="text-align:left;font-family:monospace" |<math>\exists !</math>\exists ! |
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||[[uniqueness quantification]] |
||[[uniqueness quantification]] |
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|there exists exactly |
|there exists '''exactly one''' |
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|[[first-order logic]] |
|[[first-order logic]] |
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|{{math|∃! ''x'': ''P''(''x'')}} means there |
|{{math|∃! ''x'': ''P''(''x'')}} means there exists '''exactly one''' ''x'' such that ''P''(''x'') is true. |
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|<math>\exists! n \isin \mathbb{N}: n+5=2n.</math> |
|<math>\exists! n \isin \mathbb{N}: n+5=2n.</math> |
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|- |
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|is defined as |
|is defined as |
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|everywhere |
|everywhere |
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|{{math|1=''x'' ≔ ''y''}} or {{math|''x'' ≡ ''y''}} means ''x'' is defined to be another |
|{{math|1=''x'' ≔ ''y''}} or {{math|''x'' ≡ ''y''}} means ''x'' is defined to be another name for ''y'' <br>( The symbol ≡ can also mean other things, such as [[congruence relation|congruence]]).<br /><br />{{math|''P'' :⇔ ''Q''}} means ''P'' is defined to be [[Logical equivalence|logically equivalent]] to ''Q''. |
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|<math>\cosh x := \frac {e^x + e^{-x}} {2}</math><br /><br />{{math|''A'' ⊕ ''B'' :⇔ (''A'' ∨ ''B'') ∧ ¬(''A'' ∧ ''B'')}} |
|<math>\cosh x := \frac {e^x + e^{-x}} {2}</math><br /><br />{{math|''A'' ⊕ ''B'' :⇔ (''A'' ∨ ''B'') ∧ ¬(''A'' ∧ ''B'')}} |
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|- |
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|parentheses; brackets |
|parentheses; brackets |
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|everywhere |
|everywhere |
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| Perform the operations inside the parentheses |
| Perform the operations inside the parentheses first. |
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|{{math|1=(8 ÷ 4) ÷ 2 = 2 ÷ 2 = 1}}, but {{math|1=8 ÷ (4 ÷ 2) = 8 ÷ 2 = 4}}. |
|{{math|1=(8 ÷ 4) ÷ 2 = 2 ÷ 2 = 1}}, but {{math|1=8 ÷ (4 ÷ 2) = 8 ÷ 2 = 4}}. |
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|- |
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|[[Logical consequence|proves]] |
|[[Logical consequence|proves]] |
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|[[propositional logic]], [[first-order logic]] |
|[[propositional logic]], [[first-order logic]] |
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|''x'' ⊢ ''y'' means ''x'' proves (syntactically |
|''x'' ⊢ ''y'' means ''x'' proves (syntactically entails) ''y'' |
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| (''A'' → ''B'') ⊢ (¬''B'' → ¬''A'') |
| (''A'' → ''B'') ⊢ (¬''B'' → ¬''A'') |
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|[[Logical consequence|models]] |
|[[Logical consequence|models]] |
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|[[propositional logic]], [[first-order logic]] |
|[[propositional logic]], [[first-order logic]] |
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|''x'' ⊨ ''y'' means ''x'' models (semantically |
|''x'' ⊨ ''y'' means ''x'' models (semantically entails) ''y'' |
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| (''A'' → ''B'') ⊨ (¬''B'' → ¬''A'') |
| (''A'' → ''B'') ⊨ (¬''B'' → ¬''A'') |
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|[[Sheffer stroke]], |
|[[Sheffer stroke]],<br>the sign for the NAND operator (negation of conjunction). |
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|[[Peirce arrow|Peirce Arrow]], the sign for the NOR operator (negation of disjunction). |
|[[Peirce arrow|Peirce Arrow]], <br>the sign for the NOR operator (negation of disjunction). |
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|the sign for the XNOR |
|the sign for the XNOR operator (negation of exclusive disjunction). |
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|strike out existential |
|strike out existential quantifier, same as "¬∃" |
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|modal operator for "it |
|modal operator for "it is possible that", "it is not necessarily not" or rarely "it is not probably not" (in most modal logics it is defined as "¬◻¬") |
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|Webb-operator or Peirce arrow, the sign for [[Logical NOR|NOR]]. Confusingly, "⊥" is also the sign for contradiction or absurdity. |
|Webb-operator or Peirce arrow, the sign for [[Logical NOR|NOR]].<br> Confusingly, "⊥" is also the sign for contradiction or absurdity. |
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|corner quotes, also called "Quine quotes"; for quasi-quotation, i.e. quoting specific context of unspecified ("variable") expressions;<ref>[[Willard Van Orman Quine|Quine, W.V.]] (1981): ''Mathematical Logic'', §6</ref> also used for denoting [[Gödel number]];<ref>{{citation|last=Hintikka|first=Jaakko|title=The Principles of Mathematics Revisited|url=https://books.google.com/books?id=JHBnE0EQ6VgC&pg=PA113|page=113|year=1998|publisher=Cambridge University Press|isbn=9780521624985}}.</ref> for example "⌜G⌝" denotes the Gödel number of G. (Typographical note: although the quotes appears as a "pair" in unicode (231C and 231D), they are not symmetrical in some fonts. |
|corner quotes, also called "Quine quotes"; for quasi-quotation, i.e. quoting specific context of unspecified ("variable") expressions;<ref>[[Willard Van Orman Quine|Quine, W.V.]] (1981): ''Mathematical Logic'', §6</ref> also used for denoting [[Gödel number]];<ref>{{citation|last=Hintikka|first=Jaakko|title=The Principles of Mathematics Revisited|url=https://books.google.com/books?id=JHBnE0EQ6VgC&pg=PA113|page=113|year=1998|publisher=Cambridge University Press|isbn=9780521624985}}.</ref> for example "⌜G⌝" denotes the Gödel number of G. (Typographical note: although the quotes appears as a "pair" in unicode (231C and 231D), they are not symmetrical in some fonts. In some fonts (for example Arial) they are only symmetrical in certain sizes. Alternatively the quotes can be rendered as ⌈ and ⌉ (U+2308 and U+2309) or by using a negation symbol and a reversed negation symbol ⌐ ¬ in superscript mode. ) |
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Revision as of 02:18, 27 February 2023
In logic, a set of symbols is commonly used to express logical representation. The following table lists many common symbols, together with their name, how they should be read out loud, and the related field of mathematics. Additionally, the subsequent columns contains an informal explanation, a short example, the Unicode location, the name for use in HTML documents,[1] and the LaTeX symbol.
Basic logic symbols
Symbol | Unicode value (hexadecimal) |
HTML value (decimal) |
HTML entity (named) |
LaTeX symbol |
Logic Name | Read as | Category | Explanation | Examples |
---|---|---|---|---|---|---|---|---|---|
⇒
→ ⊃ |
U+21D2 U+2192 U+2283 |
⇒ → ⊃ |
⇒ → ⊃ |
\Rightarrow
\to or \rightarrow \supset \implies |
material implication | implies; if ... then | propositional logic, Heyting algebra | is false when A is true and B is false but true otherwise. may mean the same as (the symbol may also indicate the domain and codomain of a function; see table of mathematical symbols). may mean the same as (the symbol may also mean superset). |
is true, but is in general false (since x could be −2). |
⇔
≡ ↔ |
U+21D4 U+2261 U+2194 |
⇔ ≡ ↔ |
⇔ ≡ ↔ |
\Leftrightarrow \equiv \leftrightarrow \iff |
material equivalence | if and only if; iff; means the same as | propositional logic | is true only if both A and B are false, or both A and B are true. | |
¬
˜ ! |
U+00AC U+02DC U+0021 |
¬ ˜ ! |
¬ ˜ ! |
\lnot or \neg
|
negation | not | propositional logic | The statement is true if and only if A is false. A slash placed through another operator is the same as placed in front. |
|
U+1D53B | 𝔻 | 𝔻 | \mathbb{D} | Domain of discourse | Domain of predicate | Predicate (mathematical logic) | |||
∧
· & |
U+2227 U+00B7 U+0026 |
∧ · & |
∧ · & |
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+2228 U+002B U+2225 |
∨ + ∥ |
∨
|
\lor or \vee
|
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+21AE U+2295 U+22BB
|
↮ ⊕ ⊻
|
⊕
|
\oplus
|
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. |
⊤
T 1 ■ |
U+22A4 U+25A0 |
⊤ |
⊤ | \top | Tautology | top, truth, full clause | propositional logic, Boolean algebra, first-order logic | ⊤ is unconditionally true. | ⊤(A) ⇒ A is always true. |
⊥
F 0 □ |
U+22A5 U+25A1 |
⊥ |
⊥ |
\bot | Contradiction | bottom, falsum, falsity, empty clause | propositional logic, Boolean algebra, first-order logic | ⊥ is unconditionally false. (The symbol ⊥ may also refer to perpendicular lines.) |
⊥(A) ⇒ A is always false. |
∀
() |
U+2200 |
∀ |
∀ |
\forall | 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. | |
∃
|
U+2203 | ∃ | ∃ | \exists | existential quantification | there exists | first-order logic | ∃ x: P(x) means there exists at least one x such that P(x) is true. | n is even. |
∃!
|
U+2203 U+0021 | ∃ ! | ∃! | \exists ! | uniqueness quantification | there exists exactly one | first-order logic | ∃! x: P(x) means there exists exactly one x such that P(x) is true. | |
≔
≡ :⇔ |
U+2254 (U+003A U+003D) U+2261 U+003A U+21D4 |
≔ (: =)
|
≔
|
:=
:\Leftrightarrow |
definition | is defined as | everywhere | x ≔ y or x ≡ y means x is defined to be another name for y ( The symbol ≡ can also mean other things, such as congruence). P :⇔ Q means P is defined to be logically equivalent to Q. |
A ⊕ B :⇔ (A ∨ B) ∧ ¬(A ∧ B) |
( )
|
U+0028 U+0029 | ( ) | (
) |
( ) | 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+22A2 | ⊢ | ⊢ | \vdash | turnstile | proves | propositional logic, first-order logic | x ⊢ y means x proves (syntactically entails) y | (A → B) ⊢ (¬B → ¬A) |
⊨
|
U+22A8 | ⊨ | ⊨ | \vDash, \models | double turnstile | models | propositional logic, first-order logic | x ⊨ y means x models (semantically entails) y | (A → B) ⊨ (¬B → ¬A) |
Advanced and rarely used logical symbols
These symbols are sorted by their Unicode value:
Symbol | Unicode value (hexadecimal) |
HTML value (decimal) |
HTML entity (named) |
LaTeX symbol |
Logic Name | Read as | Category | Explanation | Examples |
---|---|---|---|---|---|---|---|---|---|
̅
|
U+0305 | COMBINING OVERLINE | used format for denoting Gödel numbers.
denoting negation used primarily in electronics. |
using HTML style "4̅" is a shorthand for the standard numeral "SSSS0".
"A ∨ B" says the Gödel number of "(A ∨ B)". "A ∨ B" is the same as "¬(A ∨ B)". | |||||
↑
| |
U+2191 U+007C |
UPWARDS ARROW VERTICAL LINE |
Sheffer stroke, the sign for the NAND operator (negation of conjunction). |
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↓
|
U+2193 | DOWNWARDS ARROW | Peirce Arrow, the sign for the NOR operator (negation of disjunction). |
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⊙
|
U+2299 | \odot | CIRCLED DOT OPERATOR | the sign for the XNOR operator (negation of exclusive disjunction). | |||||
∁
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U+2201 | COMPLEMENT | |||||||
∄
|
U+2204 | ∄\nexists | THERE DOES NOT EXIST | strike out existential quantifier, same as "¬∃" | |||||
∴
|
U+2234 | ∴\therefore | THEREFORE | Therefore | |||||
∵
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U+2235 | ∵\because | BECAUSE | because | |||||
⊧
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U+22A7 | MODELS | is a model of (or "is a valuation satisfying") | ||||||
⊨
|
U+22A8 | ⊨\vDash | TRUE | is true of | |||||
⊬
|
U+22AC | ⊬\nvdash | DOES NOT PROVE | negated ⊢, the sign for "does not prove" | T ⊬ P says "P is not a theorem of T" | ||||
⊭
|
U+22AD | ⊭\nvDash | NOT TRUE | is not true of | |||||
†
|
U+2020 | DAGGER | it is true that ... | Affirmation operator | |||||
⊼
|
U+22BC | NAND | NAND operator | ||||||
⊽
|
U+22BD | NOR | NOR operator | ||||||
◇
|
U+25C7 | WHITE DIAMOND | modal operator for "it is possible that", "it is not necessarily not" or rarely "it is not probably not" (in most modal logics it is defined as "¬◻¬") | ||||||
⋆
|
U+22C6 | STAR OPERATOR | usually used for ad-hoc operators | ||||||
⊥
↓ |
U+22A5 U+2193 |
UP TACK DOWNWARDS ARROW |
Webb-operator or Peirce arrow, the sign for NOR. Confusingly, "⊥" is also the sign for contradiction or absurdity. |
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⌐
|
U+2310 | REVERSED NOT SIGN | |||||||
⌜
⌝ |
U+231C U+231D |
\ulcorner
\urcorner |
TOP LEFT CORNER TOP RIGHT CORNER |
corner quotes, also called "Quine quotes"; for quasi-quotation, i.e. quoting specific context of unspecified ("variable") expressions;[3] also used for denoting Gödel number;[4] for example "⌜G⌝" denotes the Gödel number of G. (Typographical note: although the quotes appears as a "pair" in unicode (231C and 231D), they are not symmetrical in some fonts. In some fonts (for example Arial) they are only symmetrical in certain sizes. Alternatively the quotes can be rendered as ⌈ and ⌉ (U+2308 and U+2309) or by using a negation symbol and a reversed negation symbol ⌐ ¬ in superscript mode. ) | |||||
◻
□ |
U+25FB U+25A1 |
WHITE MEDIUM SQUARE WHITE SQUARE |
modal operator for "it is necessary that" (in modal logic), or "it is provable that" (in provability logic), or "it is obligatory that" (in deontic logic), or "it is believed that" (in doxastic logic); also as empty clause (alternatives: and ⊥) |
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⟛
|
U+27DB | LEFT AND RIGHT TACK | semantic equivalent | ||||||
⟡
|
U+27E1 | WHITE CONCAVE-SIDED DIAMOND | never | modal operator | |||||
⟢
|
U+27E2 | WHITE CONCAVE-SIDED DIAMOND WITH LEFTWARDS TICK | was never | modal operator | |||||
⟣
|
U+27E3 | WHITE CONCAVE-SIDED DIAMOND WITH RIGHTWARDS TICK | will never be | modal operator | |||||
□
|
U+25A1 | WHITE SQUARE | always | modal operator | |||||
⟤
|
U+25A4 | WHITE SQUARE WITH LEFTWARDS TICK | was always | modal operator | |||||
⟥
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U+25A5 | WHITE SQUARE WITH RIGHTWARDS TIC | will always be | modal operator | |||||
⥽
|
U+297D | RIGHT FISH TAIL | sometimes used for "relation", also used for denoting various ad hoc relations (for example, for denoting "witnessing" in the context of Rosser's trick) The fish hook is also used as strict implication by C.I.Lewis ⥽ , the corresponding LaTeX macro is \strictif. See here for an image of glyph. Added to Unicode 3.2.0. | ||||||
⨇
|
U+2A07 | TWO LOGICAL AND OPERATOR |
Usage in various countries
Poland and Germany
As of 2014[update] in Poland, the universal quantifier is sometimes written ∧, and the existential quantifier as ∨. The same applies for Germany.
Japan
The ⇒ symbol is often used in text to mean "result" or "conclusion", as in "We examined whether to sell the product ⇒ We will not sell it". Also, the → symbol is often used to denote "changed to", as in the sentence "The interest rate changed. March 20% → April 21%".
See also
- Józef Maria Bocheński
- List of notation used in Principia Mathematica
- List of mathematical symbols
- Logic alphabet, a suggested set of logical symbols
- Logic gate § Symbols
- Logical connective
- Mathematical operators and symbols in Unicode
- Non-logical symbol
- Polish notation
- Truth function
- Truth table
- Wikipedia:WikiProject Logic/Standards for notation
References
- ^ "Named character references". HTML 5.1 Nightly. W3C. Retrieved 9 September 2015.
- ^ Although this character is available in LaTeX, the MediaWiki TeX system does not support it.
- ^ Quine, W.V. (1981): Mathematical Logic, §6
- ^ Hintikka, Jaakko (1998), The Principles of Mathematics Revisited, Cambridge University Press, p. 113, ISBN 9780521624985.
Further reading
- Józef Maria Bocheński (1959), A Précis of Mathematical Logic, trans., Otto Bird, from the French and German editions, Dordrecht, South Holland: D. Reidel.
External links
- Named character entities in HTML 4.0