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List of logic symbols
List of symbols used to express logical relations From Wikipedia, the free encyclopedia
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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.
This article contains logic symbols. Without proper rendering support, you may see question marks, boxes, or other symbols instead of logic symbols.
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Basic logic symbols
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Symbol | Unicode value (hexadecimal) |
HTML codes |
LaTeX symbol |
Logic Name | Read as | Category | Explanation | Examples |
---|---|---|---|---|---|---|---|---|
⇒ → ⊃ |
U+21D2 U+2192 U+2283 |
⇒ → ⊃ ⇒ |
material conditional (material implication) | implies, if P then Q, it is not the case that P and not Q |
propositional logic, Boolean algebra, Heyting algebra | (the symbol may also indicate the domain and codomain of a function; see table of mathematical symbols). |
(since x could be −2). | |
⇔ ↔ ≡ |
U+21D4 U+2194 U+2261 |
⇔ ↔ ≡ ⇔ |
material biconditional (material equivalence) | if and only if, iff, xnor | propositional logic, Boolean algebra | |||
¬ ~ ! ′ |
U+00AC U+007E U+0021 U+2032 |
¬ ˜ ! ′ ¬ |
negation | not | propositional logic, Boolean algebra | The statement A slash placed through another operator is the same as The prime symbol is placed after the negated thing, e.g. |
||
∧ · & |
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. | ||
∨ + ∥ |
U+2228 U+002B U+2225 |
∨ + ∥ ∨ |
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+2295 U+22BB U+21AE U+2262 |
⊕ ⊻ ↮ ≢ ⊕ |
exclusive disjunction | xor, either ... or ... (but not both) |
propositional logic, Boolean algebra | The statement ¬(A ↔ B), hence the symbols |
||
⊤ T 1 |
U+22A4 |
⊤
|
true (tautology) | top, truth, tautology, verum, full clause | propositional logic, Boolean algebra, first-order logic | The proposition | ||
⊥ F 0 |
U+22A5 |
⊥
⊥ |
false (contradiction) | bottom, falsity, contradiction, falsum, empty clause | propositional logic, Boolean algebra, first-order logic | The symbol ⊥ may also refer to perpendicular lines. |
The proposition | |
∀ () |
U+2200 |
∀
∀ |
universal quantification | given any, for all, for every, for each, for any | first-order logic | |||
∃ |
U+2203 | ∃
∃ |
existential quantification | there exists, for some | first-order logic | :}
| ||
∃! |
U+2203 U+0021 | ∃ !
∃! |
!}
|
uniqueness quantification | there exists exactly one | first-order logic (abbreviation) | ||
( ) |
U+0028 U+0029 | ( )
( |
precedence grouping | parentheses; brackets | almost all logic syntaxes, as well as metalanguage | Perform the operations inside the parentheses first. | (8 ÷ 4) ÷ 2 = 2 ÷ 2 = 1, but 8 ÷ (4 ÷ 2) = 8 ÷ 2 = 4. | |
U+1D53B | 𝔻
𝔻 |
\mathbb{D} | domain of discourse | domain of discourse | metalanguage (first-order logic semantics) | |||
⊢ |
U+22A2 | ⊢
⊢ |
turnstile | syntactically entails (proves) | metalanguage (metalogic) | a theorem of In other words, |
||
⊨ |
U+22A8 | ⊨
⊨ |
double turnstile | semantically entails | metalanguage (metalogic) | “in every model, it is not the case that |
||
≡ ⟚ ⇔ |
U+2261 U+27DA U+21D4 |
≡
— |
logical equivalence | is logically equivalent to | metalanguage (metalogic) | It’s when |
||
⊬ |
U+22AC | ⊬\nvdash | does not syntactically entail (does not prove) | metalanguage (metalogic) | not a theorem of In other words, |
|||
⊭ |
U+22AD | ⊭\nvDash | does not semantically entail | metalanguage (metalogic) | In other words, |
|||
□ |
U+25A1 | necessity (in a model) | box; it is necessary that | modal logic | modal operator for “it is necessary that” in alethic logic, “it is provable that” in provability logic, “it is obligatory that” in deontic logic, “it is believed that” in doxastic logic. |
|||
◇ |
U+25C7 | possibility (in a model) | diamond; it is possible that |
modal logic | modal operator for “it is possible that”, (in most modal logics it is defined as “¬□¬”, “it is not necessarily not”). | |||
∴ |
U+2234 | ∴\therefore | therefore | therefore | metalanguage | abbreviation for “therefore”. | ||
∵ |
U+2235 | ∵\because | because | because | metalanguage | abbreviation for “because”. | ||
≔ ≜ ≝ |
U+2254 U+225C U+225D |
≔
≔ |
≔ \coloneqq :=}
\scriptscriptstyle \mathrm{def}}{=} |
definition | is defined as | metalanguage |
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Advanced or rarely used logical symbols
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The following symbols are either advanced and context-sensitive or very rarely used:
More information ). The fish hook is also used as strict implication by C.I.Lewis
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Symbol | Unicode value (hexadecimal) |
HTML value (decimal) |
HTML entity (named) |
LaTeX symbol |
Logic Name | Read as | Category | Explanation |
---|---|---|---|---|---|---|---|---|
⥽ |
U+297D | \strictif | 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 | ||||
̅ |
U+0305 | combining overline | Used format for denoting Gödel numbers. Using HTML style “4̅” is an abbreviation for the standard numeral “SSSS0”.
It may also denote a negation (used primarily in electronics). | |||||
⌜ ⌝ |
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;[4] also used for denoting Gödel number;[5] 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+2204 | \nexists | there does not exist | Strike out existential quantifier. “¬∃” is recommended instead. [by whom?] | ||||
↑ | |
U+2191 U+007C |
upwards arrow vertical line |
Sheffer stroke, the sign for the NAND operator (negation of conjunction). | |||||
↓ |
U+2193 | downwards arrow | Peirce Arrow, a sign for the NOR operator (negation of disjunction). | |||||
⊼ |
U+22BC | NAND | A new symbol made specifically for the NAND operator. | |||||
⊽ |
U+22BD | NOR | A new symbol made specifically for the NOR operator. | |||||
⊙ |
U+2299 | \odot | circled dot operator | A sign for the XNOR operator (material biconditional and XNOR are the same operation). | ||||
⟛ |
U+27DB | left and right tack | “Proves and is proved by”. | |||||
⊧ |
U+22A7 | models | “Is a model of” or “is a valuation satisfying”. | |||||
⊩ |
U+22A9 | forces | One of this symbol’s uses is to mean “truthmakes” in the truthmaker theory of truth. It is also used to mean “forces” in the set theory method of forcing. | |||||
⟡ |
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+25A4 | white square with leftwards tick | was always | modal operator | ||||
⟥ |
U+25A5 | white square with rightwards tick | will always be | modal operator | ||||
⋆ |
U+22C6 | star operator | May sometimes be used for ad-hoc operators. | |||||
⌐ |
U+2310 | reversed not sign | ||||||
⨇ |
U+2A07 | two logical AND operator |
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See also
- Glossary of logic
- 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
Further reading
External links
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