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, and the LaTeX symbol.
Basic logic symbols
| Symbol | Unicodevalue(hexadecimal) | HTMLcodes | LaTeXsymbol | Logic Name | Read as | Category | Explanation | Examples |
|---|---|---|---|---|---|---|---|---|
| ⇒→⊃ | U+21D2U+2192U+2283 | ⇒→⊃ ⇒→⊃ | ⇒ {\displaystyle \Rightarrow } \Rightarrow ⟹ {\displaystyle \implies } \implies → {\displaystyle \to } \to or \rightarrow ⊃ {\displaystyle \supset } \supset | 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 | A ⇒ B {\displaystyle A\Rightarrow B} is false when A is true and B is false but true otherwise. → {\displaystyle \rightarrow } may mean the same as ⇒ {\displaystyle \Rightarrow } (the symbol may also indicate the domain and codomain of a function; see table of mathematical symbols). ⊃ {\displaystyle \supset } may mean the same as ⇒ {\displaystyle \Rightarrow } (the symbol may also mean superset). | x = 2 ⇒ x 2 = 4 {\displaystyle x=2\Rightarrow x^{2}=4} is true, but x 2 = 4 ⇒ x = 2 {\displaystyle x^{2}=4\Rightarrow x=2} is in general false (since x could be −2). |
| ⇔↔≡ | U+21D4U+2194U+2261 | ⇔↔≡ ⇔↔≡ | ⇔ {\displaystyle \Leftrightarrow } \Leftrightarrow ⟺ {\displaystyle \iff } \iff ↔ {\displaystyle \leftrightarrow } \leftrightarrow ≡ {\displaystyle \equiv } \equiv | material biconditional (material equivalence) | if and only if, iff, xnor | propositional logic, Boolean algebra | A ⇔ B {\displaystyle A\Leftrightarrow B} is true only if both A and B are false, or both A and B are true. Whether a symbol means a material biconditional or a logical equivalence, depends on the author’s style. | x + 5 = y + 2 ⇔ x + 3 = y {\displaystyle x+5=y+2\Leftrightarrow x+3=y} |
| ¬~! ′ | U+00ACU+007EU+0021U+2032 | ¬˜!′ ¬˜!′ | ¬ {\displaystyle \neg } \lnot or \neg ∼ {\displaystyle \sim } \sim ′ {\displaystyle '} ' | negation | not | propositional logic, Boolean algebra | The statement ¬ A {\displaystyle \lnot A} is true if and only if A is false.A slash placed through another operator is the same as ¬ {\displaystyle \neg } placed in front. The prime symbol is placed after the negated thing, e.g. p ′ {\displaystyle p'} 2 | ¬ ( ¬ A ) ⇔ A {\displaystyle \neg (\neg A)\Leftrightarrow A} x ≠ y ⇔ ¬ ( x = y ) {\displaystyle x\neq y\Leftrightarrow \neg (x=y)} |
| ∧·& | U+2227U+00B7U+0026 | ∧·& ∧·& | ∧ {\displaystyle \wedge } \wedge or \land ⋅ {\displaystyle \cdot } \cdot & {\displaystyle \&} \&3 | 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+2228U+002BU+2225 | ∨+∥ ∨+∥ | ∨ {\displaystyle \lor } \lor or \vee ∥ {\displaystyle \parallel } \parallel | 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+2295U+22BBU+21AEU+2262 | ⊕⊻↮≢ ⊕⊻—≢ | ⊕ {\displaystyle \oplus } \oplus ⊻ {\displaystyle \veebar } \veebar ≢ {\displaystyle \not \equiv } \not\equiv | exclusive disjunction | xor, either ... or ... (but not both) | propositional logic, Boolean algebra | The statement A ⊕ B {\displaystyle A\oplus B} is true when either A or B, but not both, are true. This is equivalent to ¬(A ↔ B), hence the symbols ↮ {\displaystyle \nleftrightarrow } and ≢ {\displaystyle \not \equiv } . | ¬ A ⊕ A {\displaystyle \lnot A\oplus A} is always true and A ⊕ A {\displaystyle A\oplus A} is always false (if vacuous truth is excluded). |
| ⊤T1 | U+22A4 | ⊤ ⊤ | ⊤ {\displaystyle \top } \top | true (tautology) | top, truth, tautology, verum, full clause | propositional logic, Boolean algebra, first-order logic | ⊤ {\displaystyle \top } denotes a proposition that is always true. | The proposition ⊤ ∨ P {\displaystyle \top \lor P} is always true since at least one of the two is unconditionally true. |
| ⊥F0 | U+22A5 | ⊥ ⊥ | ⊥ {\displaystyle \bot } \bot | false (contradiction) | bottom, falsity, contradiction, falsum, empty clause | propositional logic, Boolean algebra, first-order logic | ⊥ {\displaystyle \bot } denotes a proposition that is always false. The symbol ⊥ may also refer to perpendicular lines. | The proposition ⊥ ∧ P {\displaystyle \bot \wedge P} is always false since at least one of the two is unconditionally false. |
| ∀() | U+2200 | ∀ ∀ | ∀ {\displaystyle \forall } \forall | universal quantification | given any, for all, for every, for each, for any | first-order logic | ∀ x {\displaystyle \forall x} P ( x ) {\displaystyle P(x)} or ( x ) {\displaystyle (x)} P ( x ) {\displaystyle P(x)} says “given any x {\displaystyle x} , x {\displaystyle x} has property P {\displaystyle P} .” | ∀ n ∈ N : n 2 ≥ n . {\displaystyle \forall n\in \mathbb {N} :n^{2}\geq n.} |
| ∃ | U+2203 | ∃ ∃ | ∃ {\displaystyle \exists } \exists | existential quantification | there exists, for some | first-order logic | ∃ x {\displaystyle \exists x} P ( x ) {\displaystyle P(x)} says “there exists an x {\displaystyle x} (at least one) such that x {\displaystyle x} has property P {\displaystyle P} .” | ∃ n ∈ N : {\displaystyle \exists n\in \mathbb {N} :} n is even. |
| ∃! | U+2203 U+0021 | ∃ ! ∃! | ∃ ! {\displaystyle \exists !} \exists ! | uniqueness quantification | there exists exactly one | first-order logic (abbreviation) | ∃ ! x {\displaystyle \exists !x} P ( x ) {\displaystyle P(x)} says “there exists exactly one x {\displaystyle x} such that x {\displaystyle x} has property P {\displaystyle P} .” Only ∀ {\displaystyle \forall } and ∃ {\displaystyle \exists } are part of formal logic. ∃ ! x {\displaystyle \exists !x} P ( x ) {\displaystyle P(x)} is an abbreviation for ∃ x ∀ y ( P ( y ) ↔ y = x ) {\displaystyle \exists x\forall y(P(y)\leftrightarrow y=x)} | ∃ ! n ∈ N : n + 5 = 2 n . {\displaystyle \exists !n\in \mathbb {N} :n+5=2n.} |
| ( ) | U+0028 U+0029 | ( ) ( ) | ( ) {\displaystyle (~)} ( ) | 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. |
| D {\displaystyle \mathbb {D} } | U+1D53B | 𝔻 𝔻 | \mathbb{D} | domain of discourse | domain of discourse | metalanguage (first-order logic semantics) | D : R {\displaystyle \mathbb {D} \mathbb {:} \mathbb {R} } | |
| ⊢ | U+22A2 | ⊢ ⊢ | ⊢ {\displaystyle \vdash } \vdash | turnstile | syntactically entails (proves) | metalanguage (metalogic) | A ⊢ B {\displaystyle A\vdash B} says “ B {\displaystyle B} is a theorem of A {\displaystyle A} ”. In other words, A {\displaystyle A} proves B {\displaystyle B} via a deductive system. | ( A → B ) ⊢ ( ¬ B → ¬ A ) {\displaystyle (A\rightarrow B)\vdash (\lnot B\rightarrow \lnot A)} (eg. by using natural deduction) |
| ⊨ | U+22A8 | ⊨ ⊨ | ⊨ {\displaystyle \vDash } \vDash, \models | double turnstile | semantically entails | metalanguage (metalogic) | A ⊨ B {\displaystyle A\vDash B} says “in every model, it is not the case that A {\displaystyle A} is true and B {\displaystyle B} is false”. | ( A → B ) ⊨ ( ¬ B → ¬ A ) {\displaystyle (A\rightarrow B)\vDash (\lnot B\rightarrow \lnot A)} (eg. by using truth tables) |
| ≡⟚⇔ | U+2261U+27DAU+21D4 | ≡ —⇔≡—⇔ | ≡ {\displaystyle \equiv } \equiv ⇔ {\displaystyle \Leftrightarrow } \Leftrightarrow | logical equivalence | is logically equivalent to | metalanguage (metalogic) | It’s when A ⊨ B {\displaystyle A\vDash B} and B ⊨ A {\displaystyle B\vDash A} . Whether a symbol means a material biconditional or a logical equivalence, depends on the author’s style. | ( A → B ) ≡ ( ¬ A ∨ B ) {\displaystyle (A\rightarrow B)\equiv (\lnot A\lor B)} |
| ⊬ | U+22AC | ⊬\nvdash | does not syntactically entail (does not prove) | metalanguage (metalogic) | A ⊬ B {\displaystyle A\nvdash B} says “ B {\displaystyle B} is not a theorem of A {\displaystyle A} ”. In other words, B {\displaystyle B} is not derivable from A {\displaystyle A} via a deductive system. | A ∨ B ⊬ A ∧ B {\displaystyle A\lor B\nvdash A\wedge B} | ||
| ⊭ | U+22AD | ⊭\nvDash | does not semantically entail | metalanguage (metalogic) | A ⊭ B {\displaystyle A\nvDash B} says “ A {\displaystyle A} does not guarantee the truth of B {\displaystyle B} ”. In other words, A {\displaystyle A} does not make B {\displaystyle B} true. | A ∨ B ⊭ A ∧ B {\displaystyle A\lor B\nvDash A\wedge B} | ||
| □ | U+25A1 | ◻ {\displaystyle \Box } \Box | 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. | ◻ ∀ x P ( x ) {\displaystyle \Box \forall xP(x)} says “it is necessary that everything has property P {\displaystyle P} ” | |
| ◇ | U+25C7 | ◊ {\displaystyle \Diamond } \Diamond | 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”). | ◊ ∃ x P ( x ) {\displaystyle \Diamond \exists xP(x)} says “it is possible that something has property P {\displaystyle P} ” | |
| ∴ | U+2234 | ∴\therefore | therefore | therefore | metalanguage | abbreviation for “therefore”. | ||
| ∵ | U+2235 | ∵\because | because | because | metalanguage | abbreviation for “because”. | ||
| ≔≜≝ | U+2254U+225CU+225D | ≔ ≔ | ≔ \coloneqq := {\displaystyle :=} := ≜ {\displaystyle \triangleq } \triangleq = d e f {\displaystyle {\stackrel {\scriptscriptstyle \mathrm {def} }{=}}} \stackrel{ \scriptscriptstyle \mathrm{def}}{=} | definition | is defined as | metalanguage | a := b {\displaystyle a:=b} means "from now on, a {\displaystyle a} is defined to be another name for b {\displaystyle b} ." This is a statement in the metalanguage, not the object language. The notation a ≡ b {\displaystyle a\equiv b} may occasionally be seen in physics, meaning the same as a := b {\displaystyle a:=b} . | cosh x := e x + e − x 2 {\displaystyle \cosh x:={\frac {e^{x}+e^{-x}}{2}}} |
Advanced or rarely used logical symbols
The following symbols are either advanced and context-sensitive or very rarely used:
| Symbol | Unicodevalue(hexadecimal) | HTMLvalue(decimal) | HTMLentity(named) | LaTeXsymbol | 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 p {\displaystyle p} ⥽ q ≡ ◻ ( p → q ) {\displaystyle q\equiv \Box (p\rightarrow q)} . | ||||
| ̅ | 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+231CU+231D | \ulcorner \urcorner | top left cornertop 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+2191U+007C | upwards arrowvertical 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 |
See also
- Philosophy portal
- 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
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
References
"Named character references". HTML 5.1 Nightly. W3C. Retrieved 9 September 2015. http://www.w3.org/html/wg/drafts/html/master/syntax.html#named-character-references ↩
Virtually all Turkish high school math textbooks use p' for negation due to the books handed out by the Ministry of National Education representing it as p'. ↩
Although this character is available in LaTeX, the MediaWiki TeX system does not support it. /wiki/MediaWiki ↩
Quine, W.V. (1981): Mathematical Logic, §6 /wiki/Willard_Van_Orman_Quine ↩
Hintikka, Jaakko (1998), The Principles of Mathematics Revisited, Cambridge University Press, p. 113, ISBN 9780521624985. 9780521624985 ↩