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
codes
LaTeX
symbol
Logic Name Read as Category Explanation Examples


U+21D2

U+2192

U+2283
⇒
→
⊃

⇒
→
⊃

\Rightarrow
\implies
\to or \rightarrow
\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 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+2194

U+2261
⇔
↔
≡

⇔
↔
≡

\Leftrightarrow
\iff
\leftrightarrow
\equiv
material biconditional (material equivalence) if and only if, iff, xnor propositional logic, Boolean algebra 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, it depends on the author’s style.
¬
~
!
U+00AC

U+007E

U+0021
¬
˜
!

¬
˜
!

\lnot or \neg

\sim


negation not propositional logic, Boolean algebra 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+2227

U+00B7

U+0026
∧
·
&

∧
·
&

\wedge or \land
\cdot

\&[2]
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
&#8744;
&#43;
&#8741;

&or;
&plus;
&parallel;

\lor or \vee



\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+2295

U+22BB

U+21AE

U+2262
&#8853;
&#8891;
&#8622;
&#8802;

&oplus;
&veebar;

&nequiv;

\oplus

\veebar



\not\equiv
exclusive disjunction xor,
either ... or ... (but not both)
propositional logic, Boolean algebra The statement is true when either A or B, but not both, are true. This is equivalent to
¬(A ↔ B), hence the symbols and .
is always true and is always false (if vacuous truth is excluded).


T
1


U+22A4





&#8868;


&top;

\top



true (tautology) top, truth, tautology, verum, full clause propositional logic, Boolean algebra, first-order logic denotes a proposition that is always true.
The proposition is always true since at least one of the two is unconditionally true.


F
0


U+22A5





&#8869;

&perp;



\bot



false (contradiction) bottom, falsity, contradiction, falsum, empty clause propositional logic, Boolean algebra, first-order logic denotes a proposition that is always false.
The symbol ⊥ may also refer to perpendicular lines.
The proposition is always false since at least one of the two is unconditionally false.

()
U+2200


&#8704;

&forall;


\forall


universal quantification given any, for all, for every, for each, for any first-order logic   or
  says “given any , has property .”
U+2203 &#8707;

&exist;

\exists existential quantification there exists, for some first-order logic   says “there exists an x (at least one) such that has property .”
n is even.
∃!
U+2203 U+0021 &#8707; &#33;

&exist;!

\exists ! uniqueness quantification there exists exactly one first-order logic (abbreviation) says “there exists exactly one x such that x has property P.” Only and are part of formal logic.
is an abbreviation for
( )
U+0028 U+0029 &#40; &#41;

&lpar;
&rpar;

( ) 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 &#120123;

&Dopf;

\mathbb{D} domain of discourse domain of discourse metalanguage (first-order logic semantics)
U+22A2 &#8866;

&vdash;

\vdash turnstile syntactically entails (proves) metalanguage (metalogic) says “ is
a theorem of ”.
In other words,
proves via a deductive system.

(eg. by using natural deduction)
U+22A8 &#8872;

&vDash;

\vDash, \models double turnstile semantically entails metalanguage (metalogic) says
“in every model,
it is not the case that is true and is false”.

(eg. by using truth tables)


U+2261

U+27DA

U+21D4
&#8801;


&#8660; &equiv; — &hArr;

\equiv



\Leftrightarrow
logical equivalence is logically equivalent to metalanguage (metalogic) It’s when and . Whether a symbol means a material biconditional or a logical equivalence, it depends on the author’s style.
U+22AC ⊬\nvdash does not syntactically entail (does not prove) metalanguage (metalogic) says “ is
not a theorem of ”.
In other words,
is not derivable from via a deductive system.
U+22AD ⊭\nvDash does not semantically entail metalanguage (metalogic) says “ does not guarantee the truth of  ”.
In other words,
does not make true.
U+25A1 \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.
says “it is necessary that everything has property P”
U+25C7 \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”).
says “it is possible that something has property P”
U+2234 ∴\therefore therefore therefore metalanguage abbreviation for “therefore”.
U+2235 ∵\because because because metalanguage abbreviation for “because”.


U+2254

U+225C

U+225D
&#8788;

&coloneq;






:=

\triangleq


\stackrel{

\scriptscriptstyle \mathrm{def)){=}

definition is defined as metalanguage means "from now on, is defined to be another name for ." This is a statement in the metalanguage, not the object language. The notation may occasionally be seen in physics, meaning the same as .

Advanced or rarely used logical symbols

The following symbols are either advanced and context-sensitive or very rarely used:

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;[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+2204 ∄\nexists there does not exist Strike out existential quantifier. “¬∃” is recommended instead.

|
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

See also

References

  1. ^ "Named character references". HTML 5.1 Nightly. W3C. Retrieved 9 September 2015.
  2. ^ Although this character is available in LaTeX, the MediaWiki TeX system does not support it.
  3. ^ Quine, W.V. (1981): Mathematical Logic, §6
  4. ^ Hintikka, Jaakko (1998), The Principles of Mathematics Revisited, Cambridge University Press, p. 113, ISBN 9780521624985.

Further reading