symbols

List of logic symbols

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 This article contains logic symbols. Without proper rendering support, you may
 see question marks, boxes, or other symbols instead of logic symbols.

See also: Logical connective

In logic, a set of symbols is commonly used to express logical representation.
As logicians are familiar with these symbols, they are not explained each time
they are used. So, for students of logic, the following table lists many common
symbols together with their name, pronunciation and related field of
mathematics. Additionally, the third column contains an informal definition,
and the fourth column gives a short example.

Be aware that, outside of logic, different symbols have the same meaning, and
the same symbol has, depending on the context, different meanings.

[edit] Basic logic symbols

       Name
Symbol Should be read   Explanation    Examples      Unicode      HTML       LaTeX
             as                                       Value      Entity      symbol
             Category
                      A ⇒ B means if
                      A is true then
       material       B is also true;
       implication    if A is false
                      then nothing is
                      said about B.   x = 2  ⇒ 
                                      x^2 = 4 is
                      → may mean the  true, but
  ⇒                   same as ⇒ (the  x^2 = 4        U+21D2
       implies; if .. symbol may also   ⇒  x = 2                 ⇒   \Rightarrow\
  →         then      indicate the    is in          U+2192      →    Rightarrow
                      domain and      general                     ⊃      \to\to
  ⊃                   codomain of a   false          U+2283              \supset\supset
                      function; see   (since x
                      table of        could be
                      mathematical    −2).
        propositional symbols).
       logic, Heyting
              algebra ⊃ may mean the
                      same as ⇒ (the
                      symbol may also
                      mean superset).
       material                                                                \
       equivalence                                                       Leftrightarrow
  ⇔                   A ⇔ B means A                  U+21D4                    \
        if and only   is true if and  x + 5 = y                  ⇔  Leftrightarrow
  ≡       if; iff     only if B is     +2  ⇔  x      U+2261      ≡  \equiv\equiv
                      true.            + 3 = y                   ↔        \
  ↔     propositional                                U+2194              leftrightarrow
                logic                                                          \
                                                                         leftrightarrow

       negation       The statement ¬
                      A is true if
                      and only if A
  ¬                   is false.
            not                       ¬(¬A) ⇔ A      U+00AC       ¬    \lnot\lnot
  ˜                   A slash placed  x ≠ y  ⇔                   ˜     ˜\sim
                      through another ¬(x =  y)      U+02DC         ~
  !                   operator is the
        propositional same as "¬"
                logic placed in
                      front.

       logical
  ∧    conjunction    The statement A n < 4  ∧ 
                      ∧ B is true if  n >2  ⇔  n     U+2227               \wedge\wedge
  •         and       A and B are      = 3 when                   &and;     or \land
                      both true; else n is a         U+0026       &amp;      \&^[1]
  &     propositional it is false.    natural
                logic                 number.

       logical        The statement A
       disjunction    ∨ B is true if  n ≥ 4  ∨ 
  ∨                   A or B (or      n ≤ 2  ⇔ n
             or       both) are true;  ≠ 3 when      U+2228       &or;      \lor\lor
  +                   if both are     n is a
        propositional false, the      natural
                logic statement is    number.
                      false.
       exclusive
       disjunction    The statement A (¬A) ⊕ A
                      ⊕ B is true     is always
  ⊕         xor       when either A   true, A ⊕      U+2295               \oplus\oplus
                      or B, but not   A is                       &oplus; \veebar\veebar
  ⊻     propositional both, are true. always         U+22BB
       logic, Boolean A ⊻ B means the false.
              algebra same.

       Tautology

  ⊤         top       The statement ⊤ A ⇒ ⊤ is
                      is              always         U+22A4         T       \top\top
  T     propositional unconditionally true.
       logic, Boolean true.
  1           algebra

       Contradiction

  ⊥        bottom     The statement ⊥ ⊥ ⇒ A is
                      is              always         U+22A5      &perp;     \bot\bot
  F     propositional unconditionally true.                         F
       logic, Boolean false.
  0           algebra

       universal
       quantification ∀ x: P(x) means
  ∀     for all; for  P(x) is true    ∀ n ∈ N: n     U+2200         &    \forall\forall
       any; for each  for all x.      ^2 ≥ n.                    forall;
            predicate
                logic
       existential    ∃ x: P(x) means
       quantification there is at     ∃ n ∈ N: n
  ∃     there exists  least one x     is even.       U+2203      &exist; \exists\exists
          first-order such that P(x)
                logic is true.
       uniqueness     ∃! x: P(x)
       quantification means there is  ∃! n ∈ N:
  ∃!    there exists  exactly one x   n + 5 = 2n  U+2203 U+0021  &exist;   \exists !\
        exactly one   such that P(x)  .                             !       exists !
          first-order is true.
                logic
                      x := y or x ≡ y
                      means x is
       definition     defined to be
                      another name    cosh x :=
                      for y (but note (1/2)(exp      U+2254                  : = :=
  :=                  that ≡ can also x + exp (− (U+003A U+003D)    :=    \equiv\equiv
                      mean other      x))                          : &         \
  ≡    is defined as  things, such as                U+2261      equiv;  Leftrightarrow
                      congruence).    A XOR B :⇔                 &hArr;        \
  :⇔                                  (A ∨ B)     U+003A U+229C          Leftrightarrow
                      P :⇔ Q means P   ∧ ¬(A ∧ B
                      is defined to   )
           everywhere be logically
                      equivalent to Q
                      .
       precedence     Perform the     (8/4)/2 =
       grouping       operations      2/2 = 1,
 ( )                  inside the      but 8/(4/   U+0028 U+0029    ( )      (~) ( )
                      parentheses     2) = 8/2 =
           everywhere first.          4.

       inference
        infers or is
        derived from  x ⊢ y means y   A → B ⊢ ¬B
  ⊢     propositional is derived from → ¬A           U+22A2               \vdash\vdash
              logic,  x.
          first-order
                logic

[edit] Advanced and rarely used logical symbols

These symbols are sorted by their Unicode value:

  • U+00B7 ·​ middle dot, an outdated way for denoting AND^[citation needed],
    still in use in electronics; for example "A·B" is the same as "A&B"
  • ·: Center dot with a line above it (using HTML style). Outdated way for
    denoting NAND, for example "A·B" is the same as "A NAND B" or "A|B" or "¬(A
    & B)". See also Unicode U+22C5 ⋅​ dot operator.

  • U+0305  ̅ ​ combining overline, used as abbreviation for standard numerals.
    For example, using HTML style "4̅" is a shorthand for the standard numeral
    "SSSS0".
  • Overline, is also a rarely used format for denoting Gödel numbers, for
    example "AVB" says the Gödel number of "(AVB)"
  • Overline is also an outdated way for denoting negation, still in use in
    electronics; for example "AVB" is the same as "¬(AVB)"

  • U+2191 ↑​ upwards arrow or U+007C |​ vertical line: Sheffer stroke, the
    sign for the NAND operator.
  • U+2201 ∁​ complement
  • U+2204 ∄​ there does not exist: strike out existential quantifier same as
    "¬∃"
  • U+2234 ∴​ therefore
  • U+2235 ∵​ because
  • U+22A7 ⊧​ models: is a model of
  • U+22A8 ⊨​ true: is true of
  • U+22AC ⊬​ does not prove: negated ⊢, the sign for "does not prove", for
    example T ⊬ P says "P is not a theorem of T"
  • U+22AD ⊭​ not true: is not true of
  • U+22BC ⊼​ nand: another NAND operator, can also be rendered as ∧
  • U+22BD ⊽​ nor: another NOR operator, can also be rendered as V
  • U+22C4 ⋄​ diamond operator: modal operator for "it is possible that", "it
    is not necessarily not" or rarely "it is not provable not" (in most modal
    logics it is defined as "¬◻¬")
  • U+22C6 ⋆​ star operator: usually used for ad-hoc operators
  • U+22A5 ⊥​ up tack or U+2193 ↓​ downwards arrow: Webb-operator or Peirce
    arrow, the sign for NOR. Confusingly, "⊥" is also the sign for
    contradiction or absurdity.

  • U+2310 ⌐​ reversed not sign

  • U+231C ⌜​ top left corner and U+231D ⌝​ top right corner: corner quotes,
    also called "Quine quotes"; the standard symbol used for denoting Gödel
    number; 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. And 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 ◻​ white medium square or U+25A1 □​ 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).

Note that the following operators are rarely supported by natively installed
fonts. If you wish to use these in a web page, you should always embed the
necessary fonts so the page viewer can see the web page without having the
necessary fonts installed in their computer.

  • U+27E1 ⟡​ white concave-sided diamond
  • U+27E2 ⟢​ white concave-sided diamond with leftwards tick: modal operator
    for was never
  • U+27E3 ⟣​ white concave-sided diamond with rightwards tick: modal operator
    for will never be
  • U+27E4 ⟤​ white square with leftwards tick: modal operator for was always
  • U+27E5 ⟥​ white square with rightwards tick: modal operator for will always
    be
  • 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) See here for an image of glyph. Added to
    Unicode 3.2.0.

[edit] See also

Logical connectives Hasse diagram.svg Logic portal

  • Table of mathematical symbols
  • Polish notation
  • Logic Alphabet, a suggested set of logical symbols.
  • Unicode Mathematical Operators

[edit] Notes

 1. ^ Although this character is available in LaTeX, the Mediawiki TeX system
    doesn't support this character.

[edit] External links

  • Named character entities in HTML 4.0.

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