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.
| Symbol | Unicode value (hexadecimal) | HTML codes | LaTeX symbol | Logic Name | Read as | Category | Explanation | Examples | |||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| scope"row" align="center" | ⇒ | U+21D2 U+2192 U+2283 | ⇒ → ⊃⇒ → ⊃ | ⇒ \implies \to \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 → ⇒ (the symbol may also indicate the domain and codomain of a function; see table of mathematical symbols). \supset ⇒ | scope"row" align="center" | x=2 ⇒ x2=4 x2=4 ⇒ x=2 (since could be −2). | |||
| scope"row" align="center" | ⇔ | U+21D4 U+2194 U+2261 | ⇔ ↔ ≡⇔ ↔ ≡ | \Leftrightarrow \iff \leftrightarrow \equiv | material biconditional (material equivalence) | if and only if, iff, xnor | propositional logic, Boolean algebra | A\LeftrightarrowB | scope"row" align="center" | x+5=y+2\Leftrightarrowx+3=y | |||
| scope"row" align="center" | ¬ | U+00AC U+007E U+0021 | ¬ ˜ !¬ ˜ ! | \neg \sim | negation | not | propositional logic, Boolean algebra | The statement lnotA A slash placed through another operator is the same as \neg | scope"row" align="center" | \neg(\negA)\LeftrightarrowA x ≠ y\Leftrightarrow\neg(x=y) | |||
| scope"row" align="center" | ∧ | U+2227 U+00B7 U+0026 | ∧ · &∧ · & | \wedge ⋅ \& | logical conjunction | and | propositional logic, Boolean algebra | The statement A ∧ B is true if A and B are both true; otherwise, it is false. | scope"row" align="center" | when n is a natural number. | |||
| scope"row" align="center" | ∨ | U+2228 U+002B U+2225 | ∨ + ∥∨ + ∥ | \lor \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. | scope"row" align="center" | when n is a natural number. | |||
| scope"row" align="center" | ⊕ | U+2295 U+22BB U+21AE U+2262 | ⊕ ⊻ ↮ ≢⊕ ⊻ — ≢ | ⊕ \veebar \not\equiv | exclusive disjunction | xor, either ... or ... (but not both) | propositional logic, Boolean algebra | The statement A ⊕ B ¬(A ↔ B), hence the symbols \nleftrightarrow \not\equiv | scope"row" align="center" | lnotA ⊕ A A ⊕ A | |||
| scope"row" align="center" | ⊤ | U+22A4 | ⊤ ⊤ | \top | true (tautology) | top, truth, tautology, verum, full clause | propositional logic, Boolean algebra, first-order logic | \top | scope"row" align="center" | The proposition \top\lorP | |||
| scope"row" align="center" | ⊥ | U+22A5 | ⊥⊥ | \bot | false (contradiction) | bottom, falsity, contradiction, falsum, empty clause | propositional logic, Boolean algebra, first-order logic | \bot The symbol ⊥ may also refer to perpendicular lines. | scope"row" align="center" | The proposition \bot\wedgeP | |||
| scope"row" align="center" | ∀ | U+2200 | ∀∀ | \forall | universal quantification | given any, for all, for every, for each, for any | first-order logic | \forallx P(x) (x) P(x) x x P | scope"row" align="center" | \foralln\isinN:n2\geqn. | |||
| scope"row" align="center" | ∃ | U+2203 | ∃∃ | \exists | existential quantification | there exists, for some | first-order logic | \existsx P(x) x P | scope"row" align="center" | \existsn\isinN: | |||
| scope"row" align="center" | ∃! | U+2203 U+0021 | ∃ !∃ | \exists | \exists ! | uniqueness quantification | there exists exactly one | first-order logic (abbreviation) | \exists!x P(x) \forall \exists \exists | x P(x) \existsx\forally(P(y)\leftrightarrowy=x) | scope"row" align="center" | \exists!n\isinN:n+5=2n. | |
| scope"row" align="center" | U+0028 U+0029 | ( )( ) | (~) | precedence grouping | parentheses; brackets | almost all logic syntaxes, as well as metalanguage | Perform the operations inside the parentheses first. | scope"row" align="center" | , but . | ||||
| scope"row" align="center" | D | U+1D53B | 𝔻𝔻 | \mathbb | domain of discourse | domain of discourse | metalanguage (first-order logic semantics) | scope"row" align="center" | D:R | ||||
| scope"row" align="center" | ⊢ | U+22A2 | ⊢⊢ | \vdash | turnstile | syntactically entails (proves) | metalanguage (metalogic) | A\vdashB B a theorem of A In other words, A B | scope"row" align="center" | (A → B)\vdash(lnotB → lnotA) (eg. by using natural deduction) | |||
| scope"row" align="center" | ⊨ | U+22A8 | ⊨⊨ | \vDash | double turnstile | semantically entails | metalanguage (metalogic) | A\vDashB “in every model, it is not the case that A B | scope"row" align="center" | (A → B)\vDash(lnotB → lnotA) (eg. by using truth tables) | |||
| scope"row" align="center" | ≡ | U+2261 U+27DA U+21D4 | ≡— ⇔≡—⇔ | \equiv \Leftrightarrow | logical equivalence | is logically equivalent to | metalanguage (metalogic) | It’s when A\vDashB B\vDashA | scope"row" align="center" | (A → B)\equiv(lnotA\lorB) | |||
| scope"row" align="center" | ⊬ | U+22AC | ⊬\nvdash | does not syntactically entail (does not prove) | metalanguage (metalogic) | A\nvdashB B not a theorem of A In other words, B A | scope"row" align="center" | A\lorB\nvdashA\wedgeB | |||||
| scope"row" align="center" | ⊭ | U+22AD | ⊭\nvDash | does not semantically entail | metalanguage (metalogic) | A\nvDashB A B In other words, A B | scope"row" align="center" | A\lorB\nvDashA\wedgeB | |||||
| scope"row" align="center" | □ | 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. | scope"row" align="center" | \Box\forallxP(x) | ||||
| scope"row" align="center" | ◇ | 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”). | scope"row" align="center" | \Diamond\existsxP(x) | ||||
| scope"row" align="center" | ∴ | U+2234 | ∴\therefore | therefore | therefore | metalanguage | abbreviation for “therefore”. | ||||||
| scope"row" align="center" | ∵ | U+2235 | ∵\because | because | because | metalanguage | abbreviation for “because”. | ||||||
| scope"row" align="center" | ≔ | U+2254 U+225C U+225D | ≔≔ | := = \triangleq \stackrel{\scriptscriptstyledef \stackrel |