Logic Symbols

A comprehensive collection of the most notable symbols in formal/mathematical logic, categorized by function into tables along with each symbol's meaning and example.

In philosophy and mathematics, logic plays a key role in formalizing valid deductive inferences and other forms of reasoning. The following is a comprehensive list of the most notable symbols in logic, featuring symbols from propositional logic, predicate logic, Boolean logic and modal logic.

For readability purpose, these symbols are categorized by their function into tables. Other comprehensive lists of symbols — as categorized by subject and type — can be also found in the relevant pages below (or in the navigational panel).

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Constants

In logic, constants are often used to denote definite objects in a logical system. The following table features the most notable of these — along with their respective example and meaning.

Symbol NameExplanationExample
$a, b, c$General constants
(within a logical system)
$b \ge a_1 + a_2$
$\mathbb{B}$Boolean domainIn Boolean logic, $\mathbb{B} = \{ 0 ,1\}$.
$\top$
(or $1$ in Boolean logic)
Tautology,
Truth value ‘true’
$P \lor \lnot P \equiv \top$
$\bot$
(or $0$ in Boolean logic)
Contradiction,
Truth value ‘false’
$Q \land \lnot Q \equiv \bot$

Variables

Similar to other fields in mathematics, variables are used as placeholder symbols for varying entities in logic. The following table documents the most notable of these — along with their respective example and meaning.

Symbol NameExplanationExample
$x, y, w, z$Quantification variables$x_1 + x_2 = y$
$\mathbf{x}, \mathbf{y}, \mathbf{w}, \mathbf{z}$Metavariables for quantification variablesFor all variables $\mathbf{x}_1$ and $\mathbf{x}_2$, ‘$\mathbf{x}_1 = \mathbf{x}_2$’ is a formula.
$f, g, h$Function symbols$h\left( f_1(x), g(x, y) \right)$
$\mathbf{s}, \mathbf{t}$Metavariables for termsFor all terms $\mathbf{t}_1$ and $\mathbf{t}_2$, ‘$f(\mathbf{t}_1, \mathbf{t}_2)$’ is a term.
$P, Q, R$Propositional / Predicate symbols$P(x, a) \land Q_1(z)$
$\alpha, \beta, \gamma, \phi, \psi$Metavariables for formulasFor all formulas $\alpha$ and $\beta$, $\alpha \land \beta \equiv \beta \land \alpha$.
$\Sigma, \Phi, \Psi$Metavariables for set of sentencesIf $\Sigma$ is inconsistent, then so is $\Sigma \cup \Phi$,
$\mathcal{L}$Metavariable for formal languagesIf $\mathcal{L}$ is a language with equality and constant $a$, then ‘$a = a$’ is a formula in $\mathcal{L}$.

Operators

Operators are symbols used to denote mathematical operations, which serve to take one or multiple inputs to a similar output. In logic, these operators include logical connectives from propositional/modal logic, quantifiers from predicate logic, as well as other operators related to syntactic substitution and semantic valuation.

Unary Logical Connectives

Symbol NameExplanationExample
$\lnot P$, $\sim\!\!P$, $\overline{P}$Negation of $P$
(not $P$)
$\lnot \lnot P \equiv P$
$\Diamond P$Possibly $P$If $\Diamond P$, then $\Diamond \Diamond P$.
$\Box P$Necessarily $P$If $\Box P$, then $\neg \Diamond \neg P$.

Binary Logical Connectives

Symbol NameExplanationExample
$P \land Q$Conjunction
($P$ and $Q$)
$P \land P \equiv P$
$P \lor Q$Disjunction
($P$ or $Q$)
$\neg (P \lor Q) \equiv \\ \neg P \land \neg Q$
$P \veebar Q$, $P \oplus Q$Exclusive disjunction
($P$ xor $Q$)
$P \oplus Q \equiv \\ (P \lor Q) \land \\ \neg(P \land Q)$
$P \uparrow Q$Negation of conjunction
($P$ nand $Q$)
$P \uparrow Q \equiv \neg (P \land Q)$
$P \downarrow Q$Negation of disjunction
($P$ nor $Q$)
$P \downarrow Q \equiv \\ (\neg P \land \neg Q)$
$P \to Q$Conditional
(If $P$, then $Q$)
For all $P$, $P \to P$ is a tautology.
$P \not\to Q$Non-conditional
(Not ‘if $P$, then $Q$’)
$P \not\to Q \equiv P \land \neg Q$
$P \leftarrow Q$Converse conditional
(If $Q$, then $P$)
$Q \leftarrow (P \land Q)$
$P \not\leftarrow Q$Converse non-conditional
(Not ‘if $Q$, then $P$’)
$(P \to Q) \land \\ (P \not\leftarrow Q)$
$P \leftrightarrow Q$Biconditional
($P$ if and only if $Q$)
$P \leftrightarrow Q \equiv \\ (P \to Q) \land \\ (P \leftarrow Q)$
$P \not\leftrightarrow Q$Non-biconditional
(Not ‘$P$ if and only if $Q$’)
If $P \not\to Q$, then $P \not\leftrightarrow Q$.

Generalized Logical Connectives

Symbol NameExplanationExample
$\displaystyle \bigwedge_{i=m}^n P_i$Generalized conjunction
($P_m \land \cdots \land P_n$)
$\displaystyle \bigwedge_{i=1}^n [i \ne (i+1)]$
$\displaystyle \bigvee_{i=m}^n P_i$Generalized disjunction
($P_m \lor \cdots \lor P_n$)
$\displaystyle \neg \left(\bigvee_{i=1}^n P_i \right) = \bigwedge_{i=1}^n \neg P_i$

Quantifiers

Symbol NameExplanationExample
$\forall \mathbf{x}$Universal quantification
(For all $\mathbf{x}$)
$\forall x >0 \, (e^x > 1)$
$\exists \mathbf{x}$Existential quantification
(There exists $\mathbf{x}$)
$\exists x \forall y \, (x^2 \le y^2)$
$\exists ! \mathbf{x}$Uniqueness quantification
(There is a unique $\mathbf{x}$)
$\exists !\, q, r \in \mathbb{Z}\, \\ (n=dq+r \land \\ 0 \le|r|<d)$
$\mathrm{N} \mathbf{x}$, $\nexists \mathbf{x}$Non-existence quantification
(There is no $\mathbf{x}$)
$\mathrm{N}x P(x) \equiv \\ \forall x \, \neg P(x)$
$\exists_n \mathbf{x}$Numerical quantification
(There are exactly $n$ $\mathbf{x}$)
$\exists_3 x \in \mathbb{Z}\, (5 < x < 9)$
$\exists_{\ge n} \mathbf{x}$Numerical quantification
(There are at least $n$ $\mathbf{x}$)
$\exists_{\ge 2} x \, Q(x) \equiv \\ \exists x \exists y \, (Q(x) \land Q(y) \land x \ne y)$
$\exists_{\le n} \mathbf{x}$Numerical quantification
(There are at most $n$ $\mathbf{x}$)
$\exists_{\le 10} x \, (x^2 \le 100) \equiv \\ \neg \left(\exists_{\ge 11} x \, (x^2 \le 100)\right)$

Substitution-based Operators

Symbol NameExplanationExample
$\mathbf{t}[\mathbf{x}/\mathbf{t}_0]$Substituted term
(term $\mathbf{t}$ with occurences of $\mathbf{x}$ replaced by $\mathbf{t}_0$)
$(x^2 + y)[x/1][y/5] = 1^2 + 5$
$\mathbf{\alpha}[\mathbf{x}/\mathbf{t_0}]$Substituted formula
(formula $\mathbf{\alpha}$ with free occurrences of $\mathbf{x}$ replaced by term $\mathbf{t_0}$)
$(\forall x (x = y)) [x/a] = \\ \forall x (x = y)$

Valuation-based Operators

Symbol NameExplanationExample
$\mathbf{t}^{\sigma}$Referent of term $\mathbf{t}$ under valuation $\sigma$$\left(f(a,b)\right)^{\sigma} = \mathrm{father}(\mathrm{Al}, \mathrm{Bob})$
$\alpha^{\sigma}$Truth value of formula $\alpha$ under valuation $\sigma$If $P^{\sigma}$ is a symmetric relation, then $\left(P(x, y)\right)^{\sigma} = \\ \left(P(y, x)\right)^{\sigma}$
$\sigma (\mathbf{x}/u)$Revaluation of $\sigma$ where variable $\mathbf{x}$ is revalued as $u$$(\forall x \, \alpha)^{\sigma} = \top$ if and only if for all $u$ in the universe of discourse $U$, $\alpha^{\sigma (x/u)} = \top$.

Relational Symbols

In logic, relational symbols play a key role in turning one or multiple mathematical entities into formulas and propositions, and can occur both within a logical system or outside of it (as metalogical symbols). The following table documents the most notable of these symbols — along with their respective meaning and example.

Symbol NameExplanationExample
$\mathbf{t}_1 = \mathbf{t}_2$Identity symbol in a logical system with equality‘$\neg \left(1 = s(1) \right)$’ is a formula in the language of first-order arithmetic.
$\alpha \! \implies \! \beta$Sentence $\alpha$ implies sentence $\beta$$\forall x \, (x \ge 1) \! \implies \\ (1 \ge 1)$
$\alpha \! \impliedby \! \beta$Sentence $\alpha$ is implied by sentence $\beta$$5 \mid x \! \impliedby \! 5 \mid 7x$
$\alpha \equiv \beta$, $\alpha \Leftrightarrow \beta$, $\alpha \! \iff \! \beta$Sentence $\alpha$ and $\beta$ are logically equivalent$\neg (P \to Q) \equiv \\ P \land \neg Q$
$\sigma \models \alpha$Valuation $\sigma$ satisfies formula $\alpha$If $\phi^{\sigma} = \top$, then $\sigma \models \phi$.
$\Phi \models \phi$Set of sentences $\Phi$ entails sentence $\phi$
($\phi$ is a logical consequence of $\Phi$)
If $\Phi \models \phi$, then $\Phi \cup \Psi \models \phi$.
$\Phi \nvDash \phi$Set of sentences $\Phi$ does not entail sentence $\phi$$\{P \to Q, Q \to R \} \nvDash R$
$\models \phi$Sentence $\phi$ is a tautology$\models \forall x \, (x = x)$
$\Phi \vdash \phi$Set of sentences $\Phi$ proves sentence $\phi$$\forall x \, P(x,a) \vdash \\ P(a,a)$
$\Phi \nvdash \phi$Set of sentences $\Phi$ does not prove sentence $\phi$$\exists x \, R(x) \nvdash R(a)$
$\vdash \phi$Sentence $\phi$ is a theorem$\vdash \forall x \forall y \, (x=y \to \\ y=x)$
$\phi \because \Phi$$\phi$, because $\Phi$$A = 95^{\circ} \because \\ A+B=180^{\circ}, \\ B=85^{\circ}$
$\Phi \therefore \phi$$\Phi$, therefore $\phi$$P \lor Q, \neg P \\ \therefore Q$

For the master list of symbols, see mathematical symbols. For lists of symbols categorized by type and subject, refer to the relevant pages below for more.

Cover of Math Vault's Comprehensive List of Mathematical Symbols eBook

Prefer the PDF version instead?

Get the master summary of mathematical symbols in eBook form — along with each symbol’s usage and LaTeX code.

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