# Symbols:Logical Operators

## And

- $\land$

And. A binary operation on two propositions.

$P \land Q$ means **$P$ is true and $Q$ is also true.**

The $\LaTeX$ code for \(\land\) is `\land`

or `\wedge`

.

Some $\LaTeX$ compilers allow `\and`

(the version of MathJax used on $\mathsf{Pr} \infty \mathsf{fWiki}$ does not).

In the context of propositional logic, on $\mathsf{Pr} \infty \mathsf{fWiki}$ `\land`

is **standard**.

See Vector Algebra: Deprecated Symbols and Group Theory for alternative definitions of this symbol.

## Or

- $\lor$

Or. A binary operation on two propositions.

$P \lor Q$ means **either $P$ is true or $Q$ is true, or both.**

Its technical term is **vel**.

The $\LaTeX$ code for \(\lor\) is `\lor`

or `\vee`

.

Some $\LaTeX$ compilers allow `\or`

(the MathJax used on $\mathsf{Pr} \infty \mathsf{fWiki}$ does not).

In the context of propositional logic, on $\mathsf{Pr} \infty \mathsf{fWiki}$ `\lor`

is **standard**.

## Not

- $\neg$

Not. A unary operator on a propositions.

$\neg Q$ means **not $Q$**, the logical opposite (negation) of $Q$.

The effect of the unary operator $\neg$ is to reverse the truth value of the statement following it.

The $\LaTeX$ code for \(\neg\) is `\neg`

or `\lnot`

.

## Nand

- $\uparrow$

Logical Nand. A binary operation on two propositions.

$P \uparrow Q$ means **not $P$ and $Q$ together**.

The symbol is named the **Sheffer stroke**, after Henry Sheffer.

The $\LaTeX$ code for \(\uparrow\) is `\uparrow`

.

## Nor

- $\downarrow$

Logical Nor. A binary operation on two propositions.

$P \downarrow Q$ means **neither $P$ nor $Q$**.

The symbol is named the **Quine arrow**, after Willard Quine.

The $\LaTeX$ code for \(\downarrow\) is `\downarrow`

.

## Deprecated Symbols

### And

- $\cdot$

And. A binary operation on two propositions.

$P \cdot Q$ means **$P$ is true and $Q$ is true**.

In this usage, it is called **dot**.

An alternative to $P \land Q$, which is what is usually used by logicians.

The $\LaTeX$ code for \(\cdot\) is `\cdot`

.

See Arithmetic and Algebra, Vector Algebra and Abstract Algebra for alternative definitions of this symbol.

- $\&$

Called **ampersand**.

The $\LaTeX$ code for \(\&\) is `\&`

.

In MediaWiki $\LaTeX$, its code is `\And`

.

### Or

- $+$

Or. A binary operation on two propositions.

$P + Q$ means **either $P$ is true or $Q$ is true or both.**

An alternative to $P \lor Q$, which is what is usually used by logicians.

The $\LaTeX$ code for \(+\) is `+`

.

See Arithmetic and Algebra, Vector Algebra and Group Theory for alternative definitions of this symbol.

### Not

- $-$

Not. A binary operation on two propositions.

$-Q$ means **$Q$ is not true.**

An alternative to $\neg$, which is what is usually used by logicians.

The $\LaTeX$ code for \(-\) is `-`

.

See Arithmetic and Algebra and Set Operations and Relations for alternative definitions of this symbol.

- $\sim$

The symbol $\sim$ is also sometimes used for Not.

The $\LaTeX$ code for \(\sim\) is `\sim`

.

### Nand

- $\mid$

Logical Nand. A binary operation on two propositions.

$P \mid Q$ means **not $P$ and $Q$ together**

This is also sometimes referred to as the **Sheffer stroke**.

The $\LaTeX$ code for \(\mid\) is `\mid`

.

- $p \bar \curlywedge q$

This is derived from the symbol used by Charles Sanders Peirce to denote the Logical Nor, sometimes called the ampheck.

The $\LaTeX$ code for \(\bar \curlywedge\) is `\bar \curlywedge`

.

### Nor

- $\curlywedge$

Logical Nor. A binary operation on two propositions.

$P \curlywedge Q$ means **neither $P$ nor $Q$**.

This is the symbol used by Charles Sanders Peirce to denote the Logical Nor, and is sometimes called the ampheck.

The $\LaTeX$ code for \(\curlywedge\) is `\curlywedge`

.

The usual ways of expressing **neither $p$ nor $q$** nowadays are:

- $\neg \left({p \lor q}\right)$

- $\overline {p \lor q}$

- $p \downarrow q$