# 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 \(P \land Q\) is `P \land Q`

or `P \wedge Q`

.

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 \(P \lor Q\) is `P \lor Q`

or `P \vee Q`

.

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 \(\neg Q\) is `\neg Q`

or `\lnot Q`

.

## 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 \(P \uparrow Q\) is `P \uparrow Q`

.

## 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 \(P \downarrow Q\) is `P \downarrow Q`

.

## 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 \(P \cdot Q\) is `P \cdot Q`

.

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

- $\&$

Called **ampersand**, which is an elision which evolved via an interesting linguistic process from **and per se and**, meaning

**and (the symbol $\&$) intrinsically (is the word) and**.

The symbol $\&$ itself evolved from the Latin **et** (for **and**).

$P \mathop \& Q$ means **$P$ is true and $Q$ is true**.

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

The $\LaTeX$ code for \(P \mathop \& Q\) is `P \mathop \& Q`

or `P \mathop \And Q`

.

### 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 \(P + Q\) is `P + Q`

.

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 \(- Q\) is `- Q`

.

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.

$\sim Q$ means **$Q$ is not true.**

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

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

.

### Nand

- $\mid$

Logical Nand. A binary operation on two propositions.

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

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

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

.

- $P \mathop {\bar \curlywedge} Q$

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

$P \mathop {\bar \curlywedge} Q$ means **not both $P$ and $Q$ together**

The $\LaTeX$ code for \(P \mathop {\bar \curlywedge} Q\) is `P \mathop {\bar \curlywedge} Q`

.

The usual ways of expressing **not both $P$ and $Q$ together** nowadays are:

- $\neg \left({P \land Q}\right)$

- $\overline {P \land Q}$

- $P \uparrow Q$

### 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 \(P \curlywedge Q\) is `P \curlywedge Q`

.

The usual ways of expressing **neither $P$ nor $Q$** nowadays are:

- $\neg \left({P \lor Q}\right)$

- $\overline {P \lor Q}$

- $P \downarrow Q$