# 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 Maurice 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 Van Orman 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:

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