Symbols:Logical Operators

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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:

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