Symbols:Symbolic Logic

Symbols used in Symbolic Logic

Conjunction

$\land$

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.

Disjunction

$\lor$

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.

Logical Negation

$\neg$

Not.

$\neg Q$ 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

$\uparrow$

A binary operation on two propositions.

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

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 .

Top

$\top$

Top is a constant of propositional logic interpreted to mean the canonical, undoubted tautology whose truth nobody could possibly ever question.

The symbol used is $\top$.

The $\LaTeX$ code for $\top$ is \top .

Bottom

$\bot$

Bottom is a constant of propositional logic interpreted to mean the canonical, undoubted contradiction whose falsehood nobody could possibly ever question.

The symbol used is $\bot$.

The $\LaTeX$ code for $\bot$ is \bot .

Deprecated Symbols

This page contains symbols which may or may not be in current use, but are either non-standard in mathematics or have been superseded by their more modern variants.

Texts on logic often tend to use these symbols in preference to those used in mathematics.

However, on $\mathsf{Pr} \infty \mathsf{fWiki}$ the intention is to present a consistent style, and so these symbols are to be considered deprecated.

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 $P \land Q$, usually used by logicians.

The $\LaTeX$ code for $P \cdot Q$ is P \cdot Q .

Ampersand

$\&$

$P \mathop \& Q$ 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.

Disjunction

$+$

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 .

Minus

$-$

Not.

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

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

The $\LaTeX$ code for $-Q$ is -Q .

Tilde

$\sim$

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 .

Sheffer Stroke

$\mid$

A binary operation on two propositions.

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

This is known as the Sheffer stroke.

The $\LaTeX$ code for $P \mid Q$ is P \mid Q .

Modified Ampheck

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

A binary operation on two propositions.

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

The $\LaTeX$ code for $P \mathop {\bar \curlywedge} Q$ is P \mathop {\bar \curlywedge} Q .

Ampheck

$\curlywedge$

A binary operation on two propositions.

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

The $\LaTeX$ code for $P \curlywedge Q$ is P \curlywedge Q .