Symbols:Symbolic Logic

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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.

A unary operator on a proposition.


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page contains symbols which may or may not be in current use, but are either non-standard in mathematics or have been superseded by their more modern variants.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

A unary operation on a proposition.


$-Q$ 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.

A unary operation on a proposition.


$\sim Q$ 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

$\mid$

Logical NAND.

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

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

Logical NAND.

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

$\curlywedge$

Logical NOR.

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 .