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.

Implication

$\implies$

Implies.

A binary operation on two propositions.

$P \implies Q$ means if $P$ is true, then $Q$ is true.

The $\LaTeX$ code for \(P \implies Q\) is P \implies Q .

Biconditional

$\iff$

Biconditional.

A binary operation on two propositions.

$P \iff Q$ means $P$ is true if and only if $Q$ is true.

The $\LaTeX$ code for \(P \iff Q\) is P \iff 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 .

Therefore

$\vdash$

If statement $p$ logically implies statement $q$, then we may say:

$p$, therefore $q$

and denote it:

$p \vdash q$

The $\LaTeX$ code for \(p \vdash q\) is p \vdash q .

Necessity

$\nec$

The necessity operator $\nec$ is the unary modal operator defined for some proposition $P$ dependent on some world $w$:

$\nec P : \iff \forall w: \map P w$

The $\LaTeX$ code for \(\nec\) is \nec .

Possibility

$\nec$

The possibility operator $\pos$ is the unary modal operator defined for some proposition $P$ dependent on some world $w$:

$\pos P : \iff \exists w: \map P w$

The $\LaTeX$ code for \(\nec\) is \nec .

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.

Therefore

$\therefore$

Therefore

If statement $p$ logically implies statement $q$, then we may say:

$p$, therefore $q$

and denote it:

$p \therefore q$

An alternative to $p \vdash q$, the preferred notation on $\mathsf{Pr} \infty \mathsf{fWiki}$.

The $\LaTeX$ code for \(p \therefore q\) is p \therefore q .

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 .

Arrow

$\to$

Implies:

An alternative to $P \implies Q$.

The $\LaTeX$ code for \(P \to Q\) is P \to Q .

Hook

$\supset$

Implies.

An alternative to $P \implies Q$.

The $\LaTeX$ code for \(P \supset Q\) is P \supset Q .

Double Arrow

$\leftrightarrow$

Biconditional.

An alternative to $P \iff Q$.

The $\LaTeX$ code for \(P \leftrightarrow Q\) is P \leftrightarrow Q .

Identity

$\equiv$

Biconditional.

An alternative to $P \equiv Q$.

The $\LaTeX$ code for \(P \equiv Q\) is P \equiv 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 .