# 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`

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### 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`

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### Implication

- $\implies$

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`

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### Biconditional

- $\iff$

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`

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### 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`

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### 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`

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### NOR

- $\downarrow$

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`

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### 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`

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### 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`

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## 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`

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### 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`

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### 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`

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### Implication

- $\to$

A binary operation on two propositions.

$P \to Q$ means **if $P$ is true, then $Q$ is true.**

An alternative to $P \implies Q$, which is what is generally used nowadays by logicians.

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

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### Biconditional

- $\leftrightarrow$

A binary operation on two propositions.

$P \leftrightarrow Q$ means **$P$ is true if and only if $Q$ is true.**

An alternative to $P \iff Q$, which is what is generally used nowadays by logicians.

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

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### 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`

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### 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`

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### 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`

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### 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`

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