Definition:Bottom (Logic)

Definition

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 $\bot$.

Boolean Interpretation

There is only one boolean interpretation for $\bot$:

$\map v \bot = \F$

where $\F$ symbolises false.

Truth Table

The characteristic truth table of the bottom constant $\bot$ of propositional logic is as follows:

$\begin{array}{|c|} \hline \bot \\ \hline \F \\ \hline \end{array}$

Also denoted as

Not all sources use the $\bot$ symbol. Some use $F$ or a stylistic variant, and others write it longhand as $\text{false}$.

Also see

• Definition:Contradiction

• Definition:Top (Logic)
• Definition:Tautology

Technical note

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

Sources

• 1993: M. Ben-Ari: Mathematical Logic for Computer Science ... (previous) ... (next): Chapter $2$: Propositional Calculus: $\S 2.3$: Boolean interpretations
• 2000: Michael R.A. Huth and Mark D. Ryan: Logic in Computer Science: Modelling and reasoning about systems ... (previous) ... (next): $\S 1.2.1$: Rules for natural deduction
• 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 2.3.3$