nLab
false

Idea

In logic, the false proposition, called falsehood or falsity, is the proposition which is always false.

The faleshood is commonly denoted $\mathrm{false}$, $F$, $\perp$, or $0$.

Definitions

In classical logic

In classical logic, there are two truth values: false and true. Classical logic is perfectly symmetric between falsehood and truth; see de Morgan duality.

In constructive logic

In constructive logic, $\mathrm{false}$ is the bottom element in the poset of truth values.

Constructive logic is still two-valued in the sense that any truth value is false if it is not true.

In a topos

In terms of the internal logic of a topos (or other category), $\mathrm{false}$ is the bottom element in the poset of subobjects of any given object (where each object corresponds to a context in the internal language).

However, not every topos is two-valued, so there may be other truth values besides $\mathrm{false}$ and $\mathrm{true}$.

Examples

In the topos $\mathrm{Set}$

In the archetypical topos Set, the terminal object is the singleton set $*$ (the point) and the poset of subobjects of that is classically $\{\varnothing \hookrightarrow *\}$. Then falsehood is the empty set $\varnothing$, seen as the empty subset of the point. (See Internal logic of Set for more details).

The same is true in the archetypical (∞,1)-topos ∞Grpd. From that perspective it makes good sense to think of

• a set as a 0-truncated $\mathrm{\infty }$-groupoid: a 0-groupoid;

• a subsingleton set as a $(-1)$-truncated $\mathrm{\infty }$-groupoid: a (−1)-groupoid.

In this sense, the object $\mathrm{false}$ in Set or ∞Grpd may canonically be thought of as being the unique empty groupoid.

Related concepts

• true, unit type

• false, empty type

• contradiction, inconsistency