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

The faleshood is commonly denoted $false$, $F$, $\bot$, or $0$. These may be pronounced ‘false’ even where it would be ungrammatical for an adjective to appear in ordinary English.

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

In linear logic

In linear logic, there is both additive falsity, denoted $0$, and multiplicative falsity, denoted $\bot$. Despite the notation, it is $0$ that is the bottom element of the lattice of linear truth values. (In particular, $0 \vdash \bot$ but $\bot \nvdash 0$.)

In the archetypical toposSet, the terminal object is the singletonset$*$ (the point) and the poset of subobjects of that is classically $\{\emptyset \hookrightarrow *\}$. Then falsehood is the empty set$\emptyset$, 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