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.
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, $false$ 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 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 terms of the internal logic of a topos (or other category), $false$ 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 $false$ and $true$.
In type theory with propositions as types, falsehood is represented by the empty type.
In homotopy type theory, falsehood is represented by the empty space.
In the archetypical topos Set, the terminal object is the singleton set $*$ (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
a set as a 0-truncated $\infty$-groupoid: a 0-groupoid;
a subsingleton set as a $(-1)$-truncated $\infty$-groupoid: a (-1)-groupoid.
In this sense, the object $false$ in Set or ∞Grpd may canonically be thought of as being the unique empty groupoid.
Last revised on January 26, 2023 at 08:48:22. See the history of this page for a list of all contributions to it.