nLab
true

Idea

In logic, the true proposition, or truth, is the proposition which is always true.

The truth is commonly denoted $\mathrm{true}$, $T$, $\top$, or $1$.

Definitions

In classical logic

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

In constructive logic

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

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

In a topos

In terms of the internal logic of a topos (or other category), $\mathrm{true}$ is the top 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 $\mathrm{true}$ and $\mathrm{false}$.

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 truth is the singleton set $\{*\}$, seen as the improper subset of itself. (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;

• the singleton set as the $(-2)$-truncated $\mathrm{\infty }$-groupoid: the unique (up to equivalence) (−2)-groupoid.

In this sense, the object $\mathrm{true}$ in Set or ∞Grpd may canonically be thought of as being the unique (−2)-groupoid.