nLab falsehood

Idea

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.

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 intuitionistic logic

In intuitionistic logic, $false$ is the bottom element in the poset of truth values.

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

In natural deduction the inference rules for falsehood are given as

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \bot \; \mathrm{prop}} \quad \frac{\Gamma \vdash P \; \mathrm{prop}}{\Gamma, \bot \; \mathrm{true} \vdash P \; \mathrm{true}}$$

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 a topos

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

In type theory with propositions as types, falsehood is represented by the empty type.

In homotopy type theory

In homotopy type theory, falsehood is represented by the empty space.

Examples

In the topos $Set$

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.

Related concepts

• true, unit type

• empty type, ex falso quodlibet

• contradiction, inconsistency, consistency