module 1Lab.Classical where

# The law of excluded middle🔗

While we do not assume any classical principles in the 1Lab, we can still state them and explore their consequences.

The **law of excluded middle** (LEM) is the defining
principle of classical logic, which states that any proposition is
either true or false (in other words, decidable). Of course,
assuming this as an axiom requires giving up canonicity: we could prove
that, for example, any Turing machine either halts or does not halt, but
this would not give us any computational information.

LEM : Type LEM = ∀ (P : Ω) → Dec ∣ P ∣

Note that we cannot do without the assumption that $P$ is a proposition: the statement that all types are decidable is inconsistent with univalence.

An equivalent statement of excluded middle is the **law of
double negation elimination** (DNE):

DNE : Type DNE = ∀ (P : Ω) → ¬ ¬ ∣ P ∣ → ∣ P ∣

We show that these two statements are equivalent propositions.

LEM-is-prop : is-prop LEM LEM-is-prop = hlevel 1 DNE-is-prop : is-prop DNE DNE-is-prop = hlevel 1 LEM→DNE : LEM → DNE LEM→DNE lem P = Dec-elim _ (λ p _ → p) (λ ¬p ¬¬p → absurd (¬¬p ¬p)) (lem P) DNE→LEM : DNE → LEM DNE→LEM dne P = dne (el (Dec ∣ P ∣) (hlevel 1)) λ k → k (no λ p → k (yes p)) LEM≃DNE : LEM ≃ DNE LEM≃DNE = prop-ext LEM-is-prop DNE-is-prop LEM→DNE DNE→LEM

## Weak excluded middle🔗

The **weak law of excluded middle** (WLEM) is a slightly
weaker variant of excluded middle which asserts that every proposition
is either false or not false.

WLEM : Type WLEM = ∀ (P : Ω) → Dec (¬ ∣ P ∣)

As the name suggests, the law of excluded middle implies the weak law of excluded middle.

LEM→WLEM : LEM → WLEM LEM→WLEM lem P = lem (P →Ω ⊥Ω)

The weak law of excluded middle is also a proposition.

WLEM-is-prop : is-prop WLEM WLEM-is-prop = hlevel 1

## The axiom of choice🔗

The **axiom of choice** is a stronger classical
principle which allows us to commute propositional truncations past Π
types.

Axiom-of-choice : Typeω Axiom-of-choice = ∀ {ℓ ℓ'} {B : Type ℓ} {P : B → Type ℓ'} → is-set B → (∀ b → is-set (P b)) → (∀ b → ∥ P b ∥) → ∥ (∀ b → P b) ∥

Like before, the assumptions that $A$ is a set and $P$ is a family of sets are required to avoid running afoul of univalence.

An equivalent and sometimes useful statement is that all surjections
between sets merely have a section. This is essentially jumping to the
other side of the `fibration equivalence`

.

Surjections-split : Typeω Surjections-split = ∀ {ℓ ℓ'} {A : Type ℓ} {B : Type ℓ'} → is-set A → is-set B → (f : A → B) → is-surjective f → ∥ (∀ b → fibre f b) ∥

We show that these two statements are logically equivalent^{1}.

AC→Surjections-split : Axiom-of-choice → Surjections-split AC→Surjections-split ac Aset Bset f = ac Bset (fibre-is-hlevel 2 Aset Bset f) Surjections-split→AC : Surjections-split → Axiom-of-choice Surjections-split→AC ss {P = P} Bset Pset h = ∥-∥-map (Equiv.to (Π-cod≃ (Fibre-equiv P))) (ss (Σ-is-hlevel 2 Bset Pset) Bset fst λ b → ∥-∥-map (Equiv.from (Fibre-equiv P b)) (h b))

We can show that the axiom of choice implies the law of excluded
middle; this is sometimes known as the Diaconescu-Goodman-Myhill
theorem^{2}.

Given a proposition $P,$ we consider the suspension of $P:$ the type $ΣP$ is a set with two points and a path between them if and only if $P$ holds.

Since $ΣP$ admits a surjection from the booleans, the axiom of choice merely gives us a section $ΣP→2.$

module _ (split : Surjections-split) (P : Ω) where section : ∥ ((x : Susp ∣ P ∣) → fibre 2→Σ x) ∥ section = split (hlevel 2) (Susp-prop-is-set (hlevel 1)) 2→Σ 2→Σ-surjective

But a section is always injective, and the booleans are discrete, so we can prove that $ΣP$ is also discrete. Since the path type $N≡S$ in $ΣP$ is equivalent to $P,$ this concludes the proof.

Discrete-ΣP : Discrete (Susp ∣ P ∣) Discrete-ΣP = ∥-∥-rec (Dec-is-hlevel 1 (Susp-prop-is-set (hlevel 1) _ _)) (λ f → Discrete-inj (fst ∘ f) (right-inverse→injective 2→Σ (snd ∘ f)) Discrete-Bool) section AC→LEM : Dec ∣ P ∣ AC→LEM = Susp-prop-path (hlevel 1) <≃> Discrete-ΣP

they are also propositions, but since they live in

`Typeω`

we cannot easily say that↩︎not to be confused with Diaconescu’s theorem in topos theory↩︎