Theorem nndc 863

Description: Double negation of decidability of a formula. Intuitionistic logic refutes the negation of decidability (but does not prove decidability) of any formula.

This should not trick the reader into thinking that ¬ ¬ EXMID is provable in intuitionistic logic. Indeed, if we could quantify over formula metavariables, then generalizing nnexmid 862 over 𝜑 would give "⊢ ∀𝜑¬ ¬ DECID 𝜑", but EXMID is "∀𝜑DECID 𝜑", so proving ¬ ¬ EXMID would amount to proving "¬ ¬ ∀𝜑DECID 𝜑", which is not implied by the above theorem. Indeed, the converse of nnal 1702 does not hold. Since our system does not allow quantification over formula metavariables, we can reproduce this argument by representing formulas as subsets of 𝒫 1_o, like we do in our definition of EXMID (df-exmid 4332): then, we can prove ∀𝑥 ∈ 𝒫 1_o¬ ¬ DECID 𝑥 = 1_o but we cannot prove ¬ ¬ ∀𝑥 ∈ 𝒫 1_oDECID 𝑥 = 1_o because the converse of nnral 2540 does not hold.

Actually, ¬ ¬ EXMID is not provable in intuitionistic logic since intuitionistic logic has models satisfying ¬ EXMID and noncontradiction holds (pm3.24 705). (Contributed by BJ, 9-Oct-2019.) Add explanation on non-provability of ¬ ¬ EXMID. (Revised by BJ, 11-Aug-2024.)

Assertion

Ref	Expression
nndc	⊢ ¬ ¬ DECID 𝜑

Proof of Theorem nndc

Step	Hyp	Ref	Expression
1		nnexmid 862	. 2 ⊢ ¬ ¬ (𝜑 ∨ ¬ 𝜑)
2		df-dc 847	. . 3 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
3	2	notbii 678	. 2 ⊢ (¬ DECID 𝜑 ↔ ¬ (𝜑 ∨ ¬ 𝜑))
4	1, 3	mtbir 682	1 ⊢ ¬ ¬ DECID 𝜑

This proof depends on syntax axioms:  ¬ wn 3   ∨ wo 720  DECID wdc 846

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721

This proof depends on definitions:  df-bi 117  df-dc 847

This theorem is used by:  bj-dcdc  16787  bj-stdc  16788