Section Weak_proof_irrelevance_CCI.
Hypothesis wem : forall A:Prop, ~~A \/ ~ A.
End Weak_proof_irrelevance_CCI.
(** Remark: in the Set-impredicative CCI, Hurkens' paradox still holds with [bool] in [Set] and since [~true=false] for [true] and [false] in [bool] from [Set], we get the inconsistency of [em : forall A:Prop, {A}+{~A}] in the Set-impredicative CCI. *)
(** * Weak classical axioms *)
(** We show the following increasing in the strength of axioms: - weak excluded-middle - right distributivity of implication over disjunction and Gödel-Dummett axiom - independence of general premises and drinker's paradox - excluded-middle *)
(** ** Weak excluded-middle *)
(** The weak classical logic based on [~~A \/ ~A] is referred to with name KC in [[ChagrovZakharyaschev97]]  [[ChagrovZakharyaschev97]] Alexander Chagrov and Michael Zakharyaschev, "Modal Logic", Clarendon Press, 1997. *)
Definition weak_excluded_middle :=
forall A:Prop, ~~A \/ ~A.
(** The interest in the equivalent variant [weak_generalized_excluded_middle] is that it holds even in logic without a primitive [False] connective (like Gödel-Dummett axiom) *)
Definition weak_generalized_excluded_middle :=
forall A B:Prop, ((A -> B) -> B) \/ (A -> B).
(** ** Gödel-Dummett axiom *)
(** [(A->B) \/ (B->A)] is studied in [[Dummett59]] and is based on [[Gödel33]].  [[Dummett59]] Michael A. E. Dummett. "A Propositional Calculus with a Denumerable Matrix", In the Journal of Symbolic Logic, Vol 24 No. 2(1959), pp 97-103.  [[Gödel33]] Kurt Gödel. "Zum intuitionistischen Aussagenkalkül", Ergeb. Math. Koll. 4 (1933), pp. 34-38. *)
Definition GodelDummett := forall A B:Prop, (A -> B) \/ (B -> A).
(** [(A->B) \/ (B->A)] is equivalent to [(C -> A\/B) -> (C->A) \/ (C->B)] (proof from [[Dummett59]]) *)
Definition RightDistributivityImplicationOverDisjunction :=
forall A B C:Prop, (C -> A\/B) -> (C->A) \/ (C->B).
(** [(A->B) \/ (B->A)] is stronger than the weak excluded middle *)
(** ** Independence of general premises and drinker's paradox *)
(** Independence of general premises is the unconstrained, non constructive, version of the Independence of Premises as considered in [[Troelstra73]].  It is a generalization to predicate logic of the right distributivity of implication over disjunction (hence of Gödel-Dummett axiom) whose own constructive form (obtained by a restricting the third formula to be negative) is called Kreisel-Putnam principle [[KreiselPutnam57]].  [[KreiselPutnam57]], Georg Kreisel and Hilary Putnam. "Eine Unableitsbarkeitsbeweismethode für den intuitionistischen Aussagenkalkül". Archiv für Mathematische Logik und Graundlagenforschung, 3:74- 78, 1957.  [[Troelstra73]], Anne Troelstra, editor. Metamathematical Investigation of Intuitionistic Arithmetic and Analysis, volume 344 of Lecture Notes in Mathematics, Springer-Verlag, 1973. *)
Definition IndependenceOfGeneralPremises :=
forall (A:Type) (P:A -> Prop) (Q:Prop),
inhabited A -> (Q -> exists x, P x) -> exists x, Q -> P x.
Lemma
independence_general_premises_right_distr_implication_over_disjunction :
IndependenceOfGeneralPremises -> RightDistributivityImplicationOverDisjunction.
(** Independence of general premises is equivalent to the drinker's paradox *)
Definition DrinkerParadox :=
forall (A:Type) (P:A -> Prop),
inhabited A -> exists x, (exists x, P x) -> P x.
(** Independence of general premises is weaker than (generalized) excluded middle  Remark: generalized excluded middle is preferred here to avoid relying on the "ex falso quodlibet" property (i.e. [False -> forall A, A]) *)
Definition generalized_excluded_middle :=
forall A B:Prop, A \/ (A -> B).

Lemma excluded_middle_independence_general_premises :
generalized_excluded_middle -> DrinkerParadox.