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@@ -0,0 +1,27 @@ +Require Import Unicode.Utf8. + +Module No1. +Import Unicode.Utf8. + (*We first give the axioms of Principia +for the propositional calculus in *1.*) + +Axiom MP1_1 : ∀ P Q : Prop, + (P → Q) → P → Q. (*Modus ponens*) + + (**1.11 ommitted: it is MP for propositions containing variables. Likewise, ommitted the well-formedness rules 1.7, 1.71, 1.72*) + +Axiom Taut1_2 : ∀ P : Prop, P ∨ P→ P. (*Tautology*) + +Axiom Add1_3 : ∀ P Q : Prop, Q → P ∨ Q. (*Addition*) + +Axiom Perm1_4 : ∀ P Q : Prop, P ∨ Q → Q ∨ P. (*Permutation*) + +Axiom Assoc1_5 : ∀ P Q R : Prop, P ∨ (Q ∨ R) → Q ∨ (P ∨ R). + +Axiom Sum1_6: ∀ P Q R : Prop, (Q → R) → (P ∨ Q → P ∨ R). + (*These are all the propositional axioms of Principia Mathematica.*) + +Axiom Impl1_01 : ∀ P Q : Prop, (P → Q) = (~P ∨ Q). + (*This is a definition in Principia: there → is a defined sign and ∨, ~ are primitive ones. The purposes of giving this as an Axiom are two: first, to allow for the use of definitions in proofs, and second, to circumvent Coq's definitions of these primitive notions in Coq.*) + +End No1.
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