User interface:Kingdon/Logic

Part of the experiment into rejigging quantifiers (from a universe of sets to a more restricted universe).

Interface:Classical propositional calculus, Interface:First-order logic with quantifiability, Interface:Peano axioms.  param (CLASSICAL Interface:Classical_propositional_calculus ) param (FIRSTORDER Interface:First-order_logic_with_quantifiability (CLASSICAL) set.) param (SETS Interface:Set_theory (CLASSICAL FIRSTORDER) )

var (formula φ ψ χ θ)

kindbind (set.object object) kindbind (set.variable variable)
 * 1) Here we are defining the kinds and terms of first-order logic

var (variable a b c x y z) var (object A B C X Y Z)

def ((value a) (set.value a)) def ((= X Y) (X set.= Y))

def ((universe) (unorderedPair (∅) (singleton (∅))))

def ((∀ x φ) (set.∀ x (((value x) ∈ (universe)) → φ))) def ((∃ x φ) (¬ (∀ x (¬ φ)))) def ((≠ X Y) (¬ (X = Y))) def ((is-not-free-in x φ) (φ → (∀ x φ))) 

To think about: Is there a need to justify/prove these statements now that they are fairly simple?

To think about: the result should satisfy all the axioms of Interface:Axioms of first-order logic. Seems like this could probably be proved, but maybe believing it to be possible would suffice for now. I don't think we need that to just get something past JHilbert, though (since statements are not read from params). Also on the subject of axioms versus the full Interface:First-order logic with quantifiability is whether we need to mention  and.