Interface talk:First-order logic

Ability to move subst across logical connectives
I'm moving the following from the main page to here as we don't need them yet, and don't prove them.

We can move  across logical connectives:(ref)sbn, sbor, sban, sbim, and sbbi in metamath's set.mm, accessed February 28, 2010(/ref)

 stmt (SubstNegation  ((subst y x (¬ φ)) ↔ (¬ (subst y x φ)))) stmt (SubstDisjunction  ((subst y x (φ ∨ ψ)) ↔ ((subst y x φ) ∨ (subst y x ψ)))) stmt (SubstConjunction  ((subst y x (φ ∧ ψ)) ↔ ((subst y x φ) ∧ (subst y x ψ)))) stmt (SubstImplication  ((subst y x (φ → ψ)) ↔ ((subst y x φ) → (subst y x ψ)))) stmt (SubstBiconditional  ((subst y x (φ ↔ ψ)) ↔ ((subst y x φ) ↔ (subst y x ψ)))) 

Kingdon 19:17, 1 March 2010 (UTC)

First-order logic without equality
In an email, User:GrafZahl asks me whether we should have first-order logic without equality. Although one of the nice things about the module/interface system is that differing theories can coexist on the wiki, first-order logic without equality presents at least two problems: The second is a show-stopper; the first is just an estimate of how important it is to address this. Kingdon 14:08, 27 April 2010 (UTC)
 * 1) It is not nearly as widely used as first-order logic with equality for purposes like being the basis for set theory, arithmetic, etc.
 * 2) JHilbert, metamath, and GHilbert are not currently able to handle it. First-order logic without equality still has proper substitution, but we don't currently have any way of defining proper substitution in any way other than in terms of equality.

ForAllAddRemove -> ForAllAddRemoveNotFree
We currently have a fairly extensive theory based on. This is powerful/beautiful enough that I'm not yet suggesting that we get rid of it (or move it out to another interface), but for practical purposes it seems like versions of theorems like  with distinct variable constraints, rather than explicit freeness hypotheses, are more useful. My proposed course of action is to rename the theorems with explicit freeness hypotheses to have "NotFree" at the end. For example,  to. Then the plain names, e.g.,, are available for the distinct variable versions. This parallels some of the naming conventions in metamath (they add "v" for the distinct version, e.g. exbid versus exbidv, or add "f" for the freeness hypothesis version). I'll hold off on a mass rename until people have had a little time to respond, but I plan on using this convention for newly proved theorems (and eventually get around to the mass rename, so we can be back to a consistent convention). Kingdon 15:11, 31 May 2010 (UTC)