No, the thing in the post is three infinite conjunctions
Well yes, I meant any one of them is an infinite conjunction, which is the strange thing that some logics won’t permit.
For any triple of axiom sets (a,b,c), we can simply project those out from the OP’s axiom to prove my corresponding axiom.
This doesn’t work, for example if all three systems have tautology (1) as an axiom, let b=c=1 and let a be a singleton. Then at least all axioms of A become axioms in your system (edit: wrong), which clearly shouldn’t happen for E, and can’t be derived from that disjunction of infinite conjunctions. Like, X doesn’t follow from X∨Y.
This doesn’t work, for example if all three systems have tautology (1) as an axiom, let b=c=1 and let a be a singleton. Then at least all axioms of A become axioms in your system, which clearly shouldn’t happen for E
No. For each axiom a of A, “a or tautology” becomes an axiom. But “x or tautology” is itself a tautology for any x, so this isn’t a problem; “a or tautology” does not imply “a”.
Well yes, I meant any one of them is an infinite conjunction, which is the strange thing that some logics won’t permit.
This doesn’t work, for example if all three systems have tautology (1) as an axiom, let b=c=1 and let a be a singleton.Then at least all axioms of A become axioms in your system(edit: wrong), which clearly shouldn’t happen for E, and can’t be derived from that disjunction of infinite conjunctions. Like, X doesn’t follow from X∨Y.No. For each axiom a of A, “a or tautology” becomes an axiom. But “x or tautology” is itself a tautology for any x, so this isn’t a problem; “a or tautology” does not imply “a”.
That’s true. Sorry for not being careful, everything checks out. Something seems to have short-circuited in my mind.