Some Illustrative Examples

• These are FOL theorems, but ...

  #

  Axioms

  ⊨

  Conjecture

  RM3

  A3

  FDE  →cmi

  FDE  →con

  1

  p, ¬p

  ⊨

  q

  No

  No

  No

  No

  2

  ⊨

  p → (q ∨ ¬q)

  No

  Yes

  No

  No

  3

  ⊨

  p ∨ ¬p

  Yes

  Yes

  No

  No

  4

  q

  ⊨

  p → q

  No

  Yes

  Yes

  Yes

  5

  ¬(p → q)

  ⊨

  p

  Yes

  Yes

  Yes

  No

  6

  ⊨

  ((p → q) ∧ (q → p)) → (p ∨ q ∨ ¬(p → q) ∨ ¬(q → p) ∨ ((¬p → ¬q) ∧ (¬q → ¬p)))

  Yes

  Yes

  No

  Yes

• Comments
  • #1 shows all are sublogics of FOL ... they are paraconsistent
  • If q is replaced by p in #2, it's an RM3 and FDE theorem ... a tinge of relevance
  • #3 confirms FDE is a sublogic of A3
  • #4 places RM3 apart from A3 and FDE
  • #5 and #6 show neither FDE^→_cmi nor FDE^→_con is a sublogic of the other