Illustrative Examples

These are FOL theorems, but ...
FDE→
# Axioms  ⊨  Conjecture Ł3 RM3 A3 Łuk cmi
1    ⊨  p ∨ ¬p NO YES YES NO NO
2  ⊨  (p ∧ ¬p) → q NO NO NO NO NO
3 p, ¬p  ⊨  q YES NO NO NO NO
4 q  ⊨  p → q YES NO YES NO YES
5  ⊨  ((p → q) ∧ (q → p)) →
(p ∨ q ∨ ¬(p → q) ∨ ¬(q → p) ∨
((¬p → ¬q) ∧ (¬q → ¬p))) 		YES YES YES YES NO
6  ⊨  ¬∃y∀x (E(x,y) ↔ ¬E(x,x)) NO YES YES NO NO
  1. Glut logics prove tautologies, gap logics do not
  2. FOL ⊃ with extra truth values and conditional
  3. Ł3 is not paraconsistent
  4. →Łuk is different from →cmi
  5. →cmi is different from →Łuk
  6. No set whose members are not members of themselves