Die Logik ist keine Lehre, sondern ein Spiegelbild der Welt.
Logik ist transzendental.

- Ludwig Wittgenstein's (1921, 6.13) Tractatus Logico-Philosophicus in the original German

Logic is not a body of doctrine, but a mirror-reflection of the world.
Logic is transcendental.
- Pears/McGuinness translation

## Deontic Hexagon of Opposition

1. There are 4 logical relationships in any logical square or hexagon of opposition:
2. Contraries can both be simultaneously false, but they cannot both be true at the same time
3. Subcontraries can both be simultaneously true, but they cannot both be false at the same time
4. The superaltern implies the subaltern in a subalteration relation but not vice versa
5. Where 2 statements are contradictories, the truth of one statement implies the falsity of the other

FIG: Deontic hexagon of opposition in deontic logic
1. In non-standard deontic logic, 2 upper-case letters are used to stand for each normative status, as opposed to the single upper-case letter in standard deontic logic (SDL):
2. OBp (obligatoriness)
3. PEp (permissibility)
4. OPp (optionality)

5. We add 2 additional nodes to our deontic square of opposition to form a logical hexagon:
6. ∼OPp ⟷ OBp ⊻ ∼PEp (the action described by p is either obligatory or impermissible but not both)
7. OPp ⟷ PEp ∧ ∼OBp (the action described by p is permissible and non-obligatory)
1. OBp and ∼PEp are contraries
2. OBp and OPp are contraries
3. ∼PEp and OPp are contraries
4. PEp and ∼OBp are subcontraries
5. PEp and ∼OPp are subcontraries
6. ∼OBp and ∼OPp are subcontraries
7. OBp is the superaltern and PEp is its subaltern
8. ∼PEp is the superaltern and ∼OBp is its subaltern
9. OBp is the superaltern and ∼OPp is its subaltern
10. ∼PEp is the superaltern and ∼OPp is its subaltern
11. OPp is the superaltern and PEp is its subaltern
12. OPp is the superaltern and ∼OBp is its subaltern
13. OBp and ∼OBp are contradictories
14. PEp and ∼PEp are contradictories
15. OPp and ∼OPp are contradictories