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

## Modal 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: Modal hexagon of opposition in alethic modal logic
1. We add 2 additional nodes to our modal square of opposition to form a logical hexagon:
2. It is not-contingent that p ⟷ □p ⊻ ∼◇p (p is either necessary or impossible but not both)
3. It is contingent that p ⟷ ◇p ∧ ∼□p (p is possible and non-necessary)
1. □p and ∼◇p are contraries
2. □p and (◇p ∧ ∼□p) are contraries
3. ∼◇p and (◇p ∧ ∼□p) are contraries
4. ◇p and ∼□p are subcontraries
5. ◇p and (□p ⊻ ∼◇p) are subcontraries
6. (□p ⊻ ∼◇p) and ∼□p are subcontraries
7. □p is the superaltern and ◇p is its subaltern
8. ∼◇p is the superaltern and ∼□p is its subaltern
9. □p is the superaltern and (□p ⊻ ∼◇p) is its subaltern
10. ∼◇p is the superaltern and (□p ⊻ ∼◇p) is its subaltern
11. (◇p ∧ ∼□p) is the superaltern and ◇p is its subaltern
12. (◇p ∧ ∼□p) is the superaltern and ∼□p is its subaltern
13. □p and ∼□p are contradictories
14. ◇p and ∼◇p are contradictories
15. (◇p ∧ ∼□p) and (□p ⊻ ∼◇p) are contradictories