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 Square 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 square of opposition in alethic modal logic
1. There are 2 modal operators in alethic modal logic:
2. □ (Box) for necessity
3. ◇ (Diamond) for possibility

4. Where p denotes a propositional variable:
5. □p ⟷ ∼◇∼p (necessity)
6. ◇p ⟷ ∼□∼p (possibility)
7. □p → ◇p (necessity implies possibility)
8. ◇p ∧ ∼□p ⟷ ◇p ∧ ◇∼p (contingency)
9. ◇p ∧ ∼□p ⟷ ∼(□p ⊻ ∼◇p) (contingency)
1. Suppose that there are n possible worlds w1,w2,…, wn
2. We may denote the actual world @ in terms of wi, where i ∈ ℕ and i ∈ [1, n]

3. According to possible world semantics:
4. To claim that p is necessarily true (formally: □p) is to assert of p that it is true in all possible worlds w1,w2,…, wn
5. To claim that p is possibly true (formally: ◇p) is to assert of p that it is true in at least one possible world
6. To claim that p is contingently true (formally: ◇p ∧ ∼□p) is to assert of p that it is true in at least one possible world, though not all of them
1. □p and ∼◇p are contraries
2. ◇p and ∼□p are subcontraries
3. □p is the superaltern and ◇p is its subaltern
4. ∼◇p is the superaltern and ∼□p is its subaltern
5. □p and ∼□p are contradictories
6. ◇p and ∼◇p are contradictories