 There are 4 logical relationships in any logical square or hexagon of opposition:
 Contraries can both be simultaneously false, but they cannot both be true at the same time
 Subcontraries can both be simultaneously true, but they cannot both be false at the same time
 The superaltern implies the subaltern in a subalteration relation but not vice versa
 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

 There are 2 modal operators in alethic modal logic:
 □ (Box) for necessity
 ◇ (Diamond) for possibility
 Where p denotes a propositional variable:
 □p ⟷ ∼◇∼p (necessity)
 ◇p ⟷ ∼□∼p (possibility)
 □p → ◇p (necessity implies possibility)
 ◇p ∧ ∼□p ⟷ ◇p ∧ ◇∼p (contingency)
 ◇p ∧ ∼□p ⟷ ∼(□p ⊻ ∼◇p) (contingency)

 Suppose that there are n possible worlds w_{1},w_{2},…, w_{n}
 We may denote the actual world @ in terms of w_{i}, where i ∈ ℕ and i ∈ [1, n]
 According to possible world semantics:
 To claim that p is necessarily true (formally: □p) is to assert of p that it is true in all possible worlds w_{1},w_{2},…, w_{n}
 To claim that p is possibly true (formally: ◇p) is to assert of p that it is true in at least one possible world
 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

 □p and ∼◇p are contraries
 ◇p and ∼□p are subcontraries
 □p is the superaltern and ◇p is its subaltern
 ∼◇p is the superaltern and ∼□p is its subaltern
 □p and ∼□p are contradictories
 ◇p and ∼◇p are contradictories
