- 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
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FIG: Modal square of opposition in alethic modal logic
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- 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)
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- Suppose that there are n possible worlds w1,w2,…, wn
- We may denote the actual world @ in terms of wi, 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 w1,w2,…, wn
- 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
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- □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
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