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    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 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: Deontic square of opposition in deontic logic
    1. There are 2 deontic operators in deontic logic:
    2. O for obligatoriness
    3. P for permissibility

    5. Where p denotes a propositional variable describing the performance of an action:
    6. ∼p means that the action described by p is not performed
    7. Op ⟷ ∼P∼p (obligatoriness)
    8. Pp ⟷ ∼O∼p (permissibility)
    9. Op → Pp (obligatoriness implies permissibility)
    10. Pp ∧ ∼Op ⟷ Pp ∧ P∼p (optionality)
    11. Pp ∧ ∼Op ⟷ ∼(Op ⊻ ∼Pp) (optionality)
    1. Suppose that we live in a home world h
    2. There are n morally acceptable alternative worlds w1,w2,…, wn relative to h

    4. According to possible world semantics:
    5. To claim that the action described by p is obligatory (formally: Op) is to assert of that action that it is performed in all morally acceptable alternative worlds w1,w2,…, wn
    6. To claim that the action described by p is permissible (formally: Pp) is to assert of that action that it is performed in at least one morally acceptable alternative world
    7. To claim that the action described by p is optional (formally: Pp ∧ ∼Op) is to assert of that action that it is performed in at least one morally acceptable alternative world, though not all of them
    1. Op and ∼Pp are contraries
    2. Pp and ∼Op are subcontraries
    3. Op is the superaltern and Pp is its subaltern
    4. ∼Pp is the superaltern and ∼Op is its subaltern
    5. Op and ∼Op are contradictories
    6. Pp and ∼Pp are contradictories
     

     

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