Melvin Chen Logic

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Si les prémisses d'un syllogisme sont toutes les deux à l'indicatif,
la conclusion sera également à l'indicatif.
Pour que la conclusion pût être mise à l'impératif,
il faudrait que l'une des prémisses au moins fût elle-même à l'impératif.
- Henri Poincaré's (1913) 'La Morale et la Science' in the original French

If the premises are in the indicative mood,
then the conclusion will be in the indicative mood too.
For a conclusion in the imperative mood,
At least one premise in the imperative mood will be required.
- Personal translation

Deontic Logic (Makinson, Hansen)
David Makinson	P1: The assumed parallelism between imperative sentences and their indicative-parallel sentences does not exist. P2: There is no need to develop a separate imperative logic and we simply need to reinterpret deontic logic. Philosophers may defend the truth of P1 (contra Dubislav, Jørgensen, and Hare) in conjunction with P2 (contra Vranas) At the ΔEON '98 workshop on Deontic Logic in Computer Science in Bologna in 1998: David Makinson observed that work in deontic logic has been going on as if the distinction between norms (that can neither be true nor false) and normative propositions (that can be true or false) has never been heard of Makinson called for the development of deontic logic as a logic concerned with norms, while in accordance with the philosophical position that these norms are devoid of truth values
Deontic square of opposition in deontic logic	According to Hansen, deontic logic can be reinterpreted as a logic about imperatives: S1: {!A} ⊨ OA TRANSLATION of S1: If there is an imperative according to which A ought to be done, then the statement OA (for 'It ought to be that A') is true S2: {!A, !B} ⊨ O(A ∧ B) TRANSLATION of S2: If according to 2 imperatives, A and B ought to be done, then the statement O(A ∧ B) is true S3: {!A} ⊨ O(A ∨ B) TRANSLATION of S3: If there is an imperative according to which A ought to be done, then the statement O(A ∨ B) is true NOTE: While S3 gives rise to Ross's paradox (viz. 'It is obligatory that the letter is mailed or the letter is burnt'), there is no imperative in the set of mandatory imperatives on the L.H.S. of '⊨' that performing (A ∨ B) would satisfy S4: {!(A ∧ B)} ⊨ OA TRANSLATION of S4: If there is an imperative according to which A and B ought to be done, then the statement OA is true NOTE: While S4 gives rise to Weinberger's paradox (viz. 'Close the window and play the piano!'), there is no imperative in the set of mandatory imperatives on the L.H.S. of '⊨' that performing A alone would satisfy S5: {!(A ∧ B),!(∼A ∧ C)} ⊭ OB TRANSLATION of S5: In a conflict between imperatives (viz. (A ∧ B) and (∼A ∧ C)), the subject can still continue to reason about her obligations NOTE: It follows from S5 that a conflict does not make everything obligatory ⋮

Deontic Logic (Makinson, Hansen)

P1: The assumed parallelism between imperative sentences and their indicative-parallel sentences does not exist.
P2: There is no need to develop a separate imperative logic and we simply need to reinterpret deontic logic.
Philosophers may defend the truth of P1 (contra Dubislav, Jørgensen, and Hare) in conjunction with P2 (contra Vranas)
At the ΔEON '98 workshop on Deontic Logic in Computer Science in Bologna in 1998:
David Makinson observed that work in deontic logic has been going on as if the distinction between norms (that can neither be true nor false) and normative propositions (that can be true or false) has never been heard of
Makinson called for the development of deontic logic as a logic concerned with norms, while in accordance with the philosophical position that these norms are devoid of truth values

Deontic square of opposition in deontic logic

According to Hansen, deontic logic can be reinterpreted as a logic about imperatives:
S1: {!A} ⊨ OA
TRANSLATION of S1: If there is an imperative according to which A ought to be done, then the statement OA (for 'It ought to be that A') is true
S2: {!A, !B} ⊨ O(A ∧ B)
TRANSLATION of S2: If according to 2 imperatives, A and B ought to be done, then the statement O(A ∧ B) is true
S3: {!A} ⊨ O(A ∨ B)
TRANSLATION of S3: If there is an imperative according to which A ought to be done, then the statement O(A ∨ B) is true
NOTE: While S3 gives rise to Ross's paradox (viz. 'It is obligatory that the letter is mailed or the letter is burnt'), there is no imperative in the set of mandatory imperatives on the L.H.S. of '⊨' that performing (A ∨ B) would satisfy
S4: {!(A ∧ B)} ⊨ OA
TRANSLATION of S4: If there is an imperative according to which A and B ought to be done, then the statement OA is true
NOTE: While S4 gives rise to Weinberger's paradox (viz. 'Close the window and play the piano!'), there is no imperative in the set of mandatory imperatives on the L.H.S. of '⊨' that performing A alone would satisfy
S5: {!(A ∧ B),!(∼A ∧ C)} ⊭ OB
TRANSLATION of S5: In a conflict between imperatives (viz. (A ∧ B) and (∼A ∧ C)), the subject can still continue to reason about her obligations
NOTE: It follows from S5 that a conflict does not make everything obligatory

⋮

Deontic Logic (Makinson, Hansen)

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