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

## Parallelism (Dubislav, Jørgensen, Hare)

In Ernst Mally's (1926) The Basic Laws of Ought: Elements of the Logic of Willing, a formal system of deontic logic was proposed

## Symbolic Notation in Mally's (1926) System of Deontic Logic

!A '!' is a unary connective
'!A' means that 'A ought to be the case' or 'let A be the case'
NOTE: '!A' is now called a deontic proposition and formalized as 'OA'
A f B 'f' is a binary connective
'A f B' means that 'A requires B'

A f B ⟷ A → !B
A ∞ B '∞' is a binary connective
'A ∞ B' means that 'A and B require each other'

A ∞ B ⟷ (A f B) ∧ (B f A)
U 'U' is a sentential constant
'U' means 'the unconditionally obligatory'

U ⟷ ∼∩
'∩' is a sentential constant
'∩' means 'the unconditionally forbidden'

∩ ⟷ ∼U

1. According to Walter Dubislav, we could transfer the formalisms of indicative inferences to imperatives by using a 'trick' (Kunstgriff):
2. STEP 1: Imagine the state that the utterer of the sentence in the imperative mood (expressing commands or requests) desires realized;
3. STEP 2: Describe that state;
4. STEP 3: From this description, infer some other indicative sentence;
5. STEP 4: Interpret this other indicative sentence as describing a state that the commanding authority desires realized

1. EXAMPLE:
2. Consider INFERENCE 1:
3. P1: Thou shalt not kill.
4. C: ∴ Cain shalt not kill Abel.

5. Let P1 be denoted by !A (using Mally's notation)

6. INFERENCE 2:
7. P1′: No human being kills any other human being. (!A is transformed into A) — STEPS 1 & 2
8. P2: Cain and Abel are human beings. (2nd member of premise set)
9. C′: ∴ Cain does not kill Abel. (from P1 & P2) — STEP 3

10. Let C′ be denoted by B
11. B (or C′ of INFERENCE 2) may be transformed into !B (or C of INFERENCE 1) — STEP 4

'Roughly, we understand an imperative to be satisfied if what is commanded is the case. Thus the fiat "Let the door be closed!" is satisfied if the door is closed. It will be seen that the satisfaction of an imperative is analogous to the truth of a sentence.'
— Hofstadter & McKinsey (1939, p. 447)

1. Dubislav's Convention (for a purely imperative premise set with 1 member):
2. (DC) 'An imperative F is called derivable from an imperative E if the descriptive sentence belonging to F is derivable with the usual methods from the descriptive sentence belonging to E, whereby the identity of the commanding authority is assumed'

3. Dubislav's Modified Convention (for a purely imperative premise set with >1 member):
4. (DCM) 'An imperative F is called derivable from the imperatives E1, …,En if the descriptive sentence belonging to F is derivable with the usual methods from the descriptive sentences belonging to E1, …,En, whereby the identity of the commanding authority is assumed'

5. Dubislav's Extended Convention (for a mixed premise set with 1 imperative premise):
6. (DEC) 'An imperative F is called derivable in the extended sense from an imperative E if the descriptive sentence belonging to F is at least jointly derivable from the descriptive sentence belonging to E and true descriptive sentences that are consistent with the first, whereby the identity of the commanding authority is assumed'

7. Dubislav's Modified Extended Convention (for a mixed premise set with >1 imperative premise):
8. (DECM) 'An imperative F is called derivable in the extended sense from the imperatives E1, …,En if the descriptive sentence belonging to F is at least jointly derivable from the descriptive sentence belonging to E1, …,En and true descriptive sentences that are consistent with these, whereby the identity of the commanding authority is assumed'

1. Let !A denote an imperative sentence
2. A will denote its indicative-parallel sentence
3. Let !B denote another imperative sentence
4. B will denote its indicative-parallel sentence

5. ## Transformations under Dubislav's Convention

Inference in ordinary logic w.r.t. A & B (indicatives) Transformation Analogue inference w.r.t. !A & !B (imperatives)

⊢ ((A ∨ B) ∧ ∼A) → B

⊢ ((!A ∨ !B) ∧ !∼A) → !B

'If we command someone to use an axe or a saw, and then not to use an axe, we command him to use a saw; if we say that he will use an axe or a saw, and then that he will not use an axe, we say that he will use a saw' (Hare, 1949, p. 31)

⊢ A → (A ∨ B)

⊢ !A → !(A ∨ B)

This yields a paradox of obligation that is known as Ross's (1941) paradox

1. P1: Mail the letter.
2. C: ∴ Mail the letter or burn the letter.

⊢ (A ∧ B) → A

⊢ !(A ∧ B) → !A

This yields a paradox of obligation that is known as the paradox of the window (Stranzinger, 1978, Weinberger, 1991)

1. P1: Close the window and play the piano.
2. C: ∴ Close the window.

3. Alternatively:
4. P1: Close the window and play the piano.
5. C: ∴ Play the piano.

1. According to Jørgensen, the dilemma that bears his name may be resolved by analyzing imperative sentences into 2 factors:
1. FACTOR 1: An imperative factor — expressing the speaker's state of mind (commanding, willing, requesting, wishing, etc) and of no logical consequence;
2. FACTOR 2: An indicative factor — formulable as an indicative sentence

2. ∴ Any imperative sentence has an indicative-parallel sentence that describes the contents of the command or request
3. ∴ The ordinary rules of logic will be valid for these indicative-parallel sentences
4. ∴ There would be no reason for constructing a specific logic of imperatives

1. Hare's (1949, 1952) refinement of Jørgensen's analysis in terms of the descriptor/dictor and phrastic/neustic distinctions:
2. An imperative sentence and its indicative-parallel sentence correspond if they have the same descriptor but different dictors:
3. What is described by the descriptor is what would be the case if the sentence is true or the command is obeyed
4. This is the phrastic component in the sentence (i.e. its descriptive part)
5. The dictor is what does the saying or commanding
6. This is the neustic component in the sentence (i.e. its illocutionary part)

7. EXAMPLE (Hare, 1952):

8. P1 (indicative): You run.
9. P2 (imperative): Run.

10. P1 may be translated as 'Your running, yes'