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
Imperative Logic (Vranas)
Recall Hare's (1952) analysis in terms of phrastic (i.e. descriptive) and neustic (i.e. illocutionary) components:
P1 (indicative): You run.
TRANSLATION of P1: Your running, yes.
P2 (imperative): Run.
TRANSLATION of P2: Your running, please.
PROBLEM:
P3: If it rains, run.
TRANSLATION of P3 (according to Hare's model): Your running if it rains, please.
However, this TRANSLATION is incorrect: we are not betting that the indicative-parallel sentence 'If it rains, you run' is true
Rather, we are betting, on the condition that it rains, that you run
According to Vranas, the assumed parallelism between imperative sentences and their indicative-parallel sentences does not exist
∴ We need to develop a separate imperative logic
Each imperative sentence expresses a prescription (personal v. impersonal, single-agent v. multi-agent, conditional v. unconditional, synchronic v. diachronic, etc)
Each prescription has a satisfaction proposition and a violation proposition
P2 (imperative): Run.
Satisfaction proposition for P2: You run.
Violation proposition for P2: It is not the case that you run.
A prescription is avoided if the condition on which we are betting does not hold
P3 (imperative): If it rains, run.
P3 is satisfied if it rains (the condition on which we are betting holds) and you run
P3 is violated if it rains (the condition on which we are betting holds) and it is not the case that you run
P3 is avoided if it does not rain (the condition on which we are betting does not hold)
Let propositions be identified with sets (e.g. sets of possible worlds)
Let S denote the satisfaction set and let V denote the violation set
This allows us to introduce set-theoretic notation and talk about satisfaction, violation, and avoidance propositions in terms of satisfaction, violation, and avoidance sets
An unconditional prescription has no condition, although it has a context that is necessary
P2 (imperative): Run.
P2 has no condition, although it has a necessary context: either you run or you do not
∴ Avoidance propositions are impossible with unconditional prescriptions
∴ Satisfaction propositions and violation propositions are contradictories with unconditional prescriptions: the truth of one implies the falsity of the other
Otherwise, prescriptions are conditional
Avoidance propositions are possible with conditional prescriptions
Satisfaction and violation propositions are not contradictory with conditional prescriptions
The satisfaction, violation, and avoidance sets form a partition (e.g. the set of all relevant possible worlds) with conditional prescriptions
Each prescription may be specified in terms of its satisfaction and violation sets
The ordered pair 〈 S, V 〉 is the prescription expressed by 'If S or V is true, then let S be true'
The satisfaction proposition is (S ∨ V) ∧ S (i.e. S, since S and V are logically incompatible)
The violation proposition is (S ∨ V) ∧ ∼S (i.e. V, since S and V are logically incompatible)
Logical Connectives in Imperative Logic
Logical connective
Definition
Satisfaction table
Negation
∼〈 V, S 〉 ⟷ 〈 S, V 〉
NOTE: The law of double negation holds
∼(∼〈 V, S 〉) ⟷ ∼(〈 S, V 〉) ⟷ 〈 V, S 〉
Negated prescription (I = 〈 S, V 〉)
Sat
Av
Viol
Negation
(∼I = 〈 V, S 〉)
Viol
Av
Sat
Conjunction
V_{I ∧ I′} ⟷ V_{I} ⋃ V_{I′}
TRANSLATION: The violation set of the conjunction is the union of the violation sets of the conjuncts
AV_{I ∧ I′} ⟷ AV_{I} ⋂ AV_{I′}
TRANSLATION: The avoidance set of the conjunction is the intersection of the avoidance sets of the conjuncts
S_{I ∧ I′} ⟷ (S_{I} ⋃ S_{I′}) - (V_{I} ⋃ V_{I′)}
TRANSLATION: The conjunction is satisfied if at least one conjunct is satisfied and no conjunct is violated
More generally:
〈 S, V 〉 ⋀ 〈 S′, V′ 〉 = 〈(S ⋃ S′) - (V ⋃ V′), V ⋃ V′ 〉
Partial satisfaction table for conjunction I ∧ I′:
I
Sat
Av
Viol
I′
Sat
Sat
〈 Blank 〉
Viol
Av
〈 Blank 〉
Av
〈 Blank 〉
Viol
Viol
〈 Blank 〉
Viol
Let conjunction I ∧ I′ be used to denote 'If you love me, hug me and kiss me' (conditional)
I ∧ I′ is satisfied if both conjuncts are satisfied (i.e. you both hug me and kiss me)
I ∧ I′ is violated if at least one conjunct is violated (i.e. you don't kiss me, you don't hug me, or you neither kiss nor hug me)
I ∧ I′ is avoided if both conjuncts are avoided (i.e. if you don't love me)
NOTE: Since both conjuncts share the same context, there are no cases in which one conjunct is avoided but the other is satisfied or violated
∴ We have four 〈 Blank 〉 cells in our partial satisfaction table
Complete satisfaction table for conjunction I ∧ I′:
I
Sat
Av
Viol
I′
Sat
Sat
Sat
Viol
Av
Sat
Av
Viol
Viol
Viol
Viol
Viol
Let conjunction I ∧ I′ be used to denote 'If you love me, kiss me' and 'If you don't love me, kiss me (unconditional)
∴ I ∧ I′ effectively means 'kiss me' or 'kiss me whether or not you love me'
We can complete the satisfaction table for a conjunction:
If one of the two conjuncts is avoided, then the conjunction is satisfied if the other conjunct is satisfied
Conversely, the conjunction is violated if the other conjjunct is violated
Disjunction
〈 S, V 〉 ⋁ 〈 S′, V′ 〉 = 〈 S ⋃ S′, (V ⋃ V′) - (S ⋃ S′) 〉
I
Sat
Av
Viol
I′
Sat
Sat
Sat
Sat
Av
Sat
Av
Viol
Viol
Sat
Viol
Viol
Let disjunction I ⋁ I′ be used to denote 'kiss me or hug me'
I ⋁ I′ is satisfied if at least one disjunct is satisfied (i.e. you kiss me or you hug me)
I ⋁ I′ is violated if both disjuncts are violated (i.e. you neither kiss me nor hug me)
If one of the disjuncts is avoided, then I ⋁ I′ is satisfied if the other disjunct is satisfied