Imperative logic

Imperative logic is a field of logic that is concerned with imperatives (e.g. "Do A"). Unlike deontic logic, which is concerned with obligation and permission, imperative logic by itself is not necessarily tied to any "ought" judgments on its imperatives. However, the forms of symbolic imperative logic can help to reformulate deontic logic in a way that avoids some of the difficulties of conditional obligations, as was first examined by Hector Castañeda, and later taken up by Harry J. Gensler in his system of formal ethics.

As a symbolic extension

As an extension to propositional or predicate logic, imperative logic adds a single additional symbol, an underline on a proposition (or an agent of a predicate) to convert it from an indicative to a corresponding imperative. For example, if we take the symbol $A$ to mean the indicative "You eat an apple", then $underline\{A\}$ means the imperative "Eat an apple". It is not entirely clear what the imperative version of a proposition that does not have an obvious agent would be — if $S$ means "It is sunny outside", what does $underline\{S\}$ mean? (e.g. "Be sunny outside"?) Without further qualification, therefore, some formulae in imperative logic don't have meaningful (or at least obvious) translations into natural language (although in the belief logic that forms part of his formal ethics, Gensler would give meaning to a formula such as $u:underline\{S\}$, namely the "desire" "would that it was sunny outside"). When used as an extension to predicate logic, the agent is underlined to form an imperative, so if $Dux$ means "You give a dollar to x" then $Dunderline\{u\}x$ is the correct way to express "Give a dollar to x".

One way in which formal imperative logic is useful is that it can express certain "conditional imperatives" that are clumsy to express in natural language. For example if $H$ is "You do your homework" and $T$ is "You watch TV", then you can easily express not only::$eg\; H\; o\; eg\; underline\{T\}$"If you don't do your homework, then don't watch TV" but also the equivalent:$underline\{T\}\; o\; H$which might be rendered by the (somewhat clumsier, and perhaps ambiguous) "Watch TV, only if you do your homework". As another example, consider a conditional where both the antecedent and consequent are imperatives::$underline\{T\}\; o\; underline\{H\}$This has no translation into natural language ("If watch TV, then do you your homework"???), and again by itself may not be meaningful (but Gensler uses formulae such as this with the modal operator $square$ — $square(underline\{T\}\; o\; underline\{H\})$ = "Watch TV "entails" do your homework").

Connection to deontic logic

An often cited example of something that is difficult to express in some formulations of deontic logic is "If you smoke, then you ought to use an ashtray". If the deontic operators $O$ and $P$ only attach to indicative formulae, then it is not clear that either of the following representations is adequate::$O(mathrm\{smoke\}\; o\; mathrm\{ashtray\})$:$mathrm\{smoke\}\; o\; O(mathrm\{ashtray\})$However, by attaching the deontic operators to imperative formulae, we have unambiguously:$O(mathrm\{smoke\}\; o\; underline\{mathrm\{ashtray)$

Interpretation

Imperatives could be interpreted as demands, or as preferences. The choice of interpretation is intimately connected with the "correct" interpretation of the negation of an imperative::Demand: $eg\; underline\{A\}$ = "You may omit doing A":Preference: $eg\; underline\{A\}$ = "Don't do A" (this is the interpretation that is taken in the examples above)The first is also known as the "modal" view, and the second the "anti-modal" view.

Gensler suggests taking the preference interpretation, as we may augment the grammar with an operator $M$ which means "may", e.g. $M\; underline\{A\}$ = "You may do A" and $eg\; M\; eg\; underline\{A\}$ = "You may not omit doing A" = "You must do A", whereas it would be more difficult to unambiguously augment the demand interpretation with another operator to express preferences. Note that Gensler's $M$ is different from the deontic $P$ "permissible" operator, as "You must do A" is still an imperative, without any "ought" judgment (i.e. not the same as "You ought to do A").

Validity of arguments

Imperative logic requires a different metalogical definition for the validity of an argument, because imperatives are, by nature, not true or false. The following definition won't work:

* An argument is defined to be "valid" iff it would be logically impossible for the premises to be true and the conclusion to be false.

The following definition has been suggested:

* An argument is defined to be "valid" iff combining the premises with the negation of the conclusion would be "logically inconsistent".

The definition of logical consistency depends on the axioms and rules of a particular imperative logic system.

Note that this difficulty is not shared with deontic logic, since a deontic statement (e.g. "You ought to do A") could still be considered to be true or false in some sense.