Editing Deontic logic (section)

==Standard deontic logic==
In [[Georg Henrik von Wright]]'s first system, obligatoriness and permissibility were treated as features of ''acts''.  Soon after this, it was found that a deontic logic of ''propositions'' could be given a simple and elegant [[Kripke semantics|Kripke-style semantics]], and von Wright himself joined this movement. The deontic logic so specified came to be known as "standard deontic logic," often referred to as '''SDL''', '''KD''', or simply '''D'''. It can be axiomatized by adding the following axioms to a standard axiomatization of classical [[propositional logic]]:

: <math>(\models A) \rightarrow (\models OA)</math>
: <math>O(A \rightarrow B) \rightarrow (OA \rightarrow OB)</math>
: <math>OA\to PA</math>

In English, these axioms say, respectively:

* If A is a [[Tautology (logic)|tautology]], then it ought to be that A (necessitation rule '''N'''). In other words, [[Contradiction|contradictions]] are not permitted.
* If it ought to be that A implies B, then if it ought to be that A, it ought to be that B (modal axiom '''K''').
* If it ought to be that A, then it is permitted that A (modal axiom '''D'''). In other words, if it's not permitted that A, then it's not obligatory that A.

''FA'', meaning it is forbidden that ''A'', can be defined (equivalently) as <math>O \lnot A</math> or <math>\lnot PA</math>.

There are two main extensions of '''SDL''' that are usually considered. The first results by adding an [[Alethic modality|alethic modal operator]] <math>\Box</math> in order to express the [[Kant]]ian claim that "[[ought implies can]]":

: <math> OA \to \Diamond A. </math>

where <math>\Diamond\equiv\lnot\Box\lnot</math>. It is generally assumed that <math>\Box</math> is at least a '''KT''' operator, but most commonly it is taken to be an '''S5''' operator. In practical situations, obligations are usually assigned in anticipation of future events, in which case alethic possibilities can be hard to judge. Therefore, obligation assignments may be performed under the assumption of different conditions on different [[Temporal logic#Motivation|branches of timelines]] in the future, and past obligation assignments may be updated due to unforeseen developments that happened along the timeline.

The other main extension results by adding a "conditional obligation" operator O(A/B) read "It is obligatory that A given (or conditional on) B". Motivation for a conditional operator is given by considering the following ("Good Samaritan") case. It seems true that the starving and poor ought to be fed. But that the starving and poor are fed implies that there are starving and poor. By basic principles of '''SDL''' we can infer that there ought to be starving and poor! The argument is due to the basic K axiom of '''SDL''' together with the following principle valid in any [[normal modal logic]]:

:<math>\vdash A\to B\Rightarrow\ \vdash OA\to OB.</math>

If we introduce an intensional conditional operator then we can say that the starving ought to be fed ''only on the condition that there are in fact starving'': in symbols O(A/B). But then the following argument fails on the usual (e.g. Lewis 73) semantics for conditionals: from O(A/B) and that A implies B, infer OB.

Indeed, one might define the unary operator O in terms of the binary conditional one O(A/B) as <math>OA\equiv O(A/\top)</math>, where <math>\top</math> stands for an arbitrary [[tautology (logic)|tautology]] of the underlying logic (which, in the case of '''SDL''', is classical). 

=== Semantics of standard deontic logic ===
The [[accessibility relation]] between possible world is interpreted as ''acceptability'' relations: <math>v</math> is an acceptable world (viz. <math>wRv</math>) if and only if all the obligations in <math>w</math> are fulfilled in <math>v</math> (viz. <math>(w\models OA)\to (v\models A)</math>).

=== Anderson's deontic logic ===
Alan R. Anderson (1959) shows how to define <math>O</math> in terms of the alethic operator <math>\Box</math> and a deontic constant (i.e. 0-ary modal operator) <math>s</math> standing for some sanction (i.e. bad thing, prohibition, etc.): <math>OA\equiv\Box(\lnot A\to s)</math>. Intuitively, the right side of the biconditional says that A's failing to hold necessarily (or strictly) implies a sanction.

In addition to the usual modal axioms (necessitation rule '''N''' and distribution axiom '''K''') for the alethic operator <math>\Box</math>, Anderson's deontic logic only requires one additional axiom for the deontic constant <math>s</math>: <math>\neg \Box s\equiv \Diamond \neg s</math>, which means that there is alethically possible to fulfill all obligations and avoid the sanction. This version of the Anderson's deontic logic is equivalent to '''SDL'''.

However, when modal axiom '''T''' is included for the alethic operator (<math>\Box A\to A</math>), it can be proved in Anderson's deontic logic that <math>O(OA \to A)</math>, which is not included in '''SDL'''. Anderson's deontic logic inevitably couples the deontic operator <math>O</math> with the alethic operator <math>\Box</math>, which can be problematic in certain cases.