Deontic Logic - Standard Deontic Logic

Standard Deontic Logic

In von Wright's first system, obligatoriness and permissibility were treated as features of acts. It was found not much later that a deontic logic of propositions could be given a simple and elegant 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:

In English, these axioms say, respectively:

• If it ought to be that A implies B, then if it ought to be that A, it ought to be that B;
• If A is permissible, then it is not the case that it ought not to be that A.

FA, meaning it is forbidden that A, can be defined (equivalently) as or .

There are two main extensions of SDL that are usually considered. The first results by adding an alethic modal operator in order to express the Kantian claim that "ought implies can":

where . It is generally assumed that is at least a KT operator, but most commonly it is taken to be an S5 operator.

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:

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, where stands for an arbitrary tautology of the underlying logic (which, in the case of SDL, is classical). Similarly Alan R. Anderson (1959) shows how to define O in terms of the alethic operator and a deontic constant (i.e. 0-ary modal operator) s standing for some sanction (i.e. bad thing, prohibition, etc.): . Intuitively, the right side of the biconditional says that A's failing to hold necessarily (or strictly) implies a sanction.