Ernst Mally - Mally's Deontic Logic

Mally's Deontic Logic

Mally was the first ever logician to attempt an axiomatisation of ethics. He used five axioms, which are given below. They form a first-order theory that quantifies over propositions, and there are several predicates to understand first. !x means that x ought to be the case. Ux means that x is unconditionally obligatory, i.e. that !x is necessarily true. ∩x means that x is unconditionally forbidden, i.e. U(¬x). A f B is the binary relation A requires B, i.e. A materially implies !B. (All entailment in the axioms is material conditional.) It is defined by axiom III, whereas all other terms are defined as a preliminary.

\begin{array}{rl} \mbox{I.} & ((A\; \operatorname{f}\; B) \And (B \to C)) \to (A\; \operatorname{f}\; C) \\ \mbox{II.} & ((A\; \operatorname{f}\; B) \And (A\;\operatorname{f}\;C)) \to (A\; \operatorname{f}\; (B \And C)) \\ \mbox{III.} & (A\; \operatorname{f}\; B) \leftrightarrow\; !(A \to B) \\ \mbox{IV.} & \exists U\; !U \\ \mbox{V.} & \neg (U\; \operatorname{f}\; \cap) \end{array}

Note the implied universal quantifiers in the above axioms.

The fourth axiom has confused some logicians because its formulation is not as they would have expected, since Mally gave each axiom a description in words also, and he said that axiom IV meant "the unconditionally obligatory is obligatory", i.e. (as many logicians have insisted) UA → !A. Meanwhile, axiom 5 lacks an object to which the predicates apply, a typo. However, it turns out these are the least of Mally's worries (see below).

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