Ernst Mally

Mally was the first logician ever to attempt an axiomatization of ethics (Mally 1926). 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.

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

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).

Theorem: This axiomatization of deontic logic implies that !x if and only if x is true, OR !x is unsatisfiable. (This makes it useless to deontic logicians.) Proof: Using axiom III, axiom I may be rewritten as (!(A → B) & (B → C)) → !(A → C). Since B → C holds whenever C holds, one immediate consequence is that (!(A → B) → (C → !(A → C))). In other words, if A requires B, it requires any true statement. In the special case where A is a tautology, the theorem has consequence (!B → (C → !C)). Thus, if at least one statement ought be true, every statement must materially entail it ought be true, and so every true statement ought be true. As for the converse (i.e. if some statement ought be true then all statements that ought be true are true), consider the following logic: ((U → !A) & (A → ∩)) → (U → !∩) is a special case of axiom I, but its consequent contradicts axiom V, and so ¬((U → !A) & (A → ∩)). The result !A → A can be shown to follow from this, since !A implies that U → !A and ¬A implies that A → ∩; and, since these are not both true, we know that !A → A.

Mally thought that axiom I was self-evident, but he likely confused it with an alternative in which the implication B → C is logical, which would indeed make the axiom self-evident. The theorem above, however, would then not be demonstrable. The theorem was proven by Karl Menger, the next deontic logician. Neither Mally's original axioms nor a modification that avoids this result remains popular today. Menger did not suggest his own axioms. (See also deontic logic for more on the subsequent development of this subject.)

In metaphysics, Mally is known for introducing a distinction between two kinds of predication, a strategy better known as the dual predication approach, for solving the problem of nonexistent objects (Mally 1912).[7] He also introduced a similar strategy, the dual property strategy, but did not endorse it.[2] The dual property strategy was eventually adopted by Meinong.[2]

Mally developed a realistic approach to ontology (Mally 1935) and saw himself in opposition to the Vienna Circle and the logical positivists.[1]

Mally's metaphysical work influences some contemporary metaphysicians and logicians working in abstract object theory, especially Edward Zalta.[9]

The analytic philosopher David Kellogg Lewis argued forcefully that the name of the fictional Australian poet Ern Malley, created by James McAuley and Harold Stewart, was an allusion to Mally.[10]