Mally was the first logician ever to attempt an axiomatization 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. A f B is the binary relation A requires B, i.e. A materially implies !B. It is defined by axiom III, whereas all other terms are defined as a preliminary. 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. 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.
Failure of Mally's deontic logic
Theorem: This axiomatization of deontic logic implies that !x if and only if x is true, OR !x is unsatisfiable. Proof: Using axiom III, axiom I may be rewritten as & ) → !. Since B → C holds whenever C holds, one immediate consequence is that. 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. 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, consider the following logic: & ) → is a special case of axiom I, but its consequent contradicts axiom V, and so ¬ & ). 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.
Metaphysics
In metaphysics, Mally is known for introducing a distinction between two kinds of predication, better known as the dual copula strategy for solving the problem of nonexistent objects. He also introduced a similar strategy, the dual property strategy, but did not endorse it. The dual property strategy was eventually adopted by Meinong. Mally developed a realistic approach to ontology and saw himself in opposition to the Vienna Circle and the logical positivists.
Works
Untersuchungen zur Gegenstandstheorie des Messens , Leipzig: Barth.
Gegenstandstheoretische Grundlagen der Logik und Logistik , Leipzig: Barth.
Grundgesetze des Sollens. Elemente der Logik des Willens , Graz: Leuschner & Lubensky. Reprinted in Ernst Mally: Logische Schriften. Großes Logikfragment—Grundgesetze des Sollens, K. Wolf, P. Weingartner, Dordrecht: Reidel, 1971, 227–324.
Erlebnis und Wirklichkeit. Einleitung zur Philosophie der Natürlichen Weltauffassung , Leipzig: Julius Klinkhardt.