Ernst Mally (1879 - 1944) was an Austrian philosopher who studied under Alexius Meinong. He was one of the founders of deontic logic and is mainly known for his contributions in that field of research. 1879 (MDCCCLXXIX) was a common year starting on Wednesday (see link for calendar). ... 1944 (MCMXLIV) was a leap year starting on Saturday. ... A philosopher is a person who thinks deeply regarding people, society, the world, and/or the universe. ... Alexius Meinong Alexius Meinong (July 17, 1853 - November 27, 1920) was an Austrian philosopher. ... Deontic logic is the field of logic that is concerned with obligation, permission, and related concepts. ...

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. First-order predicate calculus or first-order logic (FOL) is a theory in symbolic logic that permits the formulation of quantified statements such as there is at least one X such that. ... This article is about the word proposition as it is used in logic, philosophy, and linguistics. ... In mathematics, a binary relation (or a dyadic relation) is an arbitrary association of elements of one set with elements of another (perhaps the same) set. ... The material conditional, also known as the truth functional conditional, expresses a property of certain conditionals in logic. ...

I. ((A f B) & (B → C)) → (A f C) II. ((A f B) & (A f C)) → (A f (B & C)) III. (A f B) ↔ !(A → B) IV. ∃U !U V. ¬(U f ∩)

Note the implied universal quantifiers in the above axioms. In predicate logic, universal quantification is an attempt to formalise the notion that something (a logical predicate) is true for everything, or every relevant thing. ...

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). A typographical error or typo is a mistake made during the typing process. ...

Failure of Mally's deontic logic

Theorem: This axiomatisation of deontic logic implies that !x iff 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. Deontic logic is the field of logic that is concerned with obligation, permission, and related concepts. ... IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation... Look up tautology in Wiktionary, the free dictionary. ...

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. Karl Menger Karl Menger (Vienna, Austria, January 13, 1902 – Highland Park, Illinois, USA, October 5, 1985) was a mathematician of great scope and depth. ... Deontic logic is the field of logic that is concerned with obligation, permission, and related concepts. ...

External links

• Entry Ernst Mally at the Stanford Encyclopedia of Philosophy
• Entry Mally's deontic logic at the Stanford Encyclopedia of Philosophy