A, given B means it is permissible that A, given B. (The notation is modeled on that used to represent conditional probability.) Dyadic deontic logic escapes some of the problems of standard (unary) deontic logic, but it is subject to some problems of its own. Other variations Many other varieties of deontic logic have been developed, including non-monotonic deontic logics, paraconsistent deontic logics, and dynamic deontic logics. History Early deontic logic Philosophers from the Indian Mimamsa school to those of Ancient Greece have remarked on the formal logical relations of deontic concepts and philosophers from the late Middle Ages compared deontic concepts with alethic ones. In his Elementa juris naturalis (written between 1669 and 1671), Gottfried Wilhelm Leibniz notes the logical relations between the licitum (permitted), the illicitum (prohibited), the debitum (obligatory), the, and the indifferens (facultative) are equivalent to those between the possibile, the impossibile, the necessarium, and the contingens respectively. Mally's first deontic logic and von Wright's first plausible deontic logic Ernst Mally, a pupil of Alexius Meinong, was the first to propose a formal system of deontic logic in his Grundgesetze des Sollens (1926) and he founded it on the syntax of Whitehead's and Russell's propositional calculus. Mally's deontic vocabulary consisted of the logical constants and , unary connective , and binary connectives and . * Mally read as "A ought to be the case".* He read as "A requires B" .* He read as "A and B require each other."* He read as "the unconditionally obligatory" .* He read as "the unconditionally forbidden". Mally defined , , and as follows: Def. Def. Def. Mally proposed five informal principles: (i) If A requires B and if B requires C, then A requires C.(ii) If A requires B and if A requires C, then A requires B and C.(iii) A requires B if and only if it is obligatory that if A then B.(iv) The unconditionally obligatory is obligatory.(v) The unconditionally obligatory does not require its own negation. He formalized these principles and took them as his axioms: I. II. III. IV. V. From these axioms Mally deduced 35 theorems, many of which he rightly considered strange. Karl Menger showed that is a theorem and thus that the introduction of the ! sign is irrelevant and that A ought to be the case if A is the case. After Menger, philosophers no longer considered Mally's system viable. Gert