Formal ethics is a formal logical system for describing and evaluating the "form" as opposed to the "content" of ethical principles. Formal ethics was introduced by Harry J. Gensler, in part in his 1990 logic textbook Symbolic Logic: Classical and Advanced Systems, but was more fully developed and justified in his 1996 book Formal Ethics. Formal ethics is related to ethical formalism in that its focus is the forms of moral judgments, but the exposition in Formal Ethics makes it clear that Gensler, unlike previous ethical formalists, does not consider formal ethics to be a complete ethical theory (such that the correct form would be necessary and sufficient for an ethical principle to be "correct"). In fact, the theorems of formal ethics could be seen as a largest common subset of most widely recognized ethical theories, in that none of its axioms (with the possible exception of rationality) is controversial among philosophers of ethics. Symbolic representation The axioms and theorems of formal ethics can be represented with the standard notation of predicate logic (but with a grammar closer to higher-order logics), augmented with imperative, deontic, belief, and modal logic symbols. Formal logic uses an underlined symbol (e.g. ) to represent an imperative. If the same symbol is used without an underline, then the plain symbol is an indicative and the underlined symbol is an imperative version of the same proposition. For example, if we take the symbol to mean the indicative "You eat an apple", then means the imperative "Eat an apple". When a proposition is given as a predicate with one or more of the arguments representing agents, the agent to which the imperative applies is underlined. For example, if means "You give a dollar to x" then is the correct way to express "Give a dollar to x". Within the system of formal ethics, an imperative is taken to represent a preference rather than a demand (called "anti-modal" view, because an underline doesn't behave like a modal operator). With this interpretation, the negation of an imperative (e.g. ) is taken to mean "Don't do A", not "You may omit A". To express demands, an imperative modal operator (for may) is defined, so that = "You may do A" and = "You may not omit doing A" = "You must do A". Note that is different from the deontic "all right" operator defined below, as "You must do A"