Formal ethics

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. formula_1) 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 formula_2 to mean the indicative "You eat an apple", then formula_1 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 formula_4 means "You give a dollar to x" then formula_5 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. formula_6) is taken to mean "Don't do A", not "You may omit A". To express demands, an imperative modal operator formula_7 (for "may") is defined, so that formula_8 = "You may do A" and formula_9 = "You may not omit doing A" = "You must do A". Note that formula_7 is different from the deontic formula_11 "all right" operator defined below, as "You must do A" is still an imperative, without any "ought" judgment (i.e. not the same as "You ought to do A").
Following Castañeda's approach, the deontic operators formula_12 (for "ought") and formula_11 (for "all right", represented formula_14 for "permissible" in some deontic logic notations) are applied to imperatives. This is opposed to many deontic logics which apply the deontic operators to indicatives. Doing so avoids a difficulty of many deontic logics to express conditional imperatives. An often given example is "If you smoke, then you ought to use an ashtray". If the deontic operators formula_12 and formula_11 only attach to indicatives, then it is not clear that either of the following representations is adequate:
However, by attaching the deontic operators to imperatives, we have unambiguously
Belief logic symbols, when combined with imperative logic, allow beliefs and desires to be expressed. The notation formula_20 is used for beliefs ("You believe A") and formula_21 for desires ("You desire A"). In formal ethics, "desire" is taken in a strong sense when the agent of the belief is the same as the agent of the imperative. The following table shows the different interpretations for formula_22 depending on the agent and the tense of the imperative:
This strong interpretation of desires precludes statements such as "I want to get out of bed (right now), but I don't act to get out of bed". It does not, however, preclude "I want to get out of bed (right now), but I don't get out of bed". Perhaps I act to get out of bed (make my best effort), but can't for some reason (e.g. I am tied down, my legs are broken, etc.).
Beliefs may be indicative, as above, or imperative (e.g. formula_23 "Believe A", formula_24 "Desire A"). They may also be combined with the deontic operators. For example, if formula_25 means "God exists", then formula_26 is "You ought to believe that God exists", and formula_27 is "Everyone ought to believe that God exists".
The modal operators formula_28 and formula_29 are used with their normal meanings in modal logic. In addition, to address the fact that logicians may disagree on what is "logically" necessary or possible, causal modal operators are separately defined to express that something is "causally" necessary or possible. The causal modal operators are represented formula_30 and formula_31. In addition, an operator formula_32 is used to mean "in every actual or hypothetic case". This is used, for example, when expressing deontic and prescriptive counterfactuals, and is weaker than formula_28. For example,
whereas
Finally, formal ethics is a higher-order logic in that it allows "properties", predicates that apply to other predicates. Properties can only be applied to actions, and the imperative notation is used (e.g. formula_36 = "action A has property F"). The only types of property that formal ethics admits are "universal properties", properties are not evaluative and do not make reference to proper names or pointer words. The following are examples of properties that are not universal properties:
Requiring a property to be universal, however, is different from requiring it to be morally relevant. formula_43, where formula_44 means "Act A is done by a black person" is a universal property, but would not be considered morally relevant to most acts in most ethical theories. Formal ethics has a definition of "relevantly similar" actions that imposes certain consistency constraints, but does not have a definition of "morally relevant properties".
The notation formula_45 is used to mean "G is a complete description of A in universal terms". Put another way, formula_25 is a logical conjunction of all universal properties that formula_1 has. The formula_45 notation is the basis for the definition of "exactly similar" actions and is used in the definition of "relevantly similar" actions.
Axioms.
Formal ethics has four axioms in addition to the axioms of predicate and modal logic. These axioms (with the possible exception of "Rationality", see below) are largely uncontroversial within ethical theory.
In natural language, the axioms might be given as follows:
Care must be taken in translating each of these natural language axioms to a symbolic representation, in order to avoid axioms that produce absurd results or contradictions. In particular, the axioms advocated by Gensler avoid "if-then" forms in favor of "don't combine" forms.