The proponent of divine command theory (DCT) claims that whatever God commands to any moral agent becomes a moral obligation. Formulations of the commands are given symbolic form by David Efird as:[1]
(RIGHT) ∀ϕ☐(Rϕ ≣ Cgϕ)
(WRONG) ∀ϕ☐(Wϕ ≣ Cg~ϕ)
(PERMITTED 1) ☐(~Eg ⊃ ∀ϕ~Wϕ)[2]
(PERMITTED 2) [(∃ϕ☐Cgϕ ∙ ∃ϕ☐Cg~ϕ)] ∙ [(∃ϕ☐~Cgϕ ∙ ∃ϕ☐~Cg~ϕ)]
The arbitrariness objection claims that [for example] if God commanded moral agents to rape then the action of committing rape would be obligatory to all moral agents.[3] The objector assumes an inference in the form of the argument stating that ∀ϕ☐(Rϕ ≣ Cgϕ) may also be applicable in the sense that ϕ could refer to rape (ρ). What would make the command arbitrary is the truth of the counterfactual: If God did command rape then there would be a moral obligation to rape.[4] If this counterfactual were true then it would serve as a defeater for DCT. The objection is not a defeater for the existence of God; it is a defeater for the DCT’s model of deontological ethics.
[1] ‘ϕ’ ranges over action types, such as going to fight at the front, and ‘∼ϕ’ stands for ‘refraining from ϕ’, ‘R’ stands for the predicate ‘__ is morally right’, ‘W’ stands for the predicate ‘__ is morally wrong’, ‘C’ stands for the relation ‘__ commands __’, where the first term of the relation stands for an agent and the second term stands for an action type, ‘g’ is a singular term standing for God, and ‘x forbids ϕ’ is true just in case ‘x commands ∼ϕ’. David Efird, “Divine Command Theory and The Semantics of Quantified Modal Logic,” in New Waves in Philosophy of Religion, Erik Wielenberg and Yujin Nagasawa, eds. (United Kingdom: Palgrave MacMillan, 2009). See http://www.york.ac.uk/depts/phil/about/staff/efird/dct_qml.pdf (November 22, 2010).
[2] This is a negative claim for permission, Karamazov’s Theorem: Necessarily, the non-existence of God implies that for every action, the action is not a wrong action. This, of course, has no deontological value, but it would a true counterfactual to the actual deontological claims of the DCT proponent.
[3] As advocated by Walter Sinnott-Armstrong in “Why Traditional Theism Cannot Provide an Adequate Foundation for Morality,” in Is Goodness without God Good Enough? (Lanham, MD: Rowman & Crutchfield, 2009), 106.
[4] The objector would need to deny necessity as the quantifier and merely suggest that the command is possible.