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:
(RIGHT) ∀ϕ☐(Rϕ ≣ Cgϕ)
(WRONG) ∀ϕ☐(Wϕ ≣ Cg~ϕ)
(PERMITTED 1) ☐(~Eg ⊃ ∀ϕ~Wϕ) 
(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. 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. 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.
 ‘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).
 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.
 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.
 The objector would need to deny necessity as the quantifier and merely suggest that the command is possible.