The Ontological Argument is a Failure

IntroductionMy opponent claims I commit a “fallacy of equivocation,” and also, that defining a being into existence is impossible. First, note that simply claiming we cannot define a being into existence is a claim without any support. It is the whimsy of my opponent, and unless my opponent can prove that claim through a formal proof, then we have no reason to take it seriously. Second, to show that the ontological argument does not commit this supposed fallacy of equivocation, and that it does prove God’s existence, I will not turn to a modern reformulation of the ontological argument in modal logic, which offers us a formal proof of God's existence as well as refuting Pro's claims that I commit an equivocation fallacy.Godel's Ontological ArgumentI will be using a somewhat simplified version of Pruss’s version of Godel’s argument, which employs third-order modal logic and a property abstraction operator. I simplify it for the sake of this debate, and will try my best to translate the formal proof for my opponent and our readers. The basic primitive notion in Godel’s argument is positive property. Formally, we can say that A(x) is a positive property iff necessarily, for all y, if y have A(x), then having A(x) in no way detracts from the greatness of y, and if y has ~A(x), then having ~A(x) does in some way detract from some respect of greatness of y. Basically, this means that a property is positive if it in no way detracts from the greatness of the entity that has the property, but its negation does detract from some respect of the greatness of the possessor. Necessary existence, essential omnipotence, essential omniscience, and essential perfect goodness, are positive properties. The following three formal axioms follow:Axiom A1: If A is positive, then ~A is not positive.Axiom A2: If A is positive and A entails B, then B is positive.Axiom A3: If S is a set of properties that are all positive, and set S entails A, then A is a positive property. (Also, keep in mind that axioms in modal logic are taken to be necessary truths)The correctness of A1 is obvious. A2 follows. If a property does not detract from the greatness of an entity, then anything it entails does not detract from it either. If a property is shown to detract from the greatness of an entity, then any property that entails that property also detracts from that entity’s greatness. Hence if ~A detracts from greatness, and A entails B, then ~B detracts from greatness, since ~B entails ~A by contraposition. This yields Axiom A2. Axiom A3 applies this same logic, but through the notion of a set of properties. This could be done by defining a set S to contain B and A. Given A1 and A2, A3 follows. We can then expand the quantity of properties within the set, so that the set is not necessarily finite. Given A1, A2, and A3, we can prove the following lemma:Lemma L1: If S is a set of properties that are all positive, then it is possible that there is an object that possesses all of the properties in set S.Here is the proof. Suppose A is one of the positive properties in set S, and that S/A is the set of properties that is obtained by removing A from S. Now, suppose it is not possible that an object possesses all of the properties in S, then, it follows necessarily that any object possessing all the properties in S/A must possess the property ~A. This is a contradiction. Why? Because S/A is a set of properties all of which are positive, A is positive, and yet S/A entails ~A, which A1 demonstrates is not positive. Given L1, it follows that an object possesses all the properties in set S, and therefore, that there is a necessarily existent, essentially omnipotent, essentially omniscient, and essentially perfectly good being. We define this object, per Round 1, as God. Thus, through a modal form of the ontological argument, we prove the necessary existence of God. The resolution is negated. Also, note that at no point in this formal proof have I equivocated between different meanings of the word God. The argument is based on the primitive notion of positive property, not on two different meanings of the word God.