Axiom 1. If a property is positive, its negation is not positive.
Axiom 2. If P is a positive property, and P entails Q, then Q is a positive property.
Axiom 3. Necessary existence is a positive property.
Definition 1. If P is a property, then EP is the property of having P essentially.
Definition 2. P is strongly positive iff it is necessary that EP is positive.
Definition 3. God-likeness is the property of having all strongly positive properties.
Axiom 4. God-likeness is a positive property.
Given an appropriate modal logic that includes S5 and assumes that properties are necessary beings, we get:
Theorem 1. If Axioms 1-4 hold, then there necessarily exists an essentially God-like being.
The hard question is whether we can give an interpretation to the notion of a "positive property" that makes Axioms 1-4 plausible and the conclusion of Theorem 1 interesting.
The following won't be needed for the proof of Theorem 1 but is a nice observation:
Lemma 1. If Axiom 2 holds, any strongly positive property is positive.
Proof of Lemma 1: EP entails P. •
The proof of Theorem 1 depends on several subsidiary results.
Lemma 2. Given Axioms 2 and 3, necessary existence is strongly positive.
Proof of Lemma 2: Let N be necessary existence. If x exists necessarily, then it is necessarily true that x exists necessarily, by S4. Hence, N entails EN. By Axioms 2 and 3, EN is positive. Moreover, this proof works necessarily (we only made use of the axioms, and axioms are assumed to hold necessarily), so EN is necessarily positive. •
Lemma 3. Given Axioms 1 and 2, every positive property is possibly exemplified.
Proof of Lemma 3: Suppose for a reductio that P is positive but not possibly exemplified. Hence, necessarily, nothing has P. Hence, necessarily and trivially, everything that has P has ~P. Hence, P entails ~P. Hence, ~P is positive as P is, by Axiom 2. But ~P is not positive, by Axiom 1. Hence absurdity ensues. •
Lemma 4. If Axiom 2 holds, and P is strongly positive, then EP is strongly positive.
Proof of Lemma 4: Necessarily, EP is positive. EP entails EEP by S4. Hence, necessarily, EEP is positive, by Axiom 2. Hence EP is strongly positive. •
Lemma 5. If P possibly is strongly positive, then P is strongly positive.
Proof of Lemma 5: Suppose possibly P is strongly positive. Then, possibly, necessarily EP is positive. By S5, necessarily, EP is positive. Hence, P is strongly positive. •
Lemma 6. Given Axioms 1 and 2, if x has God-likeness, then x essentially has God-likeness.
Proof of Lemma 6: Suppose for a reductio that x possibly lacks God-likeness. Then, possibly, there is some strongly positive property that x lacks. Since properties are necessary beings, and given S5, existential quantification over properties and possibility can be interchanged. Hence, there is a property P such that possibly P is a strongly positive property that x lacks. Let P be such. Then, possibly P is strongly positive. Hence, P is strongly positive by Lemma 5. Hence, x essentially has P. But we said that possibly x lacks P. Hence a contradiction ensues. •
Proof of Theorem 1: God-likeness is possibly exemplified, by Axiom 4 and Lemma 3. Thus, possibly, there is an x that has God-likeness. God-likeness entails necessary existence, by Lemma 2. Hence, possibly, there is a necessary being x that has God-likeness. But, by Lemma 6, anything that has God-likeness has God-likeness essentially, and this holds necessarily since it is a logical consequence of the axioms. Thus, possibly, there is a necessary being that essentially has God-likeness, since if a necessary being has an essential property, it is necessary that it have this essential property. Hence, possibly, necessarily there is an x that is essentially God-like. Hence, necessarily, there is an x that is essentially God-like, by S5. •
Axiom 1*. If a property is island positive, its negation is not island positive.
Axiom 2*. If P is an island positive property, and P entails Q, then Q is an island positive property.
Axiom 3*. Necessary existence is an island positive property.
Definition 1*. If P is a property, then EP is the property of having P essentially.
Definition 2*. P is strongly island positive iff it is necessary that EP is island positive.
Definition 3*. Island paradisiality is the property of having all strongly island positive properties.
Axiom 4*. Island paradisiality is an island positive property.
Theorem 1*. Given Axioms 1*-4*, there necessarily exists an entity that essentially has island paradisiality.
Whether this is a parody depends on what "island positive" means. If island positive entails positive (e.g., island positive = positive and compatible with being an island), then any God-like being also has island paradisiality!
Suppose, though, island positive means something like positive for an island (cf. Grim; what is positive for an island need not be positive simpliciter), so that we can add:
Axiom 5*. Being an island is strongly island positive.
Then island paradisiality entails being an island, and we have a genuine parody. But note that Axiom 5* seems to mean that the island case requires an additional assumption, over and beyond the God case. Moreover, this assumption is not obvious. For it is not clear that it is a good thing for an island to be essentially an island. If x is essentially an island, then it is impossible for there to be bridge between x and the mainland. But why should the impossibility of such a bridge be good for an island qua island?