The nine arguments presented in Part II concluded to the existence of God: at least one immaterial being who is omnipotent, omniscient, omnipresent, necessary, eternal and perfectly good. God, of course, is genderless but for the purposes of this discussion let us follow the tradition of referring to this being as, “God the Father.” That he exists is something about which Judaism, Islam and Christianity are all in agreement. But Christianity makes the further claim that there are three divine persons who depend on each other and act together in a Trinity. Granting on the evidence given that there is a God, one might reasonably ask: What reason is there for thinking that God is “tripersonal”? Swinburne suggests that there are, in fact, good a priori grounds for this belief and in what follows I will be summarising his argument. 
The Necessity of God the Son The Judaic and Islamic claim that God is a single person is sometimes called unitarianism. All Abrahamic theists, meanwhile, agree that God is morally perfect, that love is a moral perfection, and that, therefore, God is perfectly loving. The problem this inserts into the unitarian concept of God is that love is a relational property. For suppose a perfectly loving God has existed alone from eternity past. Prior to the creation of the man, whom has he loved? A being that is essentially and perfectly loving needs someone to love, and, moreover, as in a perfect marriage, a perfect love is a mutual love between equals. It is logical: If God is omnipotent he could bring about an equal to love and with whom to share all that he has and if God is all loving he would.  Let us again follow the tradition and refer to this second being as, “God the Son.” The Father has his attributes, including moral perfection, essentially: The Father can no more lack one and remain the Father than a square can lack four sides and remain a square. This means that Father could not bring the Son into being at some finite point in the past (say, a quintillion years ago) because for all eternity before that time he would have lacked moral perfection. The past-eternal existence of the Son is therefore an entailment of the moral perfection of the Father: The Father would not exist unless he caused the Son to exist and so requires the Son to exist for his own existence. And because both Father and Son are perfectly good they love each other without limit. But how is one to conceptualise this? It would be quite illogical to suppose that at some point in the past God created a being with the property of having always existed—a being that exists at all moments prior to its creation despite the fact that it has not yet been created. Instead, we should try to imagine that, for as long as the Father has existed, he has sustained the Son in existence, and since the Father has always existed, the Son has always existed too: The creation of the Son, in other words, is not a discrete event locatable in time but a continuous action that recedes with the Father into the infinite past. With this in mind we can easily understand what theologians have meant when they said that the Son is, "eternally begotten.”
The Necessity of God the Holy Spirit In creating the Son the Father shares with another all of his essential properties: The Son enjoys omnipotence, omniscience, omnipresence, eternity and moral perfection. But in a divine duality the moral good of sharing itself is the Father’s alone. Suppose, by way of illustration, that I provide you with money to pay off your debts. In doing so I have shared my money with you but the moral good of sharing itself is mine alone. However, if I provide you with enough money to pay off your debts and yourself pay off the debts of a third person I will have performed a still better action: I will have shared with you both my money and the moral good of sharing it. The love of the Father for the Son must therefore include a wish to cooperate with the Son in further total sharing with an equal: The Father and the Son must together bring into being a third divine person whom, again following the tradition, we may call the Holy Spirit. A godhead in which there is sharing but not cooperation in further total sharing is less than morally perfect. In the twelfth century Richard of St Victor made this same point and added a further argument. Consider: If you love another perfectly you will naturally seek some third person for them to love and be loved by—just as a married couple who love each other may naturally seek to produce a third person, a child, who can share in their love and whose existence allows each of the three family members to unjealously enjoy the love between the other two. In a like case a third divine person means that for each divine person there is someone besides himself for every other divine person to love and be loved by—a state of affairs that demands no less than three divine persons. And since neither the sharing of sharing nor the sharing of love can be instantiated between two, moral perfection can only be instantiated among three. The Holy Spirit, then, in common with the Son, could not enter into existence at some finite point in time before which God lacked moral perfection. The Holy Spirit must "proceed eternally" from the Father and the Son. And so Father, Son and Spirit are all coeternal and divine and, being morally perfect, love each other without limit. And this means that the Trinity has always existed: God is a society of divine beings whose essence is love.
Spheres of Activity In my previous chapter I noted that God, being morally perfect, always performs the morally best action. Faced with a situation in which there is a morally best action three morally perfect and omnipotent persons will naturally coordinate to perform it. But a problem arises in the case of what Swinburne calls "equal best" actions; that is, one of two or more actions which are all equally good. Suppose there are two people who will die without your help and you are only able to help one. All other things being equal, helping Person A and helping Person B are equal best actions. And faced with a choice of equal best actions a morally perfect person is unobligated in his choice: He can choose arbitrarily. It is here that there is need of an argument to explain how a Trinity of omnipotent and morally perfect persons avoid conflict over equal best actions. Consider the direction in which Jupiter rotates and suppose that making it rotate clockwise and counterclockwise are equal best actions in the way just described: It is possible that the Father will try to make Jupiter rotate in one direction and the Son try to make it rotate in the other. Clearly, they cannot both succeed. It follows, says Swinburne, that each of the three members of the Trinity must have different spheres of activity within which they are morally obligated to operate. In this way, each divine person would be omnipotent but, because of his perfect goodness, would never choose to perform an action incompatible with the action of another divine person. But what would determine which sphere of activity each being operated within? Swinburne finds the answer in the causal dependence of the Son and the Spirit on the Father: Persons caused to exist by another have an obligation to the person that causes them to exist. The Father, being perfectly good, will seek to avoid conflict by laying down for every other divine persons a sphere of activity; and the Son and the Spirit, being perfectly good, will recognise an obligation to conform to his adjudication.
The Ontology of the Trinity This point helps us to make further sense of the ontology of the Trinity. For if the Son and the Spirit were not caused to exist by the Father there would be no divine person with the authority to lay down a sphere of activity for every other. Divine persons might therefore find themselves in the position of two pedestrians who come face to face while walking in opposite directions and bob left and right attempting to avoid each other. Without some arbitrary rule (Always keep left) they will only avoid each other by chance—and an arbitrary rule requires an authority to impose it. From this Swinburne concludes that there could not be more than one ontologically necessary person; that is, one person whose existence is uncaused. “But,” he adds, “since the perfect goodness of the Father requires the other two divine persons to exist just as inevitably as the Father exists, they are what I will call ‘metaphysically necessary.’” In Swinburne’s taxonomy a person is metaphysically necessary if it is either ontologically necessary or inevitably caused to exist by an ontologically necessary person. Thus while the Father is the only ontologically necessary person all three members of the Trinity are metaphysically necessary persons and the Trinity as a whole is ontologically necessary because nothing else caused it to exist. Because each person exists as inevitably as every other they are, finally, all equally worthy of worship.
Properties of the Persons A property is essential to a thing if that thing cannot exist without it. Thus the property having three sides is essential to a triangle while the property of having a red hypotenuse or being drawn in pencil is not. Philosophers further recognise two kinds of essential and nonessential properties: monadic and relational. A monadic property is simply a property which a thing has apart from its relation to other things: Thus being brown or made of wood are monadic properties of my desk. A relational property is a property which a thing has in relation to other things: Thus being next to the door or made by a carpenter are relational properties of my desk. Critically, some things can share the same monadic properties and yet be distinct: It is possible that there exists a parallel universe that contains a person exactly like you in every respect: the same appearance, mental life, memory and personality. What makes a particular person who they are is not, therefore, a combination of monadic properties but something underlying those properties which philosophers sometimes call “haecceity” or “thisness.” To more easily grasp the concept of thisness it helps to consider something that lacks it: a gravitational field. Any gravitational field which had the same strength, shape and size as the one which surrounds our earth would be that gravitational field. God clearly has properties of the first four kinds. He has the essential monadic property of being omnipotent and the nonessential relational property of being the creator of the universe. But Swinburne suggests that divine persons, like gravitational fields, lack thisness: There could not be an ontologically necessary and omnimaximal being who was not the Father; nor an omnimaximal being sustained in existence by the Father who was not the Son—and so on. This point, as we shall shortly see, has important implications for the Trinity. But meanwhile, the question arises: What exactly differentiates one divine person from another? After all, being incorporeal, they lack physical features; being omniscient, they share the identical set of all true propositions; being infinitely good, they share an identical and identically perfect moral character; and being omnipotent, they can all perform the same unlimited number of actions. What does set them apart, Swinburne explains, are their “relational properties”: The Father is the Father because he has the essential property of being uncaused; the Son is the Son because he has the essential property of being caused to exist by an uncaused divine person acting alone; the Spirit is the Spirit because he is caused to exist by an uncaused divine person acting in cooperation with a divine person who is himself caused to exist by the uncaused divine person acting alone. Because the divine persons are individuated by relational properties and lack thisness there is an additional entailment for the Trinity. That human parents produce a first child does not determine who that child will be. By contrast, it was not a matter of chance or arbitrary choice by the Father which Son he caused to exist: any divine person caused to exist solely by the Father would have been the Son. And, similarly, any divine person caused to exist by Father and Son together would have been the Spirit. For this reason, Swinburne adds, creating the Son was not merely an equal best act but a unique best act—and so too, mutatis mutandis, for the Spirit.
The Limit of Three It might still be wondered why there are only three and not four or an infinite number of divine persons. For if two divine persons are better than one, and three better than two, would not four be better than three—and so on, ad infinitum? Recall first that God is morally perfect and so must perform a morally best action when there is one to perform. Sharing his divinity by creating the Son, and sharing the moral good of sharing divinity by creating the Spirit with the Son, are both actions of this kind. But again sharing the moral good of sharing divinity with a fourth person is not qualitatively different from sharing the moral good of sharing divinity with a third. And so too with Richard of St Victor’s point: The coexistence of three divine persons provides for each divine person someone other than himself for every other divine person to love and be loved by but adding a fourth divine person adds no further good state. And this means that the Father is under no obligation to create a fourth divine person. Nevertheless, one might still object that adding superfluous divine persons would increase the total good. And so however many divine persons the Father created it would be still better to create one more. To address this point we will need to make a very brief digression into a subclass of equal best actions. We have already seen that, faced with a choice of equal best actions, a morally perfect person is unobligated in his choice: He can choose arbitrarily. Another scenario in which a morally perfect person can choose arbitrarily is when he is faced with an unlimited scale of increasingly good actions. For example: People and stars are good. The universe contains finite quantities of both. Wouldn't a better universe be one with still more of each? Yes. But there is no limit to how many there could be—even if there were infinitely many people and stars there could always be a few more. And so, here too, there is no morally best action and God can satisfy the demands of his moral perfection by creating an arbitrarily finite number of people and stars (perhaps 10^11 billion and 10^19 respectively) or none at all. But what has just been said of people and stars cannot be said of divine persons owing to this critical difference: Divine persons, by definition, are metaphysically necessitated. And this means that when the Father has already fulfilled his divine nature by creating two further divine persons a fourth divine person becomes superfluous and so is not metaphysically necessary; that is, his existence would not be a necessary consequence of an ontologically necessary being and so he would not be divine. There cannot be a fourth divine person. There must and can only be three.  It is in this sense that the "one God" and "three persons" of Christian doctrine is to be spelled out. The three persons of the Trinity form a “totally integrated divine society” which “acts as one coordinated whole.” The Trinity itself is therefore a single divine tripersonal being of which there can only be one. This is an important point: While the Christian concept of God is trinitarian it is not "tritheistic" but monotheistic: There are three persons in oneGod and, “whatever one divine person is and does, God is and does.”
Conclusion We have seen that there are good a priori grounds for thinking that a perfectly loving God is tripersonal. In places Swinburne's argument was somewhat difficult but it unfolded from two very basic moral intuitions. The first was that perfect love among perfect beings requires both the total sharing of the self with an equal and the total sharing of the moral good of sharing. The second was that a perfect and perfectly unselfish love among divine beings would inevitably produce three beings so that for each being there is someone other than himself for every other to love and be loved by. In closing it is worth noting that the doctrine of the Trinity is not held by Christians solely on the basis of moral intuitions about divine love. It is also held because it is believed to have been revealed by Jesus and proclaimed as a central doctrine by the Church he founded and which exploded worldwide on the strength of the evidence for his resurrection. The point of a revelation, of course, is to tell us things we could not discover for ourselves. If, on the one hand, Jesus were God Incarnate we might expect him to reveal truths about the nature of God that are unexpected but turn out to have deep philosophical meaning—and the doctrine of the Trinity certainly has both these properties; and if, on the other hand, we have good reason to believe Jesus rose from the dead that would be good reason to believe that what he revealed is true. The a priori argument for the Trinity and the historical evidence for the resurrection are therefore mutually authenticating. Given all these facts and the coherence of an incarnation discussed in the previous chapter, together with the background evidence for bare theism presented in Part II, I conclude that Christian doctrine has a very high a priori probability.
 It may seem unlikely that anyone would develop an a priori argument for the Trinity unless they had had some contact with the Christian tradition. Swinburne concedes this point but then adds that, “Unless I had been brought up in the tradition of Western mathematics, I would be unlikely to believe that there is no greatest prime number; for I would not even have the concept of a prime number.” But, “once I have derived from tradition the relevant concepts, I am in a position to assess the proof that there is no greatest prime number.” And likewise, “we need first to be taught what a religious system claims; only then are we in a position to assess whether or not it is true.”  Augustine, writing in On Diverse Questions, suggested that if the Father wished to begetthe Son and was unable to do it, he would have been weak; and if he was able to do it but did not wish to, he would have failed to do it because of envy. Thus unitarianism is incompatible with the omnipotence and moral perfection of God.  Swinburne adds that, since it is entailed by the existence of one divine person that there will also be two others, the hypothesis that there is a Trinity is not more complicated than the hypothesis of theism whose great simplicity he has elsewhere argued. A hypothesis, he notes, “is no less simple for having complicated consequences: all the great simple scientific hypotheses have had many detailed complicated consequences.”