One of the various disagreements I had with IM Skeptical regarding my recent post, "Should Philosophy of Religion Be Ended?", concerned whether there are possible worlds in which the laws of logic do not hold. I maintain that the laws of logic must hold at every possible world if the very concept of "possible worlds" is to have any meaning whatsoever. Once it is permitted that the rules of logic are not themselves necessary truths, we are left with no means to distinguish possible truths from necessary truths, let alone possible worlds from impossible worlds (e.g., worlds that both exist and do not exist at the same time). Skeptical, in order to refute arguments for God from logic – like the "Lord of Non-Contradiction" paper by Anderson and Welty – to the contrary contends that there may be possible worlds in which rules of logic do not in fact hold.
Skeptical then suggested that despite their appeals to logic theists make special exceptions for theism, and asked me this: "Do you believe that the doctrine of the trinity is true? If you do, then how does that square with the rules of classical logic?" Now I believe that question is worthy of a considered reply. For clarity's sake my own reply to the first part of the question is simply "Yes" But of course what skeptics are more interested in is the second part: why Christians like me believe in the Trinity, especially when we claim to place such a high premium on the validity of logic in understanding God and the world he created.
Theologians have written volumes on the Trinity as a church dogma, as a description of divine ontology drawn from biblical statements, and as a model of divinity that lends itself to the activity of securing human redemption. Apologists, however, are the most interested in whether or not the Trinity is actually, or least potentially, coherent. That issue in turn concerns the logical relations among Father, Son and Spirit. According to Wayne Grudem, the set of propositions underlying the doctrine of the Trinity can be stated succinctly as
1. God is three persons [hypostases].
2. Each person is fully God.
3. There is one God.
…the problem being that these appear inconsistent. However, there are no explicit contradictions here. Additional premises would be required to create an explicit contradiction, such as
4. God is not three persons. – or –
5. Each person is less than fully God. – or –
6. There are many Gods (gods). Etc.
Now I have already mentioned Plantinga's free will defense in the context of my ongoing discussion with Skeptical, but I believe it bears mentioning again. The free will defense appears analogous to a defense of the Trinity, in that the one who argues that the problem of evil renders God's existence impossible, like the one who charges that the Trinity is illogical, bears the burden of proving that the initial set in question is formally inconsistent and not merely counterintuitive.
Clearly it would be logically problematic to say that the one God is actually three separate beings. After all, that seems to be directly translatable to 1 = 3, which is contradictory (in that something, God, is said to be both one and not-one at the same time). At issue, though, is whether the relations among the members of the godhead are absolute reflexive identity relations. To put it another way, we need to ask ourselves: what exactly does it mean to say "God is Father, Son and Spirit," and to also say, "Father, Son, and Spirit are God"? If it's right to say that God is strictly equal to – nothing more and nothing less – all three members of the godhead, and vice-versa, then we are saying that one equals three and effectively speaking nonsense. But I don't think it's necessarily true that the relations among the members of the godhead are absolute reflexive identity relations.
Some theologians, for example, have suggested these are "relative identities," wherein identity relations are still logically valid but in terms other than shared properties. Deutsch comments in the Stanford Encyclopedia: "It is possible for objects x and y to be the same F and yet not the same G, (where F and G are predicates representing kinds of things (apples, ships, passengers) rather than merely properties of things (colors, shapes)). In such a case ‘same’ cannot mean absolute identity. For example, the same person might be two different passengers, since one person may be counted twice as a passenger." Now this "passenger" analogy, like most other analogies, does not apply all that well to the Trinity, but for present purposes the fact that relative identities are possible is enough to undercut the argument that the Trinity is explicitly illogical.
While analogies proposed for the Trinity are typically imperfect (as analogies are generally), they do often serve to underscore that the Trinity is a mystery in need of an explanation rather than an example of explicit illogic. It should not surprise anyone that there are few, if any, applicable worldly analogies for a spiritual reality. And skeptics, at least those familiar with scientific theories, should know that on naturalism, nature houses numerous mysteries of its own: How an entropic universe can come into existence unaided; how quantum mechanics can be reconciled with general relativity; how chemical evolution can take place apart from replicators that only operate within living systems. Etc. Meanwhile there are some serious Trinitarian models that purport to provide solutions – or at least potential solutions – beyond merely pointing out that the Trinity is not formally contradictory. Those will have to be addressed by someone else, or at least at some other time.