Further Thoughts about God, Modality, and Necessary Existence

In a recent post, I argued that God’s existence is not “logically” necessary but should instead be thought of as “metaphysically” necessary. I also argued there that nothing exists out of logical necessity on the grounds that an ontologically empty world (a null world) is a logically coherent possibility. I subsequently got into a Facebook discussion about related matters and was pressed to consider recent work on the ontological argument and the concept of necessary existence. So I spent a few days looking over Necessary Existence (Oxford, 2018), a book co-authored by philosophers Alexander Pruss and Josh Rasmussen. Their book prompted some rethinking on my part, mainly about the relation between “logical” and “metaphysical” modality. It turns out that my views on modality and God’s necessary existence are closer to those held by fans of the modal ontological argument than I had previously thought. The differences, it turns out, are more terminological than substantive. Roughly speaking, I prefer to reserve the term “metaphysical” necessity for something other than what Pruss, Rasmussen, et al. mean by the term.

In this post I’m going to talk about modality in general and how different types of constraints give rise to different types of possibility. This will lead into a discussion of modal terminology, especially the logical/metaphysical distinction. I’m going to propose some more precise labels and then contrast two different conceptions of metaphysical necessity.

Modality and Constraints

Imagine a space of “possibilities”, S_null, that is completely unconstrained. Since S_null has no constraints, nothing is too weird, absurd, or crazy to count as “possible” according to S_null, not even contradictions. According to S_null everything is “possible”, nothing is “impossible”, and nothing is “necessary”. As such, S_null is a pre-modal space. It doesn’t have enough structure to support serious reasoning about modality.

To get the necessary structure we have to add constraints, C, to form a new space, S_C. C represents a stipulation that some of the contents of S_null are to be regarded as not merely “possible” but “necessary” and that everything in S_null that conflicts with C is to be regarded as “impossible”. But what it is for something in S_null to conflict with C? For S_C to be a truly modal space we need a way of excluding internal conflicts, and so C has to at the very least include some notion of implication and enforce a rule of non-contradiction. With this we have the beginnings of a notion of logical possibility. But some disambiguation is still needed since there are several different notions of implication, both formal and informal, and hugely many different systems of logic (first-order, second-order, free, modal, relevant, dialethic, tense, etc.). Which system of logic, if any, deserves to define logical possibility? Are there many different kinds of logical possibility?

There is much that could be said in response to these questions, but I think the main distinction to draw is between purely formal and at least partly material or non-formal constraints on logical possibility. This distinction gives rise to two categories of modal systems:

1. Formal logical necessity: C = the axioms of some consistent and complete formal logical system plus whatever is strictly provable given those axioms.
2. Informal logical necessity (aka abstract metaphysical necessity): C = all “truths of reason” (in some idealized sense of “reason”) + complete information about the natures or essences of things.

Following Alvin Plantinga in The Nature of Necessity, most modern analytic philosophers refer to both (1) and (2) as types of “logical” necessity, with the difference that (1) is “narrowly logical” because the constraints it imposes are strictly formal, whereas (2) is “broadly logical” because the constraints include lots of information that can’t readily be formalized, such as that “no prime minister is a prime number” or that “water is H₂O”—truths one can grasp as necessary only if one understands (to some extent) the natures or essences of the things in question.

Also following Plantinga, many modern philosophers refer to (2) as “metaphysical” necessity. I have sympathy for that label because understanding the natures or essences of things is metaphysically relevant. But I also have reservations about the label because I think (2) also excludes information that is metaphysically relevant, and so doesn’t give us a complete picture of what’s metaphysically possible and what’s not. Insofar as (2) is a kind of metaphysical necessity, it is an abstract metaphysical necessity. I’ll have more to say on this below.

The Need for Material (Non-Formal) Constraints

As Pruss and Rasmussen point out (ch. 2), a purely formal understanding of modality runs into problems with Gödel’s Incompleteness Theorems. The first of the two theorems states that for any formal system strong enough to include basic arithmetic, there are theorems (i.e., necessary truths) that are not provable in the system. The second incompleteness theorem states that such formal systems cannot even prove their own consistency. Given (1) and Gödel’s results there is no way to conclusively rule out scenarios in which both a mathematical claim and its negation count as “possible” simply because neither the claim nor its denial are provable in the formal system. But then we have to countenance the possibility of contradictions within the constrained possibility space S_C. This is not good. What’s worse, most systems of formal logic allow contradictions to propagate in the sense that from a contradiction anything and everything follows.

Now, maybe the prospects for a purely formal notion of modality are not quite so bad. Such systems may work fine in restricted contexts as long as we don’t countenance any self-referential Gödelian-type propositions. But to systematically block self-reference we’ll need to introduce non-formal or material constraints in order to determine if the content of some referring expression includes itself. Alternatively, we could adopt a dialethic system of logic that allows for true contradictions but keeps them tightly corralled. Yet another option is to prevent contradictions from propagating by restricting certain inference rules. For example, the claim that everything follows from a contradiction depends on the logical rule of addition according to which from any proposition P we can infer P ∨ Q (“P or Q”), where Q is any proposition whatsoever. This rule makes sense if we think of “or” (∨) truth-functionally, but it doesn’t make good sense in ordinary language. For example, suppose person A says to person B, “You were early today. Either you are early or you are late.” According to the rule of addition, the second sentence logically follows from the first. If the first sentence is true, then so is the second. But in ordinary language the second sentence is arguably false irrespective of the first sentence. After all, person B could easily have been neither early nor late but on time. We have here a conflict between two different interpretations of “or”. In the truth-functional sense, “or” just means “here are some alternatives: at least one of these happens to be true.” But in ordinary language “or” often means something stronger and modally loaded, such as “here are some alternatives: they exhaust the (relevant) possibilities, so at least one of them must be true.” This stronger reading is the basis for recognizing the “false alternative” fallacy in informal logic. When pointing out the fallacy, one objects to an either–or statement because it overlooks relevant possibilities. Consequently, it can be argued from the perspective of informal logic that we should eliminate, or at least restrict, the logical rule of addition. Some “relevant logics” do just this. But relevance is a notion that can’t be fully captured in formal terms. The “false alternative” is a fallacy in informal logic because it depends on the content of the alternatives and whether they do in fact exhaust the relevant possibilities. So again it seems like we need non-formal constraints on possibility in order to do full justice to our intuitive grasp of what’s possible and what’s not.

Non-Formal Constraints and Necessary Existence

Now, if we’re going to add non-formal or material constraints it would seem rather arbitrary to include information about only some natures or essences and not all. That’s why (2) speaks of “complete information” about the natures or essences of things. The “things” in question are mere possibilia. They are things referenced or described in the possibility space and need not be things that actually exist. Now, most (if not all) essences are generic in nature and not existence-entailing, e.g., human nature doesn’t guarantee that there are any humans. Some philosophers also think there are individual essences or haecceities that are rigidly bound to unique individuals. I think haecceities are a rather bad idea, but nevertheless if they exist, then they would constrain possibility according to (2). Like generic essences, however, individual essences aren’t normally existence-entailing. My individual essence (assuming I have one) surely doesn’t entail that I exist.

But what about God? If any essence is existence-entailing, the divine nature seems like a good candidate because the concept of God plausibly includes the concept of a necessarily existent being, one who cannot fail to exist. This is where debates over the modal ontological argument are usually joined. Does God’s essence entail God’s existence? If so, is the divine nature so understood actually possible, or could it harbor a hidden contradiction? And is the notion of an ontologically empty or null world actually possible given the modal constraints in (2), or might it harbor a hidden contradiction? These aren’t easy questions to answer. Indeed, both ideas (that God’s essence entails His existence and that of a null world) seem possible to me when I consider them individually. (Side note: one shouldn’t dismiss the null world idea on the grounds that if there were a null world then it would be “true” that there is a null world. Using the truth predicate to do ontological boot-strapping is a cheat. As I explained in my previous post, it represents a failure to take the null world idea seriously on its own terms.)

In any case, because the modal constraints in (2) are invariant, consisting entirely of necessary “truths of reason” and conceptually necessary relations among abstract or hypothetical essences, it seems that the right modal logic for (2) must be S5. (Modal system S5 says that necessity and possibility are invariant across “possible worlds”, or maximal consistent scenarios. So if something is possible in one world then it’s possible in all worlds.) Given S5, the modal ontological argument works as long as God’s existence-entailing essence harbors no hidden contradictions. If such a God possibly exists—i.e., if God exists in some “possible world”—then God necessarily exists in that world and thus, by S5, God necessarily exists in all worlds, including the actual world. It follows that God’s existence is “logically necessary” in Plantinga’s broad sense or in the informal sense as I’ve dubbed it. And since Plantinga equates broadly logical necessity with “metaphysical” necessity, it also follows that God’s existence is “metaphysically necessary” in an abstract sense.

From Abstract to Concrete Necessity

As I indicated above, I don’t like calling (2) “metaphysical” necessity because I believe it leaves out or abstracts from metaphysically relevant information, information that includes not just the essences of things, but their actual dispositions insofar as those are modally relevant. So if we’re going to call the kind of necessity articulated by (2) a type of “metaphysical” necessity, then I want to qualify it and call it abstract metaphysical necessity in distinction from concrete metaphysical necessity:

3. Concrete metaphysical necessity (aka real or causal necessity): C = all “truths of reason” (in some idealized sense of “reason”) + complete information about the natures or essences of things + complete information about the actual causal disposition of reality.

We might also call (3) real necessity or causal necessity. Why do we need (3)? Because if we want a full account of metaphysical modality we shouldn’t suppose that all modally relevant information is invariant. Common-sense tells us otherwise. Prior to the 2020 U.S. Presidential election, it was possible for Trump to win a second consecutive term. But after the election, or at least after Biden was officially certified as the winner, it ceased to be possible for Trump to win a second consecutive term. Likewise, Christian theology says that God could have not been a creator. Prior to God’s creative decree the proposition <possibly, God never creates> was true. But once the decree occurs, that same proposition becomes false. Once God has created He cannot fail to have created. The past actuality of God’s decree constrains what is possible for God moving forward. In short, what’s really possible is not invariant but dynamic. The modal landscape changes as the causal landscape of reality changes.

While S5 is arguably the right modal logic for abstract metaphysical necessity, it’s not the right logic for concrete metaphysical necessity. Possibility according to (3) means, roughly, causally reachable from the present state of reality. Since some outcomes that once were reachable no longer are, what’s possible in this sense is not necessarily possible, so S5 fails. For concrete metaphysical necessity, modal system S4 is more appropriate. According to S4, necessity carries forward, but possibility need not. What’s necessary is necessarily necessary, but what’s possible isn’t necessarily possible. Without S5, of course, a modal ontological argument for God’s concrete metaphysical necessity isn’t going to work. Nevertheless, God’s existence may still be concretely metaphysically necessary in the sense I articulated in my previous post. In this sense, to say that God necessarily exists is to say that (a) God exists, (b) God never came into being, and (c) God cannot cease to be.

Summing Up

On my view,

• God’s existence is not logically necessary in the “narrowly logical” or “strictly formal” sense.
• God’s existence may be metaphysically necessary in the “broadly logical” or “abstract metaphysical” sense. His existence is necessary in this sense IF His essence is existence-entailing without any hidden internal contradictions, in which case the modal ontological argument is sound.
• God’s existence is metaphysically necessary in the “concrete metaphysical” sense provided that God actually exists.
• If God’s existence is abstractly metaphysically necessary, then it is concretely metaphysically necessary, but the converse need not hold. So even if the modal ontological argument fails, God’s existence could still be concretely metaphysically necessary.
• S5 holds for strictly formal and abstract metaphysical necessity but fails for concrete metaphysical necessity, where S4 is the more appropriate modal system.