Strange Bedfellows

Kevin Harris: Entering the world of Dr. William Lane Craig is entering a world of things like counterfactuals, conditionals, the subjunctive mood, middle knowledge, confirmational holism, and now a new one on me: impredicative definitions. Now, before we explain that, let me give some background. Jeff Lowder has an article on Secular’s Substack called Odd Bedfellows: Why William Lane Craig and Gender Theorists Share the Same Logic.^[1] Now that's an intriguing title, Bill. There’s a word about Jeff. The very first email that I ever sent was to my father. The second was to Jeff Lowder. I discovered him and the Secular Web at the start of all the 1990s Internet battles that we were in, the Internet Infidels against the Christian apologists. And I've always been impressed by Jeff, his knowledge, and his generosity. He is on the secular side. Jeff has spilled a lot of cyber ink on you, Bill. I think the two of you have met.

Dr. Craig: Yes. Years ago, Jan and I visited the Air Force Academy in Colorado Springs where Jeff was, at that time, I think, a young lieutenant. He met with us and gave us a tour of the entire campus, showed us all around. He was very congenial, and even though I was sorry to hear about his involvement in the Internet Infidels, nevertheless he was a gracious host, and so we've had a good relationship ever since.

Kevin Harris: When the Internet came out, the first thing I did was start to look for Christian apologetics and opportunities to use it on this new online world. He’s one of the first people that I ran into. He was trying to do the same thing with the Secular Web on the other side. Jeff’s post that we’re going to be looking at is citing the book that you and Erik Wielenberg published in 2020, A Debate on God and Morality, and it’s based on your 2018 debate at North Carolina State. Talk to us about that book, Bill.

Dr. Craig: This was an excellent debate. Erik Wielenberg, as I said to him after the debate was over, is an excellent debater as well as an excellent philosopher. He defended his secular ethics that he calls “godless normative realism,” and I defended a divine command theory of ethics based in theism, and then offered criticisms of his godless normative realism. So it was an outstanding debate and has been converted into a good book.

Kevin Harris: And now you and David Baggett have a new book on this topic scheduled to come out in May. Give us an overview, if you would, please.

Dr. Craig: This book is called Reasonable Moral Faith, and it is on the moral argument for the existence of God. David Baggett is the most published Christian philosopher on this subject, and the book comprises four sections. First, a history of the moral argument for God's existence that gives little thumbnail sketches of the principal advocates of this argument down through history. Then the second part, the real meat of the book, is a defense of moral realism over against various moral anti-realisms. And then finally, in the third part, there is a fivefold moral argument presented for God's existence. And then it closes with a critique of atheistic Platonic realism of the variety that Erik Wielenberg espouses.

Kevin Harris: And we are working on getting David to join us on the podcast when that book comes out, and maybe we can talk about it, do a podcast on it. But let’s get to the post by Jeff Lowder. He begins, quote:

An impredicative definition is one that invokes the very set being defined to define itself. The classic example is the barber who shaves everyone in town who does not shave themselves. (Does he shave himself? If he does, he doesn’t; if he doesn’t, he does. Paradox ensues).

Kevin Harris: Let’s stop right there and make sure we have our terms right. Bill, it sounds like we’re dealing with circular reasoning of some kind. The barber shaves everyone who does not shave himself. And if the barber does not shave himself, then he is included in the set of everyone who does not shave himself. But if the barber shaves everyone who does not shave himself, that includes him. And that means he does shave himself, and he’s no longer in the set of those who do not shave themselves. So that’s one of the classic illustrations. Can you elaborate on that illustration and the definition of impredicative definitions?

Dr. Craig: Yes. An impredicative definition is a definition that quantifies over a collection or totality that includes the very entity being defined. And this can lead to self-contradictions, as in the example of the barber. Jeff actually misstates the barber illustration, however. It’s not enough to say that the barber shaves everyone who does not shave himself. You have to add that the barber shaves only those who do not shave themselves. So he shaves all and only those who do not shave themselves. And that will lead then to this self-contradiction that you explained. But impredicative definitions do not need to lead to self-contradiction. For example, we could talk about the set of all things mentioned in this podcast. Well, that set is itself mentioned in this podcast. I just referred to it. So the set of everything mentioned in this podcast is itself mentioned in this podcast. And these impredicative definitions are extremely important. They are familiar in logic, in mathematics, and in philosophy. In fact, it was the paradoxes of naive set theory caused by impredicative definitions that brought down the set theory invented by Georg Cantor in the 19th century. And as a result, theorists gave up on the effort to define what a set is. Instead, they simply adopted a system of axioms as the basis for set theory, leaving the notion of a set undefined but having these axioms that govern the behavior of sets. And this became then the genesis of Zermelo-Fraenkel set theory. So these impredicative definitions are not some sort of arcane subject. This is something that’s familiar in logic, math, and philosophy.

Kevin Harris: Jeff continues, quoting:

Because of this, philosophers and logicians usually treat circular, self-referential definitions with extreme suspicion. They are often viewed as vacuous, empty, or logically vicious.

But in a strange twist of meta-ethical fate, one of the world’s leading conservative Christian philosophers, William Lane Craig, has rigorously defended impredicative definitions. Why? To save the foundations of morality from infinite regress. The irony—and the subject of this post—is that by defending this logic for God, Craig has inadvertently validated the logical structure of a concept he almost certainly rejects: gender self-identification.

Kevin Harris: First, gender self-identification is a political hot potato as well as a philosophical one. The rally cry from those theorists is, “A woman is anyone who identifies as a woman.” But before we get to the logical structure of this, do you reject gender self-identification as Jeff suggests you probably do?

Dr. Craig: If you mean by that that you can choose your gender, yes, I do reject it. I do not think that gender is a conventional notion. I think that God has created us as male and female, man and woman, just as the Genesis creation story describes.

Kevin Harris: So Jeff argues that the same logical structure that applies to your divine command theory of morality can be applied to gender self-identification claims. Let’s see why. Jeff writes, quote:

The dilemma is classic. If morality is based on God’s commands, we eventually have to ask: “Why are we obligated to obey God’s commands?” If you say, “Because it is good to obey Him,” you are appealing to a standard of goodness outside of God (placing a moral law above Him). If you say, “Because He commanded us to obey Him,” you seem to be stuck in a circle. Or worse, an infinite regress . . .

In a fascinating passage responding to philosopher Michael Huemer, Craig decides to bite the bullet. He argues that the circularity isn’t vicious—it’s impredicative.

And then he quotes from your response. Here you are, Bill:

DCT’s [Divine Command Theory’s] definition of moral obligation should be understood as an impredicative definition, one that includes itself... For example, the set of all things mentioned in this book is itself mentioned in this book. No paradox there! So suppose God has issued to us the following command: ‘Obey everything I command you!’... The command to obey everything God has commanded us is one of the things God has commanded us to do.

Jeff admits your point is logical, Bill. God’s authority is a closed, self-referential loop that is logically consistent. Now, are there any corrections or additions before we get to the gender theorist stuff?

Dr. Craig: Yes, I think that Jeff’s explanation of my theory is incomplete. What the theory says is that duties arise from imperatives issued by a qualified authority. And God, as the supreme Good, is qualified to issue moral commands. Therefore, our moral duties are determined by God’s commands. Now, someone might say, “All right, fine. My moral obligations are determined by God’s commands. But why are we obligated to obey God’s commands?” There are two possible responses that one could give to this question. First, you could say, “God commands me to obey his commands.” But then that will lead to an infinite regress because you would then ask, “Why should I obey his command to obey his commands?” And that will lead to a third response, “God commands us to obey his command to obey his commands.”. This sort of infinite regress really isn’t objectionable because it’s not vicious. At every level, what you’re saying is perfectly consistent. It’s rather like an infinite regress that a proposition is true, and it is true that the proposition is true, and it is true that it is true that the proposition is true, to infinity. Not every sort of infinite regress is a vicious one. I think that would be a possibility here. The other response to this objection would be to say every moral theory has its explanatory ultimate or stopping point beyond which you cannot go, and my explanatory stopping point just is God’s commands. Again, I think there is no problem with that sort of response. It is correct that any moral theory will reach an explanatory ultimate beyond which one cannot go, and you can say my theory simply takes God’s commands as the explanatory ultimate. But then, as I pondered this question, it occurred to me that there is actually a third answer, namely God’s command, “Obey everything I command you,” includes itself. This is not biting the bullet as Jeff suggests. This appeal to an impredicative understanding of God’s command, I think, is frankly a brilliant solution to the question of why we should obey God’s commands.

Kevin Harris: Wait a minute. Let me ask you this before we keep going. Break down the word for us. We know what a predicate is, but impredicative?

Dr. Craig: I wouldn’t try to discern the meaning of the word etymologically by combining predication with impredication. That would make it sound as though it’s not a predicate. The idea is, as I said before, that it concerns a collection or totality that the thing being defined is itself a member of.

Kevin Harris: Okay. Proceeding from there, Jeff writes:

Now, let’s look at perhaps the most controversial definition in modern social theory. A common critique of the view that “a woman is anyone who identifies as a woman” is that it is circular or vacuous. Critics often argue that you cannot use the word “woman” to define the word “woman.” It lacks independent criteria. . . . But wait . . . If we apply William Lane Craig’s defense of impredicativity, this is not a paradox. . . . The definition creates a closed, self-referential loop where the authority to constitute the category comes from the subject itself.

If Craig is right that impredicativity is logically benign, then the standard objection to gender self-identification—that it is circular or logically empty—fails. Structurally, the definition “Woman = Identifies as a woman” is just as robust as “Obligation = God commands us to obey commands.”

Before we look at any ramifications, are they structurally or logically the same?

Dr. Craig: No, they are not. This definition of a woman is not an example of an impredicative definition. I think Jeff is simply mistaken here. I think he’s confusing a circular definition with an impredicative definition. A circular definition is one in which the term that is being defined or a synonym of it reappears in the definition. For example, if I were to say a circle is a round shape, that would be a circular definition because what you mean by round is circular. So the definition would be vacuous, uninformative, and circular. But that’s not what an impredicative definition is. An impredicative definition is one that includes itself. And the definition given of a woman doesn’t refer to that definition. So even if it is circular, I do not think it’s impredicative.

Kevin Harris: I was thinking at this point, yeah, but the gender identity theorist is not God. Sure enough, Jeff writes, quote:

Craig, of course, is a sharp philosopher. He anticipates that if he allows this loop, he allows it for everyone. If God can pull Himself up by his own bootstraps, why can’t Michael Huemer? Why can’t I? Craig writes:

Huemer further says that if this answer is acceptable, ‘then the following would also be an acceptable explanation: you’re morally obligated to obey my commands, because I command you to do so.’ That all depends on who is speaking in the first person. If it is God, then the theory is acceptable; but if it is Michael Huemer, the theory is utterly implausible, since it is not the paradigm of goodness who speaks...

And here lies the rub. Craig admits that the logic is available to anyone. To stop Michael Huemer (or the gender theorist) from using it, Craig has to pivot from logic to ontology. He claims the loop only works for God because God is the ‘paradigm of Goodness.’

That sounds right to me. God’s very ontological status and paradigm must be taken into account. Wouldn’t comparing God and finite persons be a category error?

Dr. Craig: I think Jeff is just confused here. What the theory says is that duties arise from imperatives issued by a qualified authority. And God, as the supreme good, is qualified to issue moral commands, but a mere human being is not. Notice this has nothing to do with an impredicative definition. The questioning is about the qualifications of the person issuing the moral imperatives.

Kevin Harris: Jeff offers an objection. He writes, quote:

But here is where the argument eats its own tail. On Craig’s metaethical theory (what I call “Divine Nature Theory”), what is the paradigm of Goodness? It isn’t an external standard. It is God Himself.

So the argument for God’s authority ultimately looks like this:

1. Why obey God? Because He commanded it (Impredicativity).

2. Why does His impredicativity work? Because He is the Good.

3. What is the Good? Resemblance to God.

The ‘paradigm of Goodness’ isn’t a ladder out of the hole; it’s just a bigger shovel. God is the being who defines His own authority by asserting it, and defines the Good by existing.

So are you just moving the issue back a step, Bill, with the bigger shovel?

Dr. Craig: No, I think it’s pretty clear that Jeff has got the explanatory order wrong. It really goes like this. First, why do God’s commands constitute our moral duties? Answer: because God is the supreme Good and duties arise from imperatives issued by a qualified authority. Second, why are we obligated to obey God’s commands? Answer: because God’s command to do everything that he commands us includes that very commandment.

Kevin Harris: Jeff elaborates more on whether you’re moving the issue back a step, but let me summarize it. He writes, quote:

When Craig says that God’s nature is the “paradigm of goodness,” the word “paradigm” is key. It means that God (or His nature) just is the Good, just as at one time the meter bar in Paris was the paradigm of the meter. On this view, then, to be “good” is simply to resemble God, just as “to be one meter long” used to be “be as long as the meter bar in Paris.” So, when Craig appeals to God’s nature to validate God’s circular command, he is just drawing a bigger circle:

. . .

Craig’s Defense: Why is He the Paradigm? Because He is God.

The “authority” Craig appeals to is just another instance of the subject constituting the category by existing.

Kevin Harris: Seems like Jeff presses the issue here. Despite God’s being the paradigm, it’s still just another instance of the subject constituting the category by existing, he says. Can we stop the presses here?

Dr. Craig: I think Jeff errs when he says, “Craig appeals to God’s nature to validate God’s circular command.” God’s command is not circular; it’s impredicative. I appeal to God’s nature to explain why he is qualified to issue moral commands. As the supreme Good, God is qualified to issue moral commands, which then become our moral duties. That’s the heart of the theory.

Kevin Harris: Let me ask you: Is the Euthyphro dilemma trying to sneak in here?

Dr. Craig: I didn’t mention that, but I think you are absolutely right. I think that Jeff is conflating the Euthyphro dilemma with this use of impredicativity, and they are not the same thing. So you are absolutely right. The question of why God commands what he does is a prior question that is related to my construing God as the paradigm of goodness.

Kevin Harris: Jeff finishes with what he thinks is the real conclusion. Quote:

This brings us to the uncomfortable reality. The critic is right that Craig is not committed to the truth of gender self-identification. He can still deny it. But he has lost the ability to dismiss it as circular. He cannot say, “You can’t just define reality by pointing to yourself!” because that is exactly what his God does. He cannot say, “You need an external standard to validate your identity!” because he explicitly denies that God needs one.

Ultimately, Craig’s (presumed) argument against the gender theorist’s definition of “woman” is not that their logic is flawed, viz., their definition is impredicative. It is simply that he believes his subject (God) has the power to self-constitute reality, while their subject (the individual) does not.

That is a theological claim, one which Craig almost certainly believes to be true. But it is not a logical victory. It is an admission that the battle is no longer about whether self-reference is valid, but about whose self-referential circle is allowed.

Dr. Craig: I hope as a result of our conversation today, you’ve come to see that the logical structure here is not the same. The gender theorist’s definition of a woman is not impredicative. At most, it is circular, though contrary to Jeff, I’ve never even suggested that. Indeed, I don’t think that it is circular. A circular definition would be something like a woman is a female person. That would be uninformative. But it is very informative to say that a woman is anyone who identifies as a woman. So it’s not clear to me that this definition is circular to begin with. But in any case, the logical structure is quite different. It’s not impredicative.

Kevin Harris: As we wrap it up, just to clarify, would you still tell the person who defines a woman as someone who identifies as a woman that their definition is flawed? Are you a logical bedfellow with gender self-identification theorists?

Dr. Craig: This is what’s so bizarre about this whole blog. I’ve never said anywhere that such a definition is flawed or circular. Jeff just imagines the whole thing. I just disagree with the definition because I don’t think that one’s gender is self-determined or conventional. So the problem isn’t with the logic of the definition. It’s just a bad definition.

Kevin Harris: Are you looking for a deep dive into a philosophical systematic theology? Dr. William Lane Craig’s Systematic Philosophical Theology, Volume 1 and 2A, are now available. Subsequent volumes will be coming over the coming months and years. And your financial support helps make this possible. If you haven’t picked up your copy, be sure to visit wiley.com, Amazon, or wherever academic books are sold. And thank you.^[2]