Anselm and Brouwer

I was reading Anselm’s replies to Gaunilo, and was struck by this:

Furthermore: if it can be conceived at all, it must exist. For no one who denies or doubts the existence of a being than which a greater is inconceivable, denies or doubts that if it did exist, its non-existence, either in reality or in the understanding, would be impossible. For otherwise it would not be a being than which a greater cannot be conceived. But as to whatever can be conceived, but does not exist – if there were such a being, its non-existence, either in reality or in the understanding, would be possible. Therefore if a being than which a greater is inconceivable can be even conceived, it cannot be nonexistent.

Let → indicate subjunctive conditionals. Let E!(x) say that x exists.

1. E!(God) → □E!(God).

2. ∀x[if Conceivable(x) and  ∼ E!(x), then: E!(x) → ⋄ ∼ E!(x)].

3. So, not: (Conceivable(God) and  ∼ E!(God)).

4. So, if Conceivable(God), then E!(God).

The ∀x quantifier in (2) is problematic, since it ranges over beings that don’t exist. Perhaps we can read it substitutionally. Let’s suppose we can finesse this issue.

What interests me in (2) is that big conditional in it is most plausibly as seen as a special case of:

5. If C(p) and q, then p → ⋄q,

where C(p) says “conceivably p”, which may or may not be the same as “possibly p”.

We can prove (5) from the Brouwer Axiom

6. If q, then □⋄q,

where L is necessity, and the following principle about subjunctive conditionals:

7. If C(p) and Lr, then p → r,

namely that necessities would still hold no matter what conceivable things happened (to get (5) from (6) and (7), let r be ⋄q). Principle (7) is very plausible if conceivability is possibility: if a possible thing happened, anything necessary would still be true. It’s less plausible if conceivability is not possibility.

And, of course, the Brouwer Axiom is controversial, albeit not quite as much as S5. I initially hoped that the use of the subjunctive conditional in (2) allowed Anselm to get by with something weaker than Brouwer. But not so if the route to (2) goes through (5) and possibility implies conceivability (PIC). For we get Brouwer from (5), PIC and the very plausible principle:

8. If p is possible, then we do not have p → r and p →  ∼ r.

For suppose that contrary to Brouwer we are at a world where q is true, but ⋄q is false at some accessible world. By PIC,  ∼ ⋄q is conceivable. Let p be  ∼ ⋄q. Then C(p) and  ∼ p. But clearly  ∼ ⋄q →  ∼ ⋄q. If we had (5), we would have  ∼ ⋄q → ⋄q, and contradict (8).

So, if Anselm’s argument for (2) goes through (5), we don’t have an improvement over Brouwer. But we can still get (2), given some very plausible assumptions, and the following special case of (5):

9. If C(∼q) and q, then  ∼ q → ⋄q.

And I feel that (9) has some plausibility above (6), at least if conceivability is the same as possibility. For suppose q is true and  ∼ q is possible. Then it seems somewhat plausible that q is possible in the  ∼ q-worlds that are closest to the actual world. Maybe. But maybe there is some way to derive Brouwer from (9) and additional plausible premises.

Kant's 20 thalers objection to the Ontological Argument

I’ve been thinking of this way of putting one of Kant’s “20 thalers” objection to Anselm’s ontological argument:

1, When we say “x is greater than y”, what we mean is that what x would be like if it existed is greater than what y would be like if it existed.

Now, Anselm claims that a perfect being that exists in thought and reality is greater than a perfect being that exists only in thought. But this does not seem true. For what we need to compare is what (a) the perfect being that exists in thought and reality would be like if it existed to what (b) the perfect being that exists only in thought would be like if it existed. But the answer here is that the two perfect beings would be exactly the same under the hypothetical condition that they both existed in reality.

(A query: Wouldn’t the being in (b) be a self-contradictory being if it existed in reality, since it would be a being that exists only in thought and yet that exists in reality? This depends on how the counterfactual is resolved.)

Note that this objection does not apply to the necessary being versions of the argument (like, perhaps, Anselm’s “cannot be conceived not to exist” version). For a perfect being who is a necessary being would be greater, if it existed, than a perfect being who isn’t a necessary being, if that one existed.

Why did God become man?

One way to put the question of the theory of the atonement might seem be to ask with Anselm: "Why did God become man?"

I do not think that that is the right way, though. For God is omnirational. This means that whenever he acts, he is moved by all the unexcluded good reasons for the action. But almost any good that is achieved by a morally permissible action will provide God with an unexcluded good reason for that action. Very high up among the goods achieved by God through the Incarnation is ensuring that we can be forgiven with justice. But there are many other goods achieved by the Incarnation: Aquinas lists nine other great goods. And besides the great goods, there are far lesser but nonetheless genuine goods, such as cooperating with Joseph in the making of tables. Plausibly, there are many goods in between the really great ones and the far lesser ones, such as providing Israel with a king, bringing the nations to Israel in worship, and healing certain lepers through a face-to-face interaction.

One might try to run some kind of a distinction between the goods wrought in the Incarnation, so as to delineate the goods that are the proper subject of a theory of the atonement or the proper answers to "Why did God become man?" Let's try a few such distinctions.

First, there is a distinction between those goods that could be achieved without God becoming man and those goods that can only be achieved by God becoming man. But this won't draw the distinguishing line in the right place. First, pace Anselm, it is far from clear that we couldn't be forgiven with justice simply by divine fiat. Certainly, Aquinas thought we could. It would be fitting for the penalty to be paid by Christ on our behalf, but it would not be unjust for God simply to release us. Second, some of the goods lower down on the list require the Incarnation. Thus, the good of God's cooperating with Joseph in carpentry (this cooperation makes me think of Enoch walking with God, but in the case of Joseph there is a greater literalness) requires an Incarnation. (God can, by his omnipotent power, create tables ex nihilo, but such creation is not carpentry.) Likewise, there is a value to God healing the lepers through a face-to-face interaction of a sort that can only be had by means of the Incarnation.

Second, perhaps we could distinguish between reasons that are such that if they were present on their own, God would still have decided on the Incarnation, and reasons that were only contributory. Thus, while omnirationality implies that God became man in part in order to work with Joseph, surely if no other goods were realized by the Incarnation than God's engaging in carpentry with Joseph, then there would be no Incarnation. But why think that? God could have done it. Maybe it would be an unlikely scenario, but surely a possible one. Why accept the counterfactual of free will that had God had working with Joseph as the only reason to be Incarnate then he wouldn't have done it? Such a counterfactual seems meaningless (or trivially false) to me, and even Molinists need not extend their view to counterfactuals of divine freedom.

Third, perhaps we can distinguish reasons such that were they absent, the Incarnation would not have occurred. But again we get into dubious counterfactuals about divine decisions. Take one of the "big" reasons for the Incarnation. I see no reason to accept the counterfactual that had that reason been absent, there would have been no Incarnation. God might still have become incarnate for the other reasons.

There is, of course, a distinction as to the weight of the reasons. The "big" reasons are much better reasons. But this distinction is one of degree.

So it seems to me that the answer to the question of why God became man is simply a list. Aquinas gives ten items to put on the list. There are many more. We can prioritize the items on the list, of course. But each of the items on the list will be a reason that God was acting in the light of, since God acts in the light of all the unexcluded good reasons in favor of his action.

This does not imply that there is no such thing as a theory of the atonement, just that the question "Why did God become man?" doesn't delineate the theory precisely enough.

Penal substitution theories of the atonement

According to the penal substitution theory of the atonement, Christ's sufferings satisfy justice in place of our being punished. That is, basically, the theory as found in Anselm's Cur Deus Homo.

Some contemporary Christians, mainly Protestant, add the claim that Christ was punished by the Father, and his punishment substitutes for our punishment. We can call the resulting theory punishment by punishment substitution (PBPS). PBPS isn't Anselm's theory, and as Mark Murphy has pointed out it may even be incoherent, since a part of punishing is the showing of disapproval at the person being punished, while God cannot show disapproval at an innocent person.

The Heidelberg Catechism explicitly only says that Christ satisfies for us. But it says in the answer to Question 14 that no mere creature can satisfy for us because "God will not punish any other creature for the sin which man has committed", which may implicate that satisfaction involves being punished. Still, it does not say that it does so in the case of Christ.

In any case, it seems to me that the biblical theory is not that the punishment of Christ substitutes for our punishment, but that the sacrifice of Christ substitutes for our punishment. Old Testament sacrifices for our sins were not punishments of the animals, except in the extended sense of the word as when we speak of "the punishing heat of Texas summer." It is central to the idea of sacrifice in the Old Testament that it is the best that is sacrificed. To sacrifice something is to treat it as the best that is available. But when someone is being punished, then he is far from being treated as the best—he is being treated as one of the worst. Thus, the biblical picture of Christ as sacrificed is in serious tension with PBPS.

That the sacrifice of Christ substitutes for our punishment isn't yet a theory of the atonement. To make it a theory of the atonement one would have to say how it does so.