"On the same grounds"

Each of Alice and Seabiscuit is a human or a horse. But Alice is a human or a horse “on other grounds” than Seabiscuit is a human or a horse. In Alice’s case, it’s because she is a human and in Seabiscuit’s it’s because he’s a horse.

The concept of satisfying a predicate “on other grounds” is a difficult one to make precise, but I think it is potentially a useful one. For instance, one way to formulate a doctrine of analogical predication is to say that whenever the same positive predicate applies to God and a creature, the predicate applies on other grounds in the two cases.

The “on other/same grounds” operator can be used in two different ways. To see the difference, consider:

1. Alice is Alice or a human.

2. Bob is Alice or a human.

In one sense, these hold on the same grounds: (1) is grounded in Alice being human and (2) is grounded in Bob being human. In another sense, they hold on different grounds: for the grounds of (1) also include Alice’s being Alice while the grounds of (2) do not include Bob’s being Alice (or even Bob’s being Bob).

Stipulatively, I’ll go for the weaker sense of “on the same grounds” and the stronger sense of “on different grounds”: as long as there is at least one way of grounding “in the same way”, I will count two claims as grounded the same way. This lets me say that Christ knows that 2 + 2 = 4 on the same grounds as the Father does, namely by the divine nature, even though there is another way in which Christ knows it, which the Father does not share, namely by humanity.

Even with this clarification, it is still kind of difficult to come up with a precise account of “on other/same grounds”. For it’s not the case that the grounds are literally the same. We want to say that the claims that Bob is human and that Carl is human hold on the same grounds. But the grounding is literally different. The grounds of the former is Bob’s possession of a human nature while the grounds of the latter is Carl’s possession of a human nature. Moreover, if trope theory is correct, then the two human natures are numerically different. What we want to say is something like this: the grounds are qualitatively the same. But how exactly to account for the “qualitatively sameness” is something I don’t know.

There is a lot of room for interesting research here.

Grace and theories of time

1. All grace received is given through Christ’s work of salvation.
2. Christ’s work of salvation happened in the first centuries AD and BC.
3. One cannot give something through something that does not exist.
4. Abraham received grace prior to the first century BC.
5. So, Abraham’s grace was given through Christ’s work of salvation.
6. So, it was true to say that Christ’s work of salvation exists even when it was yet in the future.
7. So, presentism and growing block are false.

Junia/Junias and the base rate fallacy

I think it would be useful to apply more Bayesian analyses to textual scholarship.

In Romans 16:7, Junia or Junias is described as “famous among the apostles”. Without accent marks (which were not present in the original manuscript) it is not possible to tell purely textually if it’s Junia, a woman, or Junias, a man. Moreover, “among the apostles” can mean “as being an apostle” or “to the apostles”. There seems to be, however, some reason to think that the name Junia is more common than Junias in the early Christian population, and the reading of “among” as implying membership seems more natural, and so the text gets used as support for women’s ordination.

This post is an example of how one might go about analyzing this claim in a Bayesian way. However, since I am not a Biblical scholar, I will work with some made-up numbers. A scholarly contribution would need to replace these with numbers better based in data (and I invite any reader who knows more Biblical scholarship to write such a contribution). Nonetheless, this schematic analysis will suggest that even assuming that there really were female apostles, it is more likely than not that Junia/s is one.

Let’s grant that in the early Christian population, “Junia” outnumbers “Junias” by a factor of 9:1. Let’s also generously grant that the uses of “famous among” where the individual is implied to be a member of the group outnumber the uses where the individual is merely known to the group by a factor of 9:1. One might think that this yields a probably of 0.9 × 0.9 = 0.81 that the text affirms Junia/s to be an apostle.

But that would be to commit the infamous base rate fallacy in statistical reasoning. We should think of a text that praises a Junia/s as “famous among the apostles” as like a positive medical test result for the hypothesis that the individual praised is a female apostle. The false positive rate on that test is about 0.19 given the above data. For to get a true positive, two things have to happen: we have to have Junia, probability 0.9, and we have to use “among” in the membership-implying sense, probability 0.9, with an overall probability of 0.81 assuming independence. So the false positive rate on the test is 1 − 0.81 = 0.19. In other words, of people who are not female apostles, 19 percent of them will score positive on tests like this.

But we have very good reason to think that even if there were any female apostles in the early church, they are quite rare. Our initial sample of apostles includes the 12 apostles chosen by Jesus, and then one more chosen to replace Judas, and none of these were women. Thus, we have reason to think that fewer than 1/13 of the apostles were women. So let’s assume that about 1/13 of the apostles were female. If there were any female apostles, they were unlikely to be much more common than that, since then that would probably have been more widely noted in the early Church.

Moreover, not everyone that Paul praises are apostles. “Apostle” is a very special position of authority for Paul, as is clear from the force of his emphases on his own status as one. Let’s say that apostles are the subjects of 1/3 of Pauline praises (this is something that it would be moderately easy to get a more precise number on).

Thus, the chance that a randomly chosen person that Paul praises is a female apostle even given the existence of female apostles is only about (1/13)×(1/3) or about three percent.

If we imagine Paul writing lots and lots of such praises, there will be a lot of Junia/s mentioned as “famous among the apostles”, some of whom will be male, some female, and some of whom will be apostles and some not.
All of these are the “positive test results”. Of these positive test results, the 97% percent of people praised by Paul who aren’t female apostles will contribute a proportion of 0.19 × 97%=18% of the positive test results. These will be false positives. The 3% people who are female apostles will contribute at most 3% of the positive test results. These will be true positives. In other words, among the positive test results, approximately the ratio 18:3 obtains between the false and true positives, or 6:1.

In other words, even assuming that some apostles are female, the probability that Junia/s is a female apostle is at most about 14%, once one takes into account the low base rate of women among apostles and apostles among those mentioned by Paul.

But the numbers above are made-up. Someone should re-do the analysis with real data. We need four data points:

• Relative prevalence of Junia vs. Junias in the early Christian population.

• Relative prevalence of the two senses of “famous among” in Greek texts of the period.

• Reasonable bounds on the prevalence of women among apostles.

• Prevalence of apostles among the subjects of Pauline praise.

And without such numbers and Bayesian analysis, I think scholarly discussion is apt to fall into the base rate fallacy.

Is eternalism compatible with the actualization of potentiality?

Every so often, someone claims to me that there is a difficulty in reconciling the Aristotelian idea of the actualization of potential with eternalism, the view that past, present and future are equally real. I am puzzled by this question, because I can’t see the difficulty. On the contrary, there is a tension between presentism, the view that only present things exist, and this Aristotelian thesis:

1. Some present events are the actualization of a no-longer present potentiality.

2. A non-existent thing is not actualized.

3. Therefore, some no-longer present potentialities exist.

4. Therefore, something that is no longer present exists.

5. Therefore, presentism is false.

One might say: Yes, the potentiality doesn’t exist, but it did exist, and it was actualized. But then:

6. Some present potentialities are actualized in not yet present events.

7. A non-existent thing does not actualize anything.

8. So, there exist some not yet present things.

9. So, presentism is false.

Of course, this is the old problem of transtemporal relations for presentism as applied to the actualization relation.

So, what about the question whether eternalists can have actualization of potentials? Here may be the problem. On eternalism plus Aristotelianism, it seems that the past unactualized potential exists even though it is now actualized. This seems to be a contradiction: how can an unactualized potential be actualized?

A first answer is that a potential is actualized at a time t provided that its actualization exists at t. Thus, the potential is unactualized at t₁ but actualized at a later time t₂, because its actualization exists at t₂ but not at t₁. But, the objector can continue, by eternalism at t₁ isn’t it the case that the actualization exists? Yes: but the eternalist distinguishes:

10. It is true at t₁ that B exists.

11. B exists at t₁.

Claim (11), for spatiotemporal objects, means something like this: the three-dimensional spacetime hypersurface corresponding to t = t₁ intersects B. Claim (10) means that B exists simpliciter, somewhere in spacetime (assuming it’s a spatiotemporal object). There is no contradiction in saying that the actualization doesn’t exist at t₁, even though it is true at t₁ that it exists simpliciter.

The second answer is that Aristotelianism does not need actualizations of unactualized potentials. Causation is the actualization of a potential. But Aristotle and Aquinas both believed in the possibility of simultaneous causation. In simultaneous causation, an event B is the actualization of a simultaneous potential A. At the time of the simultaneous causation, nobody, whether presentist or eternalist, can say that B is the actualization of unactualized potential, since then the potential would be actualized and unactualized at the same time. Thus, one can have causation, and actualization of potential, where the potential and the actualization are simultaneously real, and hence where the actualization is not of an unactualized potential. The eternalist could—but does not have to—say that transtemporal cases are like this, too: they are actualizations of a potential, but not of an unactualized potential.

Final and efficient causation

It is sometimes said that:

1. One can have p explain q and q explain p when the types of explanation are different.

I think (1) is mistaken, but in this post I want to focus not on arguing against (1), but simply on arguing against one particular and fairly common form of argument for (1):

2. In cases of Aristotelian final causation, it typically happens that y is a final cause of its own efficient cause.

3. If y is a final cause of x, then that y occurred finally explains that x occurred.

4. If x is an efficient cause of y, then that x occurred efficiently explains that y occurred.

5. So, it’s possible to have p explain q and q explain p when the types of explanation are final and efficient, respectively.

I want to argue that this argument fails (bracketing the interpretive question whether Aristotle or Aquinas accepts its premises).

First, explanation is factive: if p explains q, then both p and q are true. This is because explanations provide correct answers to why questions, and a false answer isn’t correct. But final explanations are not factive. I can offer an argument in order to convince you and yet fail to convince you. (Indeed, perhaps this post is an example.) Therefore, (3) is not always true. That doesn’t show that (3) is false in the case that the argument needs. But it is plausible that an action that fails for extrinsic reasons has exactly the same explanation as a successful action. The failed action cannot be explained by its achieving its goal, since it doesn’t achieve its goal. Therefore, the successful action cannot be explained in terms of its achieving its goal, either.

Second, efficient causation is a relation between tokens. If I turn on the lights in order to alert the burglars, then my token turning-on-the-lights is the efficient cause of the token alerting-the-burglars. But final causation is not a relation between tokens. For suppose that I fail to alert the burglars, say because the burglars are blindfolded (they were challenged to rob me blind, and parsed that phrase wrong) and don’t see the lights. Then there are infinitely many possible tokens of the alerting-the-burglars type any one of which would pretty much equally well serve my goals. For instance, I could alert the burglars at 10:44:22.001, at 10:44:22.002, etc. In the case of action failure, no one of these tokens can be distinguished as “the final cause”, the token I am aiming at. Indeed, if one particular possible token a₀ were the final cause, then if I happened to produce another token, say a₇, my action would have been a failure—which is absurd. Thus, either all the infinitely many possible tokens serve as the final causes of the action or none of them do. It seems wrong to say that there are infinitely many final causes of the action, so none of the tokens is.

Given that explanation of the failed action is the same as of the successful action, it follows that even in the successful case, none of the tokens provides the final cause.

Therefore, we should see final causation as a relation between a type, say alerting the burglars at some time or other near 10:44:22, and a token, say my particular turning on of the lights. But if so, then (2) is false, for it is false that in the successful case the same things are related by final and efficient causation: the final causation relates the outcome type with a productive token and efficient causation relates the productive token with the outcome token.

As I said, this doesn’t show that (1) is false, but it does show that efficient and final explanation do not provide a case of (1).

Acknowledgments: I am grateful to Tim Pawl for discussion of these questions.

Truly going beyond the binary in marriage?

There is an interesting sense in which standard polygamy (i.e., polygyny or polyandry) presupposes the binarity of marriage. In standard polygamy, there is one individual, A, who stands in a marriage relationship to each of a plurality of other individuals, the Bs. But the marriage relationships themselves are binary: A is married to each of the Bs, and the Bs are not married to each other (they have a different kind of relationship).

The same would be true with more complex graph theoretic structures than the simple star-shaped structure of polygyny and polyandry (with A at the center and the Bs at the periphery). If Alice is married to Bob and Carl, and Bob and Carl are each married to Davita, the quadrilateral graph-theoretic structure of this relationship is still constituted by a four binary marriage relationships.

Thus, in these kinds of cases, what we would have are not a plural marriage, but a plurality of binary marriages with overlap. This, I think, makes for more precise terminology. The moral and political questions normally considered under the head of “plural marriages” are about the possibility or morality of overlap between binary marriages.

To truly go beyond the binary would require a relationship that irreducibly contains more than two people, a relationship not constituted by pairwise relationships. I think a pretty good case can be made that even if one accepts overlapping binary marriages, as in standard polygamy, as genuine marriages (I am not sure one should), irreducibly non-binary relationships would still not be marriages (just as unary relationships wouldn't be). The structure of the relationship is just radically different.

God and analogy

According to Aquinas, whenever we correctly say something non-negative of God, we speak analogically.

It is correct to say that Socrates is wise and God is wise. But being humanly wise and divinely wise are different—the most fundamental difference being that, by divine simplicity, God doesn’t have his wisdom, but is his wisdom. But this leads to:

1. The predicate “is humanly wise or is divinely wise” applies literally to both Socrates and God.

And yet this disjunctive predicate is not negative, so (1) seems to provide a counterexample to Aquinas’ theory.

But this is too fast. Claim (1) only provides a counterexample to Aquinas’ theory if:

2. Applying analogically and applying literally are incompatible.

But I think Aquinas can, and should, say that (2) is false. If he does that, then he can affirm both (1) and:

3. The predicate “is humanly wise or is divinely wise” applies analogically to both Socrates and God.

In fact, I think Aquinas can say that the relevant kind of analogical application of predicates is a special case of literal predication.

I think that Aquinas is not really making a claim about literal and non-literal use of words when he is talking of analogical predication. Instead, I think he is making a claim about grounding, somewhat like:

4. The predicate “is F” is used analogically between entities x and y just in case the propositions that x is F and that y is F have a relevantly different grounding structure.

On this account, disjunctive predicates like “is a human or a dog” are used analogically: for the grounding structure of the proposition that Alice is a human or a dog is that it’s grounded in Alice being a human, while the grounding structure of the proposition that Fido is a human or a dog is that it’s grounded in Fido being a dog. And similarly, “is humanly wise or is divinely wise” is used analogically, since in the case of Socrates the grounds of applicability are Socrates having wisdom and in the case of God the grounds are God being (his) wisdom.

Notice that on this story, Aquinas’ claim about analogical predication is not so much a linguistic claim as a metaphysical claim about the truth grounds.

The story makes clear why negative predicates are not used analogically: for the grounding structure of the truth that God is not a bicycle and the truth that Alice is not a bicycle is relevantly the same—both are grounded in not being arranged bicycle-wise.

So far, our reconstruction of Aquinas’ theory of predication is:

5. A predicate that applies to God is negative or is used analogically.

But that’s not quite right. Here is one counterexample: “is not a bicycle or is both a bicycle and a non-bicycle.” This predicate is not negative but disjunctive. But it applies to God and to Socrates in the same way—by both not being bicycles.

I think the issue here is this. Just as analogical predication is a metaphysical and not linguistic notion, so negative predication is a metaphysical and not linguistic notion. We might say something like this:

6. The predicate “is F” is used negatively of entity x just in case what grounds x being F is the non-obtaining of some state of affairs.

Thus, “is not a bicycle or is both a bicycle and non-bicycle” is used negatively of both God and Socrates, because what grounds its application in both cases are respectively the non-obtaining of the states of affairs of God being arranged bicycle-wise and of Socrates beng arranged bicycle-wise. On the other hand, the disjunctive predicate “is Athenian or not Greek” is used negatively of God and non-negatively of Socrates. Interestingly, this case shows that the disjunction in (5) is not exclusive. For “is Athenian or not Greek” is used both negatively of God and is used analogically between Socrates and God, since the structure of the grounds of application is relevantly different.

The problems haven’t all gone away. A necessary condition for “is F” to be used analogically of God and a creature is that “is F” applies to God and a creature, and hence a predicate that applies only to God cannot be used analogically. But suppose that in fact no one other than God knows whether the Continuum Hypothesis is true. Then the predicate “knows whether the Continuum Hypothesis is true” is not used analogically, since it only applies to God. But then we have a counterexample to (5).

We could try to modalize (4): a predicate is used analogically provided that it could have one ground as applied to God and another as applied to something other than God. But, again, it’s not hard to come up with a counterexample: “knows that 2 + 2 and is not a creature.” For that predicate can only apply to God.

We could also weaken (5) to merely apply to those predicates that apply (or could apply) to both God and a creature. This may seem to be an undue weakening: now one can escape from Aquinas’ doctrine of analogical predication simply by saying things that only apply (or could only apply) to God. But perhaps one can supplement the weakened (5) with:

7. Any predicate that applies to God is built out of predicates that apply both to God and to a creature.

I am not too happy about this.