Knowledge and induction

Assume that in fact all ravens are black. Suppose you are sequentiallly observing ravens, and noting each one to be black. After observing n ravens, your evidence that the next raven is black will typically be significantly better than the evidence that all ravens are black. Now at some point, say after observing n_A ravens, your evidence that all ravens are black will rise to the level of knowledge. Thus, plausibly, at an earlier point in the sequence, call it n_N, your evidence that the next raven is black will have risen to the level of knowledge.

Suppose now you have observed n_A − 1 ravens, and you have been handed a raven in an opaque box, which you are certain you are about to open. Since n_N < n_A, at this point you have reached n_N. Hence:

1. You do not know that all ravens are black.

2. You do know that the next raven is black.

3. You know that when you observe the next raven, you will have sufficient evidence for knowledge that all ravens are black.

But note that while you know you will have sufficient evidence for knowledge that all ravens are black, you don’t know that you will know that all ravens are black. There is nothing deeply surprising about this distinction. We might well say about someone who has been subjected to misleading or Gettiered evidence that they have sufficient evidence to know something but nonetheless they don’t know, though the case at hand feels different.

One interesting thing about this case, as I read it, is that it contradicts the thesis that K = E, i.e., that knowledge is evidence. For if knowledge is evidence, and you know that the next raven is black, then you already have the evidence you will gain by observing the next raven, and hence you are already in the position to know that all ravens are black.

Another interesting thing is that it shows that you can know something and nonetheless it be rational for you to investigate it. For you know that the next raven is black, but it’s worth investigating further, since it is only upon observation that your knowledge of the next raven’s blackness turns into the kind of evidence that gives you knowledge that all ravens are black.

All this might make one think that I have misconstrued the epistemic facts, and it is false that there can be a point n_N prior to n_A at which you know that the next raven is black. Here is one way to back up my intuition that there can be such a point n_N < n_A. Suppose that we know for sure we live in a world where the color distributions of birds are always uncorrelated between the males and the females of the species, so that information about the color of members of one sex are irrelevant to the color of members of the other sex. Also assume that you know for sure that ravens have equal numbers of each sex, that you are observing ravens in an alternative female-male-female-male-… sequence, and that your priors for the color distributions of the two sexes of ravens are the same. Then if p_M and p_F are the probabilities that all male ravens are black and all female ravens are black, and p_A is the probability that all ravens are black, then at any given point in the observation sequence p_A = p_Mp_F. Let n_M and n_F be the points in the sequence where you know that all male and all female ravens are black, respectively. Then, n_M < n_A and n_F < n_A, since p_A = p_Mp_F is significantly smaller than either p_M and p_F at all points in the sequence except when we’ve observed all the ravens of one sex, and since p_M and p_F rise fairly gradually as we go through the sequence. Thus, at the point n_A − 1, we will have already reached knowledge that all the male ravens are black and the knowledge that all the female ravens are black. In particular, then, we know that the next raven is black, since if n_A is even, the n_Ath raven is male and we know all male ravens are black, and if n_A is odd, then the n_Ath raven is female, and we know that all female ravens are black.

Epistemic rationality and Pascal's Wager

Pascal’s Wager is an argument that it is prudentially rational to engage in theistic belief promotion practices (TBPP), namely practices apt to promote one’s belief in God.

My interest this post is the standard epistemic rationality objection to the Wager, that engaging in TBPPs is irrational—a kind of brainwashing of oneself. Let’s think about the objection with a bit more care. Consider a specific TBPP Q, say Pascal’s example of going to Mass. If Q is indeed a TBPP, one expects engagement in TBPP to make it more likely that one believes in God. But how does one expect Q to achieve that goal? There are two possibilities. Either Q is expected to achieve that goal by providing one with evidence for theism or in some non-evidential way (or a combination of the two).

Suppose Q is expected to work evidentially. Then we already have the expectation of a higher credence given Q. This expectation is either rational or not. If it is not rational, then we don’t actually have good reason to engage in Q. If it is rational, however, then we should rationally raise our credence in theism right now, without having to bother engaging in Q, and for reasons having nothing to do with any wager. But if Q promotes belief in God non-rationally, then we should not engage in Q for the sake of such promotion of belief—we should not aim to non-rationally promote beliefs.

Let me make the first horn of the dilemma—namely, that the expectation of a higher credence is rational—a bit more precise. We can distinguish two (not mutually exclusive) ways in which a practice rationally increases one’s credence in a hypothesis H. One way is purely epistemic, by uncovering facts about reality. This is the usual way. But if that’s the way we expect to increase our credence in theism by engaging in Q, then we already have evidence that there are such theism-indicating facts to be discovered, and so we should already increase our credence in Q. The other way is practical, by promoting the hypothesis H in a way that shows up to us. The second way is a bit unusual, but here is an example: one way to increase your credence that you will not die of heart disease is to live a healthy life. For if you live a healthy life, you are less likely to die of heart disease, and since you will notice signs of improved cardiac health (e.g., lower resting heart rate, less huffing and puffing on stairs, etc.), your credence that you won’t die of heart diseases will also increase. But this practical way of increasing credence is utterly irrelevant in the case of theism, since nothing we can do can make God more or less likely to exist! So the only rational way that remains is the evidence-based way, and evidence-of-evidence is already evidence.

I used to be quite impressed by the worry that Pascal’s Wager leads to self-deception. I am less impressed. Here is why. There is a serious technical flaw in the argument for the first horn of the dilemma. A simple model for the relation of credence and belief is that you believe a proposition if and only if your credence is above some threshold β. This model might be false, but an analogue of what I will say should apply on more sophisticated models as well.

Here is the point. Consider a case where one is thinking about observing (and suppose this is a simple non-Newcombian observation that does not affect the hypothesis) whether some event E evidentially relevant to H has obtained. Then one’s expected posterior credence is:

1. C(E)C(H∣E) + C(∼E)C(H∣∼E),

where C is one’s credence function. But if one is a good Bayesian reasoner, then by total probability the value in (1) is simply equal to one’s prior C(H). Thus the value of one’s credence has no expectation of change upon observation when one is being rational. This seems to support the idea that if you expect your credence to go up, you should already raise it.

But in fact it’s not so simple. For even though the expected posterior credence equals one’s current credence, it could well be that it is more likely that the expected posterior credence exceeds the threshold β if you make the observation than if you don’t. Indeed, cases are obvious. Suppose the belief threshold β is 0.9, and you tossed a coin out of my sight. Suppose I have a prior credence 0.5 that this coin is fair and a prior credence 0.5 that it is double-headed. Then currently I don’t believe (or disbelieve) that the coin is fair. But if I look at the coin and I see tails, I will believe that it is fair—indeed, I will have posterior credence 1 in its fairness. But if I don’t look at the coin, I am not going to get any evidence, and I will continue not to believe that the coin is fair. If I look at the coin, the probability that I will see tails is 0.25 (I have credence 0.5 that it’s fair, and if it’s fair, the chance of tails is 0.5), and so the probability that I will believe if I look is 0.25 (since if I look and see tails, my posterior will be 1 which is bigger than β = 0.9), and the probability that I will believe if I don’t look is 0. Of course, if I don’t see tails, my credence that the coin is fair will go down. But while it will go down, it won’t affect whether I believe that the coin is fair—for I already don’t believe it (in the sense of not-believe, rather than in the sense of believe-not). And there is no irrationality of any sort in looking at the coin in this case.

In other words, the point is that while one’s rational credence has no positive or negative rational expectation of change upon observation, whether one’s rational credence is above a threshold certainly can have a positive or negative rational expection of change.

How could this work in a Pascal’s Wager situation? Let’s talk through one possibility. Take Pascal’s example of going regularly to Mass. Suppose, as Pascal says, your current credence in God is 0.5. You might think that if God exists, going to Mass has a decent chance, say 0.2, of resulting in an evident radical transformation E of your life, so evident and radical that updating your credence in theism on E will push your credence in God to above the threshold β. Of course, you might go to Mass it might not produce any such evident radical transformation (this is true even if God always improves the hearts of people who go to Mass, since he might do so more gradually or less evidently), and in that case your rational credence in God will go below 0.5. But going below 0.5 won’t affect whether you believe in God, since 0.5 is already, I assume, far below the belief threshold β. On the other hand, maybe if you don’t go to Mass, the probability that you will get evidence that will push your credence in theism above β is pretty small—smaller than 0.2. Very likely, your credence will just oscillate a little around 0.5 in the non-Mass-going case. Thus if there is a payoff you get for your credence exceeding the threshold β, it will be worth going to Mass, without there being any epistemic irrationality in the reasoning.

Thought of in this way, we get some practical guidance as to which TBPPs the agnostic or atheist should engage in. They should look for practices that, if God exists, have a decent chance of producing evidence for theism sufficient to push them above β.

I am a bit doubtful that Pascal meant us to think in the above way. He may well have been recommending TBPPs on the grounds of their non-rational effect on belief. My argument above does not defend that.

What's a properly epistemic difference?

There are different types of knowledge: mathematical, geological, biological, a priori, a posteriori, de se, interpersonal, hard-earned, esoteric, very well justified, professional, certain, firm, testimonial, propositional, etc.

When we divide up knowledge, some of the divisions are more properly epistemic than others. For instance, the difference between hard-earned knowledge and easy-come knowledge is not an epistemic difference at all—it is just a difference between how one came by it. If you and I live in villages on the opposite side of a mountain ridge, my knowledge of the waterfall outside our village is easy-come knowledge, while you had to work hard to earn that knowledge, climbing over the mountain ridge. But we are epistemically on par with respect to the knowledge of the waterfall. So, some types or classifications of knowledge do not divide up the knowledge epistemically at all. We might call these “wholly accidental” classifications of knowledge.

Other classifications of knowledge divide up the knowledge by means of the type of justification. For instance, the classic distinction between a priori and a posteriori is like this. Notice, however, that the very same thing can be known a priori and a posteriori. If you use a calculator to find out what 117 × 19 is, you learn it a posteriori, but if you use figure it out in your head, it’s a priori. And when the evidence is equal in strength and the content is the same, we do not have an epistemic difference to mark. (In the past, we might have said that a priori knowledge is better evidenced and maybe even infallible, but we now know that this is not in general true.) I am not sure whether the difference between a priori and a posteriori knowledge is a genuine epistemic difference. After all, the difference between knowledge gained by reading books in French and reading books in German doesn’t seem to be.

Yet other classifications of knowledge divide up the knowledge by the quality or degree of justification. Knowledge comes with better and worse justifications. This is a genuinely epistemic division.

The division of knowledge based on degree of certitude is ambiguous. By degree of certitude one could just mean a psychological feature of the agent, in which can this is not a properly epistemic division, or one could mean the degree of justification, in which case it is, or one could mean a combination of the two (“well-justified confidence”).

Another properly epistemic division of knowledge is with respect to content. Some differences of content do not seem to mark a genuine epistemic difference—say, the difference between biological and geological knowledge. But others do, like the difference between propositional and interpersonal knowledge do. At the same time, some of the content-based classifications include both an accidental and a properly epistemic element. Thus, it might be that interpersonal knowledge by definition must be gained through interpersonal interaction. But the epistemic benefits of interpersonal knowledge can be had without interpersonal interaction. So the concept of interpersonal knowledge may contain some epistemically accidental features about how the knowledge was in fact gained and some properly epistemic features as to the content.

This is all a bit of a mess.

Second person knowledge and reciprocal interaction

A number of philosophers have posited a special “second-person knowledge”, expressible by phrases like “Bob and Alice know each other”. It is often claims that second-person knowledge requires reciprocal interpersonal interaction.

I think this is false. Suppose Alice and Bob live on Earth but have not yet had any interaction. There is a Twin Earth, where by chance everything happens like on Earth, and there are Twin-Alice and Twin-Bob there. Suddenly Earth and Twin-Earth diverge, as follows.

• Twin-Alice and Twin-Bob are teleported away for a vacation on another planet. They don’t enter into our story after this.

• Bob is teleported to Twin-Earth, where he takes the place of Twin-Bob, without noticing anything being different.

• On Earth, Bob is replaced by Robo-Bob, who is a robot who looks just like Bob, but operates on non-intelligent software that generates random human-like behavior.

• On Twin-Earth, Twin-Alice is replaced by Robo-Alice, who looks just like Alice, but operates on non-intelligent software that generates random human-like behavior.

• On Earth, Alice meets Robo-Bob, and they have what to all appearances is a reciprocal interpersonal relationship of the sort that generates mutual second-person knowledge.

• On Twin-Earth, Bob meets Robo-Alice, and they have a similar “relationship”, with Bob behaving normally for his character.

• In these “relationships”, the behavior of Alice and Bob is such that it would be normal for them (given their respective characters) in the context of a normal interpersonal relationships.

At this point, Alice and Bob think that the above “relationships” yield second-person knowledge, but they don’t, because these relationships are with a randomly behaving non-intelligent robot. Let’s add this:

• By chance, on Earth, Robo-Bob happens to behave just like Bob behaves on Twin-Earth.

• By chance, on Twin-Earth, Robo-Alice happens to behave just like Alice behaves on Earth.

Still, Alice and Bob don’t have second-person knowledge in virtue of the “relationships” with robots. For one, they don’t even know with whom that relationship would be. But there is a way in which they have something like Gettiered second-person knowledge with an unknown person. Finally, one more thing happens, a big reveal.

• Alice and Bob are informed of everything that happened above, and they are given knowledge of who the real Alice and Bob are, say by being shown genuine properly labeled photographs of one another.

At this point, I think a case can be made that Alice’s knowledge of Bob is sufficiently close to the kind of knowledge that she would have had if she had been interacting with Bob rather than Robo-Bob. Alice might say: “Since I know that the real Bob was behaving to Robo-Alice just as Robo-Bob was behaving to me, while Robo-Alice was responding to Bob just as I was to Robo-Bob, and both persons were behaving in the normal way for their character, I know as much about Bob from Robo-Bob’s behavior as I would have had I had a normal relationship with Bob.” And Bob might say the same thing, mutatis mutandis.

But there was no reciprocity. Alice’s behavior does affect Bob once Bob learns that Alice behaved just like Robo-Alice, and Bob’s behavior does affect Alice once Alice learns about Bob’s actual behavior, but this is a pair of one-way interactions rather than reciprocal.

There are three things one might say here:

1. Alice and Bob have second-person knowledge of each other.

2. Alice and Bob don’t have second-person knowledge of each other, but what they have has all of the epistemic value of second-person knowledge.

3. There is something complex about interaction that I have a hard time putting my finger on that makes for a relevant difference.

A long walk

Alice has lived forever in a universe with an infinite road that has a beginning and no end, and is marked every mile. Every day of her life, by an irresistable longing, she has followed these rules:

1. If somehow she’s not on the road, she goes to mile zero on the road.

2. If she is on the road, she walks a mile in the endless direction of the road.

3. Besides the movement required by 1 and 2, she stays in place.

Where is Alice now? Nowhere! There is no possible present scenario compatible with the rules. She can’t either be at mile zero or off the road, because if two days ago she was on the road, she would now be on the road now be past mile zero, and if two days ago she wasn’t on the road, she would now be on the road past mile zero. She can’t be at mile n, because then she would have to have been off the road some time back, which violates the previous argument.

The rules are all coherent. A person who has to follow these rules seems to be possible. Yet the story is impossible. What went wrong? The neatest explanation seems to be that a (causally connected) infinite past is impossible.

(This is inspired by a recent infinite past Grim Reaper story I read from Rob Koons.)

A theological argument that justified true belief is not knowledge

My 13-year-old daughter came up with a rather nice argument against taking knowledge to be nothing but justified true belief.

Jesus tells us that no one knows the day or the hour of his return. But now for each of the 24 possible hours, we can arrange for someone to have reason to believe that that hour is the hour of return. One of these 24 people will then have a justified true belief as to the hour when Jesus returns. If justified true belief is knowledge, then this would contradict what Jesus told us.

Of course, likely, when Jesus said that no one knows the hour, he was probably talking of a specific hour on a specific date—next Tuesday noon, say, rather than a noon in general. But the argument adapts. If the second coming is somewhere in the next 900,000 years, we could divide up the 8 billion people on earth, give each one reason to believe the second coming is during a specific hour during the next 900,000 years, and then one would be right, and would know, if knowledge is justified true belief.

That said, Jesus didn’t say that no one can know the hour, but only that no one knows the hour. Thus as long as no one actually lucks out and has a justified true belief, we have no contradiction to Scripture even if knowledge is justified true belief.

However, apart from the theological ramifications, I think this argument shows that pretty much anything that could be put into language could be known if knowledge is justified true belief. And that is implausible.

This line of argument also damages this line of thought of mine.

A two-sorted logic and the Trinity

For a while this spring I’ve been thinking about ways of avoiding quaternity views of God.

The problem is that, first,

1. Father ≠ Son, Son ≠ Spirit and Father ≠ Spirit

and, second, we seem to have to choose between the following two options:

A. Father = God, Son = God and Spirit = God, or

B. Father ≠ God, Son ≠ God and Spirit ≠ God.

Add:

2. Father is divine, Son is divine, Spirit is divine, and God is divine.

If we accept (1) and (A), then we have a logical contradiction given classical identity.

If on, other hand, we accept (1), (B) and (2), then there are four that are divine, a quaternity!

A now-standard solution to problems like this is to go for a version of a relative identity theory rather than classical identity. Then the (negated) = in (1) and the = in (A)/(B) are different identity relations (e.g., sameness of person versus sameness of essence).

In this post, I want to consider a somewhat different take on non-classical identity. Suppose we take a variant on two-sorted logic. On a many-sorted logic, bound variables and names come with sorts, and predicates have grammatical restrictions on the sorts of terms that can be their arguments. Usually, the restrictions say for each argument place what sort of term can go in that place. But we can have a more complicated kind of sort restriction.

Suppose, then, we have two sorts: essence and hypostasis, and that = has the classical rules of inference, but has the sort restriction that only terms of the same sort can go on the two sides of =. Suppose that “Father”, “Son” and “Spirit” are of the hypostasis sort, and “God” is of the essence sort. Then we can have (1), but neither (A) nor (B) will express a truth, as both (A) and (B) will be ungrammatical. Even if = has the classical rules of inference, I think there will be no way for us to derive that there are four that are divine. Indeed, in the two-sorted logic, the only way to say “there are four that are divine” is:

3. There are four essences that are divine or there are four hypostases that are divine.

For we cannot mix the essence-variables and hypostasis-variables in an “=” formula.

Interestingly, I think we can have an even stronger non-classical logic of identity while avoiding incoherence and quaternity, though I don’t know if this will appeal to anyone. Make the sort restriction on = be that u = v is grammatical if and only if either u and v are both of the same sort or u is a hypostasis term and v is an essence term. Next, specify that the =-elimination says that from the sentence a = b and a formula ϕ(a), we are allowed to infer ϕ(b), but only if “ϕ(b)” is grammatically correct.

In this stronger logic, we can have all three of (1), (A) and (2), apparently without contradiction. The crucial thing is that from Father=God it is impossible to conclude God=Father, because the latter is ungrammatical. It seems to me to be the Holy Grail of Trinitarian logic to be able to affirm all of (1), (A) and (2).

That said, I don’t like the asymmetric sort restriction on =.

Non-local real presence of Christ

Aquinas’s account of location for material substance is as follows. Ordinary material substances have a special accident—one more fundamental than all other matter-related accidents—he calls “dimensive quantity”, but which I will just call “dimensions”. This accident makes the object have a specific shape and size. The substance is then in a place provided that its dimensions are “commensurate with” that place—namely, provided that the dimensions are fitted into the place.

We can now say that a substance x is present in a place z in virtue of the following two presentness facts:

1. x is present in its dimensions D

2. the dimensions D are present in place z.

Furthermore, in the ordinary case, we can have an account of what “present” means in (a) and (b). A substance’s being present in dimensions D is just the substance’s having the dimensions D as an accident. And the dimensions being present in a place z is just their being commensurate with z. (This last one would bear more analysis, but that’s not my interest here.) When we have (a) and (b) with “present” grounded in this way—by having and commensuration respectively—we have what St Thomas call “local presence” or what we might call “ordinary physical location”.

Now, in the Eucharist the following happens according to Thomas. The substance of the bread turns into the body of Christ. The accidents of the bread miraculously remain in existence, but are no longer the accidents of any existing substance (whether bread or Christ). In particular, the dimensions D of the bread remain (and are a subject for the other accidents). And Christ’s body is present on the altar (say) because of the following two presentness facts:

3. Christ’s body is present in the dimensions D which are formerly of the bread

4. the dimensions D are present on the altar.

The ground of (d) just like that of an ordinary case of (b): the dimensions are commensurate with the place. But since Aquinas insists that Christ does not take on the accidents of the bread, the ground of (c) must be different from the ground of the ordinary case of (a): Christ does not have D. (In particular, Christ is not round and thin when transsubstantiation happens in a western Catholic Church.) Instead, Thomas says about (c) that Christ is “substantially” in the “foreign” dimensions D, but is not a subject of these dimensions, i.e., does not have them. As a result, we have the same structure of presence as in the ordinary case—it is mediated by dimensions—but because the grounds of the substance’s presence in the dimensions are different, this is not ordinary physical location any more.

In my 2008 paper, I gave up on figuring out what is meant by “substantial presence”, and indeed suggested that the problem is insoluble, and we should go for a different solution, one on which Christ has ordinary physical location on the altar. That solution may well be right, but I want to try out a more Thomistic solution—though the full story does not fit with everything Thomas says.

Consider the relationship between Seabiscuit and his accident of swiftness. This relationship has two features which make for interdependence:

1. Seabiscuit’s swiftness ontologically depends on Seabiscuit, i.e., Seabiscuit ontologically sustains his swiftness

2. Seabiscuit is qualified by his swiftness.

In all ordinary cases, the relations of ontologically sustaining and being qualified by between a substance and an accident are coextensive. Seabiscuit sustains all his accidents and they all qualify him, and similarly for all substances. To have an accident is then for the accident to ontologically depend on one and for one to be qualified by it.

Expanded out this way, we have a richer story as to what grounds an ordinary substance being present in its dimensions: the substance ontologically sustains the dimensions and is qualified by them.

Now, Thomas’s denial that Christ is “subject to” the dimensions of the bread means one aspect of the ordinary relationship between a substance and its accident is absent here—Christ is not qualified by the dimensions. However, that still leaves the possibility that the other aspect of the relationship is present. In other words, we can suppose that miraculously Christ’s body ontologically sustains “foreign” accidents that this body is not qualified by. This ontological sustenance relationship makes Christ’s body be substantially in the accidents, including in the dimensions.

Thus, the expanded account is:

1. Christ’s body is present in the dimensions D formerly of the bread by ontologically sustaining these dimensions in the way that a substance sustains its accidents but without D qualifying Christ’s body and hence without D becoming its accident.

2. The dimensions D are commensurate with a place, just as in ordinary physical location.

The presence in (1) is a special case of a type of presence that Thomas recognizes in his account of divine omnipresence. Thomas says that God is present to all things “by his essence”, namely by directly being their cause. This causation is, of course, divine sustenance. Thus, Christ’s body’s being “substantially” in the dimensions by sustaining them ontologically is like God’s being “by essence” present to all things by sustaining them. This is a recognized and metaphysically serious mode of presence, and hence it plausibly counts as a real presence.

At this point, it may seem that I have solved the problem of what Aquinas means by the substantial presence of Christ’s body in the dimensions of (the former) bread. But there is one hitch. I think Aquinas disagrees with my account. When Aquinas discusses how the accidents of bread and wine can remain without their substances, his answer is not that the body of Christ sustains them, but that God sustains them, because anything that can be done by creaturely causes can be done by God. St Thomas’s phrasing very much sounds like he thinks the sustenance of the accidents is done directly by the power of God.

The account I am offering requires that God miraculously bestow on Christ’s body the power to sustain accidents foreign to it (without being qualified by them). I don’t see any good reason to think this can’t happen. We thus have an extension of Thomas’s account, but it is one that I think is compatible with other aspects of his metaphysics and theology.

I am still not completely convinced that I should abandon my account on which Christ’s body is present in the Eucharist by ordinary physical location. My account clearly makes Christ’s body by really present. The modified Thomistic account may do that, but it may not.

I want to end with a consideration in favor of the Thomistic view that Christ’s presence in the Eucharist is not ordinary physical location. Many Protestants think that Christ’s body is “spiritually present”, and historically the Reformed wing of the Protestant tradition has taken spiritual presence quite seriously—not just as symbol—while denying ordinary physical presence (I am grateful to one of my graduate students for pointing this out). An account of Christ’s real presence that makes Christ not be ordinary physically present thus has an advantage: it fits with the intuitions not only of many Catholic thinkers but also of many non-Catholic ones. Perhaps the modified Thomistic account just is what spiritual presence is, and hence we have a way of moving Catholics and some Protestants closer together through St Thomas.

Human ridiculousness

1. Humans are ridiculous.

2. Humans are only ridiculous if there is a being much greater than humans.

3. So, there is a being much greater than humans.

A counterexample to Weak Supplementation

The Company axiom of mereology holds that an object cannot have only one proper part. This is a weaker version of the (Weak) Supplementation axiom which holds that if an object has a proper part, it has another proper part that doesn’t overlap the first.

Say that an object is simple at a time t provided that its instantaneous temporal part at t is simple.

Suppose we accept that:

1. fundamental particles have instantaneous temporal parts at every time at which they exist

2. fundamental particles are simple at every time at which they exist

3. there is no contingent identity.

Now, suppose x is a fundamental particle that comes into existence at time t₁ and persists until t₂ > t₁. Then it has an instantaneous temporal part y at t₁. Then y is a proper part of x: it is a part of x and distinct from x. Consider a world w that is just like ours up to and including t₁, but x comes to an end at t₁ (maybe time itself comes to an end at t₁, if one wants to be extreme). Then in w, y is still a part of x. And it must still be distinct from x, since identity cannot be contingent (this argument uses the Brouwer axiom).

Since x exists only at t₁ in w, any part it has at w is a part it has at t₁. The only candidates for such parts are x and y. Thus, in w, y is the only proper part of x. So, contrary to Company (and Supplementation) x has a proper part y and no other proper part.

I am inclined to deny (a). But I am also inclined to deny Company.

The Trinity and classical identity, again

Let me re-phrase an argument from an earlier post.

Suppose “=” is governed by the classical rules of identity, and G, F, S, and H are names for God, the Father, the Son and the Holy Spirit. Let x ≠ y abbreviate Not(x = y). Let D(x) say that x is divine. There then are three predicates A, B and C such that:

1. A(S) and not A(F)

2. B(H) and not B(S)

3. C(S) and not C(F).

For instance, we can let A(x) say that x is begotten, B(x) say that x is not begotten, and C(x) say that x proceeds.

Then add:

4. Either (a) F = G and S = G and H = G or (b) F ≠ G and S ≠ G and H ≠ G.

5. D(F)

6. D(S)

7. D(H)

8. D(G)

Premise 4 says that the Father, Son and Holy Spirit are on par with respect to being God. Either each is classically identical with God, or classical identity does not apply to any person and God (in which case we explain “The Father is God” as something other than classical identity).

It follows from (1)-(3):

9. F ≠ S and S ≠ H and H ≠ S.

It then follows that (4)(a) is false, so we must have:

10. F ≠ G and S ≠ G and H ≠ G.

It then follows from (9) and (10) and classical identity being an equivalence relation that:

11. ∃w∃x∃y∃z(D(w)∧D(x)∧D(y)∧D(z)∧w≠x∧w≠y∧w≠z∧x≠y∧x≠z∧y≠z).

But (11) is the standard classical logic translation of:

12. There are at least four that are divine.

Heresy!

What are the ways out? We can’t reject (1)-(3). Even if we have some quibbles about the specific examples I chose for A, B and C, every orthodox Trinitarian agrees that for each pair of persons there is something that is truly predicated of one that isn’t truly predicated of the other.

Rejecting any of (5)-(7) is a non-starter: one isn’t a trinitarian if one does not say that the Father is divine, the Son is divine and the Holy Spirit is divine.

That leaves (4) and (8). Start with (8). That seems as uncontroversial as anything about God can be. God is divine!

But here is one way of rejecting (8): reject the presupposition that there is a name, “G”, for God. If we do that, we also end up rejecting (4), of course, but not in a way that threatens the parity of the three persons of the Trinity with respect to being God. I think there are two ways of doing this:

1. Reject the claim that there is a proper name for God as such.

2. Reject the very existence of classical identity.

It is tempting to say that instead of rejecting the very existence of classical identity to God, one can reject its applicability to God. But that can’t be done. It is part of the very concept of classical identity that it applies to everything, that for any name N it is axiomatic that N = N and that it is a theorem that ∀x(x=x).

What about (I)? Surely this is a non-starter. Doesn’t the Christian tradition constantly talk about names of God? Well, yes, but there are names and proper names. What if we say this? There are proper names for the Father, the Son and the Holy Spirit. But there is no proper name for God. Instead, what we have is something like a definite description like “the divine one”.

We still haven’t solved the problem. The normal way to understand definite descriptions is the Russellian way. When you say “The divine one created the world”, you are saying:

13. ∃x(D(x)∧∀y(D(y)→y=x)∧C(x,W)).

That won’t do, however. For (13) leads to a contradiction when it is combined with (1) together with the non-negotiable Trinitarian claim

14. D(F) and D(S)

that the Father is divine and the Son is divine, as well as classical inference rules for identity.

So, what do we do? Here is a suggestion. Let R be an equivalence relation. Then we can have an R-based article “the_R”, and sentences with “the_R” are translated in the Russellian way except with R in place of =.

(Compare how Aquinas makes the distinction between talking in the neuter and talking in the masculine of God, and where when one applies substantives in the neuter, one is talking of the divine essence. One can think of “the_E” as a neuter article, which English doesn’t distinguish from the personal—masculine or feminine—articles, and which Latin lacks altogether, since it lacks articles.)

Thus, “The divine one created the world” translates to:

15. ∃x(D(x)∧∀y(D(y)→yEx)∧C(x,W)).

No contradiction results from (1)-(3) and (14)-(15).

We can call “the divine one” an E-definite description, while “the begetter” is an =-definite description. Aquinas at times in his discussion of the Trinity makes a distinction between substantives used in the neuter and substantives used in the masculine—the masculine is personal in a way that the neuter is impersonal and more suited to when we talk of the divine essence.

Thus, on our present theory, God as such has no proper name, but he does have E-definite descriptions.

Now, what are we to make of the truth value of the following?

16. The Son is identical with the divine one.

If “The Son” is just a proper name and “the divine one” is “the_E divine one”, then (16) is:

17. ∃x(D(x)∧∀y(D(y)→yEx)∧x=S).

And this is false, because it contradicts (5), (6) and (9). So, on the theory under consideration, we have to deny (16). That sounds kind of bad. But perhaps it’s not bad if we realize that “identical” here is classical identity, and we think that classical identity comes to “is the same hypostasis as”, since in the divine case, same hypostasis means same person, and it sounds wrong to say that the Son is the same person as God.

So, I think there is a way of holding on to classical identity while defending the Trinity, but it is costly: we need to say that there is no proper name for God as such and that definite descriptions for God are E-definite descriptions. But denying classical identity is also costly.

Transsubstantiation and the conversion of bread into Christ's body

One of the philosophical challenges of Aquinas’ account of transsubstantiation is his insistence that the bread and wine are not merely annihilated and replaced by Christ’s body and blood, but that they are changed into Christ’s body and blood.

Now, it is easy to see how bread could be changed into a part of Christ’s body. That routinely happened when Christ ate bread in his earthly life. But Aquinas thinks that Christ is wholly present in the Eucharist, so that can’t be the account. But it is very puzzling what it would mean for an item B to be changed into an individual item C that already existed prior to the change. What would it mean, for instance, for the chair I am sitting on to change into the laptop I am typing this on? It is easy to imagine God moving the fundamental particles of the chair into positions such that they constitute a laptop. But that would be a case of the chair changing into a second laptop, not into the laptop that I am typing this on. Indeed, it seems like it’s impossible for something to change into something that already exists, simply because the thing already exists.

Aquinas is well aware of this objection, and has a fascinating response:

A form cannot be changed into another form, or one [designated] matter into another [designated] matter, by the power of a finite agent. However, such a conversion can be effected by the power of an infinite agent, which has an action on the whole entity. For the common nature of being belongs to each form and to each [designated] matter, and the author of being (auctor entis) is able to convert what there is of being in the one (id quod entitatis est una) into what there is of being in the other (id quod est entitatis in altera), by removing that by which it was distinguished from the latter.

Here is what I think is going on. Like many other philosophers before and after him, Aquinas thinks that individual objects need something whereby they are individuated—something that distinguishes them from other things. The project of figuring out what individuates things from other things is indeed a major part of Aristotelian metaphysics. Aquinas’ point seems to be this. God wields a very fine scalpel at the level of being, a scalpel so infinitely sharp that no finite being can wield it. That scalpel allows God to slice off an individual B that which distinguishes B from an individual C. When God slices that off, B literally loses its identity, and becomes C, as there is then nothing whereby B can be distinguished from C. (That God has such a fine scalpel is also indicated by the way that in the Eucharist he can slice a substance away from its accidents, and have the accidents remain without the substance.)

Let’s explore this account. First note that it seems to commit Aquinas to a different account of the individuation of material objects from his usual one. Aristotelians normally think that material objects are distinguished either by having different forms or, when the form is the same, by having different matter. Now, the bread on the altar and the body of Christ do have different forms: one is bread and the other is a human body. So on the usual Aristotelian account of what makes the bread different from the body of Christ, it is the bread’s bready form. But slicing away the bready form does not turn the bread into the body of Christ, or indeed into any human body. It just turns the bread into a formless lump of matter.

Perhaps we should suppose, however, that there is more than just literal removal going on. Maybe what happens is that God removes the bready form and replaces it with the form of the human body. But great as that miracle would be, that would just turn bread into a human, not into this human, Jesus Christ.

What if we suppose that God removes the bready form and replaces it with the form of Jesus Christ (namely, the soul of Jesus Christ)? But now the bread is simply becoming a new part of the body of Christ (in a miraculous verison of the way that the bread you may have for lunch may become new cells in you), and so only a part of Christ is present in the Eucharist.

But perhaps what I have described doesn’t slice away enough. Suppose the following happens. God slices the bready form away from the bread. That still leaves the bread’s matter. And the matter of the bread is distinct from the matter of Christ’s body. God continues removing the grounds of distinctness. He wields his infinitely sharp scalpel and carefully removes that in the matter of the bread which makes it be distinct from the matter of Christ’s body. The result is that that now the matter of the bread is not distinct from the matter of Christ’s body. Indeed, the matter of the bread literally converts into the matter of Christ’s body, not merely a new part of Christ’s body.

A major problem with this interpretation is that the form of bread is annihilated, whereas Thomas thinks the form of bread is also converted into the form of Christ’s body (admittedly with a qualification; see ST III.75.A6repl2 for details).

But perhaps we should make another move. Suppose that we have a non-Aristotelian account of individuation that works as follows: for any two created things, B and C, there is a relation that B has to C that individuates B from C and a relation that C has to B that individuates C from B. We can imagine each created thing having a vast number of labels. Somehow Alice has written into her being “I am not Bob” and “I am not Seabiscuit” and “I am not Oak Tree #18289”, and Bob has written into his being “I am not Alice” and “I am not Seabiscuit” and “I am not Oak Tree #18289”. This relational account of individuation does not require form or matter. It is not very Aristotelian. But it has a great theological merit: it makes the individuation of creatures be an image of the individuation of persons in the Trinity, which also proceeds (according to Western Christians) by opposed relations. Now imagine that God slices off of Alice the label “I am not Seabiscuit.” Instantly, Alice is converted into Seabiscuit. (Of course, it’s not right to say that Alice is Seabiscuit now. In this respect, it’s like when Bucephalus turned into a cadaver: Bucephalus and the horse-shaped cadaver are distinct entities.)

The exegetical problem with this interpretation is that it forces one to reject the standard Aristotelian story about individuation across species being by form and within a species being by matter. Instead, individuation is always by “individuating relations”. I am happy with this, because I never liked the standard Aristotelian story. But it makes it unlikely that the story is what Thomas has in mind.

But suppose one wants this to be more Aristotelian. Here is a way to do this. Take the orthodox Aristotelian account that across species individuation is by form and within species by matter. This account leaves unanswered the question of what makes a human form and a horse form different, as well as the question of what makes Peter’s matter different from Paul’s matter. Suppose we answer these questions by the relational account, thereby combining the Aristotelian account with the relational. Thus, a human form has (perhaps primitive) distinctness relations to all other kinds of forms, and Peter’s matter has a (perhaps primitive) distinctness relation to all other chunks of matter.

We can now imagine the following happening. There is bread on the altar. At the moment of consecration, God (a) removes from the bready form that which distinguishes it from a human form and (b) removes from the bread’s matter that which distinguishes it from Christ’s matter. Step (a) ensures that now the bread has human form, while step (b) ensures that the human form is that of Christ, since within a species the numerical distinction of forms is due to matter.

This last account is quite Aristotelian, and only requires that we go one step further than Aristotle by supposing an answer to the question of what makes different kinds of forms different and distinct chunks of matter distinct. It’s too Aristotelian for my taste—I don’t want matter to play that much of a metaphysical role. But it is a cool account, I think. And it could be Aquinas’.

A four-dimensional model of Eucharistic presence

In yesterday’s post, I discussed the Real Presence in the context of relativistic time. There, I made the assumption that when Christ is really present in the Eucharist, it is Christ at a specific time of life (intuitively, the current time, but that notion is tricky given Relativity). It is, in particular, an adult glorified Christ and not the toddler Christ who is present in the Eucharist.

But after discussion with my Aquinas seminar grad students, I think there is something rather appealing about denying that assumption. What if instead we say that the whole of the four-dimensional Christ is present in the Eucharist? Aquinas apparently thinks that the whole Christ is present in every potential “part” of the consecrated host. This suggests (but does not entail) the idea of a three-dimensional entity present at a single point in space. Why, then, can’t a four-dimensional entity be present at a single point of spacetime? This would require a distinction between internal and external time. During an instant of external time there would be a positive (indeed, infinite, in the case of a being that lives forever) length of internal time. This is just as the whole-presence of a three-dimensionally Christ in the Eucharist requires a distinction between internal and external space: there may be five feet (say) of internal space between Christ’s head and Christ’s toes, but both are present in the external space of two inches—or much less if Aquinas is right that Christ is present in every potential part.

Is there any point to such a supposition? Yes.

First, the Tradition holds that Christ is wholly present in the Eucharist. Given four-dimensionalism, a literal metaphysical reading of that requires the whole of the four-dimensional extent of Christ to be present. Granted, I think this is an overreading of the Tradition: even if four-dimensionalism is true, it is plausible that the doctrinal pronouncements on this only refer to the whole three-dimensional extent of Christ. But, still, supposing the four-dimensionalism, it is certainly in the spirit of the teaching on Christ being wholly present, even if not required by it, to suppose the whole four-dimensional extent of Christ to be present.

Second, in Q73.A4, Aquinas has a beautiful discussion of the threefold temporal signification of the Eucharist. With respect to the past, it commemorates Christ’s passion. With respect to the present, it brings all the members of the Church together. With respect to the future, it prefigures the enjoyment of God in heaven. If we think that we are united with Christ in his full four-dimensional extent, this deepens and underscores this threefold signification.

Third, the Catholic tradition holds that the Mass is a re-presentation of the sacrifice of Calvary. This is a mysterious doctrine, and a four-dimensional whole-presence model of the Eucharist gives us a precise account of that doctrine as well: Christ as hanging on the Cross is present in the Eucharist.

That said, there are three things that make me uncomfortable about this four-dimensional extension of the doctrine that the whole of Christ is present in the Eucharist. The first is simply that it is a new theological theory (as far as I know), and most new theological theories are heretical.

The second is that it feels important to me that it is the glorified Jesus who is present in the Eucharist. But perhaps I am wrong about this feeling, and in having this feeling I am underplaying the commemorative aspect of the past temporal aspect of the Eucharist. Perhaps a justification for my feeling of discomfort is given by the Church’s emphasis on the Eucharist as an unbloody re-presentation of the sacrifice of Christ—but if Christ’s mangled crucified body is present in the Eucharist, then this the unbloodiness is merely a matter of appearances.

The third is that it is difficult what to make of the period when Christ was dead. Aquinas thinks God was still incarnate in the dead body of Christ, and if the Apostles had celebrated the Eucharist then, a dead body would have come to be present. If Aquinas’s reasons are good ones, which I am not confident of, then on the four-dimensional whole presence model we should say that the dead body of Christ is present. But this doesn’t seem right. I worry both about the apparently unfitting gruesomeness of this, as well as about the idea that there is something in the Eucharist other than Christ’s body, blood, soul and divinity (a dead body is not a body!). That said, I am suspicious of Aquinas’s view of the Incarnation and the dead body of Christ. But even if Aquinas is wrong, we have another problem. If the whole temporal extent of Christ is to be present, the soul of Christ as it was when Christ was dead needs to be present. (Especially if, as I think, survivalism is correct.) But a soul is only present in a spatial location insofar as it is united to a body that is in that location. But the soul of Christ as it was when he was dead was not united to a body, so it seems that there is no way for it to be present in the Eucharist. If this problem cannot be solved, the account may yield the whole four-dimensional extent of Christ being present, but not the whole temporal extent of Christ being present (the soul is temporal but not spatial, and hence not four-dimensional). Thus, the account may not achieve quite as much as it seems to.

One can resolve the second and third problems by supposing a moderate version of the view: Christ’s whole glorified four-dimensional self is present in the Eucharist—namely, Christ in the Eucharist is all of Christ from the time of his resurrection. This loses some of the advantages of the view, and it is not clear that what remains is sufficiently compelling. But, on the other hand, it’s also not clear that there is any serious disadvantage to that view over a three-dimensional-slice view of the Real Presence, except maybe the novelty. And it has the advantage of there not having to be a fact about the exact correlation of times between heaven and earth.

The Real Presence and Relativity Theory

Jesus Christ like all human beings has an internal clock. One can measure that clock in heartbeats or in lower level physical interactions or in some other way. Let’s measure it in “internal years”. If Jesus was born in 4 BC, then in 4 BC, his internal clock was at about a year (he was conceived about 0.75 years before he was born). In 1 BC, it was at 4 years, in 1 AD, it was at 5 years, and in 30 AD, it was at 34 years.

I don’t know what Jesus’s internal clock was at in 100 AD, and it’s not immediately obvious that the question makes sense. For it is not immediately obvious that there is a correlation between time on earth and time in heaven of such a sort it makes sense to ask “What is happening in heaven right now?” After all, according to Relativity Theory, it doesn’t make sense to ask “What is happening right now in the Andromeda Galaxy” without specifying a reference frame for the “right now”, and it’s not immediately clear that there is a common reference frame between heaven and earth.

However, the real presence of Jesus in the Eucharist does provide a temporal correlation between heaven and earth. Around 22:45 UTC today, Jesus will come to be present in our campus parish. Moreover, Jesus will be present as an adult glorified human, not as the three-year-old he was in 1 BC. There thus appears to be a fact of the matter as to what his internal clock will be showing when he comes to be present at 22:45 UTC in Waco today.

Interestingly, this gives a temporal ordering on events scattered across the earth apparently independent of our ordinary relativistic reference frames. For if the Eucharist is celebrated around 22:45 UTC in Waco and around 22:45 UTC in London, there is a fact of the matter whether Christ as present in Waco is older or younger or at the same age (according to his internal clock), and this fact provides a reference-frame independent temporal ordering between these two Eucharistic celebrations.

Indeed, since according to Catholic and Orthodox faith, Christ remains Eucharistically present in the tabernacles across the world, we constantly have a temporal ordering between events scattered spatially across the world. In principle, this defines a theologically privileged reference frame between scattered events—a Eucharistic reference frame. Events at locations z₁ and z₂ in spacetime are Eucharistically simultaneous, we might say, provided that Christ as Eucharistically present at z₁ and at z₂ has the same value of the internal clock.

Of course, some philosophers of time think there is an objective reference frame in the physical world. If they are right, then very likely the theologically privileged frame is the same as the objective one.

All that said, it is not completely clear to me that Christ as Eucharistically present has to have a well-defined value of his internal clock. But I suspect so, because of the intuition that it is the adult and not toddler Jesus who is Eucharistically present.

Gettiered by degrees

Consider a standard Gettier case. A cutout of a sheep in a field hides a sheep behind it. At that distance, the cutout looks just like a sheep. You have a justified true belief that there is a sheep, but you don’t know it (or so the story goes).

Now imagine that cutout is to some degree transparent, so some of the whiteness you see is in fact from the sheep, and some from the cutout. Consider the continuum of cases as the cutout goes from fully opaque to full transparent. Perhaps it fades from opaque to transparent as you’re looking—all without you knowing that it is fading. When it’s fully or nearly opaque, you are Gettiered and don’t know there is a sheep. When it’s fully or nearly fully transparent, you know there is a sheep.

Supposing that knowledge has a distinctive value over and beyond the value of justified true belief, it seems plausible to think that this value increases monotonically with the transparency of the cutout. If the cutout is becoming more and more transparent before your eyes, you are gaining epistemic value, without noticing you are doing so.

It’s an interesting question: What kind of a function is there from cutout-transparency to value? Is it continuous, or is there a transparency threshold for knowledge at which it jumps discontinuously? If it is continuous, is it linear?

I have to confess that these kinds of questions seem a bit silly, and this gives some ammunitition to the thought that knowledge does not have a distinctive value.