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 ontologicallt 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.