Qualitative haecceities

A haecceity H of x is a property of an entity such that necessarily x exists if and only if x instantiates H.

Haecceities are normally thought of as non-qualitative properties. But one could also have qualitative haecceities. Of course, if an entity has a qualitative haecceity then it cannot be duplicated, so one can only suppose that everything has a qualitative haecceity provided one is willing to agree with Leibniz’s Identity of Indiscernibles.

I am personally drawn to the idea that everything does have a qualitative haecceity, and specifically that the qualitative haecceity of x encapsulates x’s qualitative causal history: a complete qualitative description of x’s explanatorily initial state and of all of its causal antecedents. One might call such properties “qualitative origins”. The view that every entity has a qualitative origin is a haecceity is a particularly strong version of the essentiality of origins: everything in an entity’s causal history is essential to it, and the causal history is sufficient for the entity’s existence.

I suppose the main reason not to accept this view is that it implies that two distinct objects couldn’t have the same qualitative origin, but it seems possible that God could create two objects ex nihilo with the same qualitative initial state Q. I am not so sure, though. How would God do that? “Let there be two things satisfying Q?” But this is too indeterminate (I disagree with van Inwagen’s idea that God can issue indeterminate decrees). If there can be two, there can be three, so God would have to specify which two things satisfying Q to create. But that would require a way of securing numerical reference to specific individuals prior to their creation, and that in turn would require haecceities, in this case non-qualitative haecceities. So the objection to the view requires non-qualitative haecceities.

But what started us on this objection was the thought that God could say “Let there be two things satisfying Q.” But if God could say that, why couldn’t he say “Let there be two things satisfying H”, where H is a non-qualitative haecceity? I suppose one will say that this is nonsense, because it is nonsense to suppose two things share a non-qualitative haecceity. But isn’t there a double-standard here? If it is nonsense to suppose two things share a non-qualitative haecceity, why can’t it be nonsense to suppose two things share a qualitative haecceity? It seems that “what does the explaining” of why two things can’t share a non-qualitative haecceity is the obscurity of non-qualitative haecceities, and that’s not really an explanation.

So perhaps we can just say: Having a distinct qualitative origin is what it is to be a thing, and it is impossible for two things to share one. This does indeed restrict the space of possible worlds. No exactly similar iron spheres or anything like that. That’s admittedly a little counterintuitive. But on the other hand, we have a lovely explanation of intra- and inter-world identity of objects, as well as a reduction of de re modality to de dicto, all without the mystery of non-qualitative haecceities. Plus we have Leibniz’s zero/one picture of the world on which all of reality is described by zeroes and ones: we put a zero beside an uninstantiated qualitative haecceity and a one besides an initiated one, and then that tells us everything that exists. This is all very appealing to me.

Distinguishing between properties

Some philosophers worry about “principles of individuation” that make two things of one kind be different from another. Suppose we share that worry. Then we should be worried about Platonism. For it is very hard to say what make two fundamental Platonic entities of the same sort different, say being positively charged from being negatively charged, or saltiness from sweetness.

However, the light-weight Platonist, who denies that predication is to be grounded in possession of universals, has a nice story to tell about the above kinds of cases. For here is a qualitative difference between saltiness and sweetness:

• saltiness is necessarily had by all and only salty things, but

• sweetness is not necessarily had by all and only salty things.

But for the heavy-weight Platonist to tell this story would involve circularity, for what it is for a thing to be salty will be to exemplify saltiness.

Of course, this story only works for properties that aren’t necessarily coextensive. But it’s some progress.

Presentism and referring to past individuals

It seems to me that the presentist can only de re refer to past (or future—but that's less of a problem) individuals if there are haecceities or the identity of indiscernibles is true.

Can presentists say someone will have infinitely many descendants?

In an earlier post, I showed that presentists can count infinities—i.e., that presentists can give a paraphrase for sentences like "There have ever been aleph-0 horses." I did this by an ersatzist construction. I then left it open whether some such construction could work in general to give presentist paraphrase.

The problem is basically the problem of transtemporal quantification. If haecceitism is true, then it's easy. The presentist just replaces talk transtemporal talk of individuals with talk of haecceities. Likewise, if the presentist accepts the impossibility of exact intrinsic duplicates—for then one can replace talk of individuals with talk of individual-types. The interesting question is whether this can be done if you're a presentist who is not a haecceitist and who thinks there can be exact intrinsic duplicates.

I have a sentence that a non-haecceitist presentist who accepts intrinsic duplicates may have difficulty giving finite truth conditions for:

1. Somebody will have infinitely many descendants.

I don't know if presentist truth conditions for (1) are possible.

If we allow infinite sentences, it can be done. But that's cheating. :-) Or is it?

Adverbial ontology and dispensing with parts

Once one has an adverbial ontology, like the one I used to help with the Incarnation, one no longer needs the parts of a substance in one's ontology. "I have two hands." That's made true by my being two-handed. "My right hand has five fingers." That's made true by my having a right hand five-fingeredly. More explicitly, there is a mode m in virtue of which I have a right hand. (According to my Incarnation post, I have m indirectly, as m is a mode of my humanity.) Then we have two moves we can make. We could say that there are five modes of m, which each of which is a different way of the hand's being fingered. If we go that route, then we are forced into identity of indiscernibles for fingers, and by extension for any other parts. I welcome that consequence myself, since I'm anyway pulled to identity of indiscernibles by my theory of transworld identity. But alternately we could simply posit a mode of being five-fingered, perhaps a mode relational to the number five.

To my mind there are three main uses of parts:

1. Some properties of wholes are grounded in properties of parts. "I have the property of having heart-beat in virtue of having as a part a heart that in turn has the property of beating."
2. Parts have location and help explain partial location. "I am partly in this room and partly in that, because one of my legs is here and the other there."
3. Some parts are widely thought to be able to move between substances. "Several hours after you ate the apple, a carbon atom that used to be a part of an apple tree has become a part of you."

The adverbial mode ontology does the first two tasks well.

1: There is a mode in virtue of having which I have heart-beat. But I have that mode indirectly: that mode modifies my being hearted, which in turn modifies my humanity. So properties are divided up. But an advantage of the mode way is that we get to uniformly divide properties not just by parts, but by functional subsystems. Some functional subsystems correspond to parts, but likely not all. In a computer running several processes at once on the same processor core, the processes may correspond to different functional systems—say, one doing Fourier transforms of microphone data and another watching for user input events—but the processes may be implemented by overlapping sets of physical parts (and a computer has no others), and we could easily imagine that there is no distinct set of parts corresponding to each process. It seems likely that something like that happens in the brain, and even if it does not, the possibility should be accounted for in our ontology. The adverbial mode ontology apportions properties had in virtue of a functional subsystem in the same way that it apportions those properties had in virtue of a physical part, and that strikes me as exactly right.

2: This is really just a special case of 1. "My right leg is located in this room" is true in virtue of my being right-leg-possessed this-roomly.

But unless we posit that modes can move between substances—I've heard Rob Koons speculate in that direction and Aquinas's account of transsubstantiation famously allows modes to survive the destruction of their underlying substance—it's harder to handle 3. On general Aristotelian grounds, I think one can just bite the bullet. The identity of a part, if there are parts, is going to be dependent on the whole. There is no carbon atom that was a part of the apple tree and is now a part of you. There are (in the eternalist sense of "are"), at best, two carbon atoms, one that was identity dependent on the tree and the other that is identity dependent on you, and the first caused the second. This is counterintuitive.

So what our adverbial ontologists should say about 3 is that the apple tree has some mode m₁ that makes it count as having had a certain carbon atom once, and you have some mode m₂ that makes you count as having a certain carbon atom. There is a continuity of location (see 2) between the one mode (perhaps with some other intervening modes, depending on the ontological status of the apple as such) and the other. Moreover, m₁ is a cause of m₂. Or, if we prefer (and I think we should), the apple tree as modified by m₁ caused you to have m₂. I.e., there is a mode c₁ of causation had by m₁, which is a causation of m₂, or of me as having m₂. But the numerical identity of the particles, that we need to give up on. However, since giving up on parts dissolves the problem of material constitution, and since every other solution to the problem of material constitution has something else counterintuitive about it, we are in this regard no worse off here than any view on which there are parts.

A challenge for the view is to distinguish between those modes that correspond to parts and those modes that don't. But one might just reject the distinction. Or one might go like this. It might be that all and only the modes that have a location mode are parts. But don't non-part subsystems have a location mode? Maybe not. Rather, they may be modes--or joint modes (maybe a mode can be a mode of more than one mode--or maybe even more than one substance--and maybe that is how relations are to be handled)--of one or more parts, and the parts are what have a location mode. The non-part subsystems, then, have a location in a derivative sense.

Carnap's probability measure

Carnap's objective prior probability measure was designed to make induction possible. Almost nobody uses Carnap's probability measure any more—the only exception I am aware of is Tooley in his debate book with Plantinga on evil. I have no idea why Tooley is using the Carnap measure—I thought it was out of date. In any case, it's easy to point out at least two things that are wrong with the Carnap measure, and hence why Tooley's arguments based on it need to be reworked. To explain the problems with the Carnap measure, I need some details. If you're familiar with Carnap measure, you can skip ahead to "Problem 1".

Carnap's prior probability measure is best seen as a measure for the probability of claims made by sentences of a truth-functional language with n names, a₁,...,a_n, and k unary predicates, Q₁,...,Q_k. Let N be the set of names, Q the set of predicates and T the set {True, False}. Call the language L(Q,N). Say that a state s is a function from the Cartesian product QxN to T, and let S be the set of all states. There is a natural way of saying whether a sentence u of L(N,P) is true at a state s. Basically, you say that the sentence Q_i(a_j) is true at s if and only if s(Q_i,a_j)=True, and then extend truth-functionally to all states.

There is a natural probability measure on S, which I will call the "Wittgenstein measure", defined by P_W(A)=|A|/|S| for every subset A of S, where |X| is the cardinality of the set X. This probability measure assigns equal probability to every state. Given a probability measure P on states, we get a probability measure for the sentences of L(Q,N). If u is such a sentence, define the subset u^T={s:u is true at s} of S. Then, we can let P(u)=P(u^T). The Wittgenstein measure does not allow induction. Suppose that we have three names, and two predicates, Raven and Black. Our evidence E is: Raven(a₁), Raven(a₂), Raven(a₃), Black(a₁) and Black(a₂). Then, P_W(Black(a₃)|E)=1/2=P_W(Black(a₃)), as can be easily verified, because all states are equally likely, and hence the state that makes all the a_i be black ravens is no more likely than the state that makes all the a_i be ravens but with only a₁ and a₂ black.

So, Carnap wanted to come up with a probability measure that allows induction but is still fairly natural. What he did was this. Instead of assigning equal probability to each state, he assigned equal probability to each equivalence class of states. Say that s~t for states s and t if there is some permutation p of the names N such that s(R,p(a))=t(R,a) for every predicate R and every name a. Let [s] be the equivalence class of s under this relation: [s]={t:t~s}. Let S* be the set of these equivalence classes. Then, if s is a state, we define: P_C({s})=1/(|[s]||S*|). In other words, each state in an equivalence class has equal probability, and each equivalence class has equal probability. If A is any subset of S, we then define P_C(A) as the sum of P_C({a}) as a ranges over the elements of A.

The merit of Carnap measure is that it assigns a greater probability to more uniform states. Thus, P_C(Black(a₃)|E) should be greater than 1/2 (I haven't actually worked the numbers).

Problem 1: Carnap measure is not invariant under increase of the number of predicates. Intuitively, adding irrelevant predicates to the language, predicates that do not appear in either the evidence or the hypothesis, should not change the degree of confirmation. But it does. In fact, we have the following theorem. Let u be any sentence of L(Q,N). Let Q_r be Q with r additional predicates thrown in. Let u_r be a sentence of L(Q_r,N) which is just like u (i.e., u_r is u considered qua sentence of L(Q_r,N)).

Theorem 1: P_C(u_r) tends to P_W(u) as r tends to infinity.

In other words, as one increases the number of predicates, one loses the ability to do induction, since P_W is no good for induction. The proof (which is non-trivial, but not insanely hard) is left to the reader.

Problem 2: Let d be a sentence of L(Q,N) saying that indiscernibles are identical. For instance, let d_ij be the disjunction ~(Q₁(a_i) iff Q₁(a_j)) or ... or ~(Q_k(a_i) iff Q_k(a_j)), and let d be the conjunction of the d_ij for all distinct i and j.

Theorem 2: P_C(u|d)=P_W(u|d).

Thus, when we condition on the identity of indiscernibles, Carnap measure collapses to Wittgenstein measure. But Wittgenstein measure is worthless for induction. And often the identity of indiscernibles holds. For instance, suppose we have a₁,a₂,a₃ as our individuals, and our evidence is this: a₁,a₂,a₃ are each a raven, a₁ and a₂ are black. So far so good, we can do induction and we get some confirmation of a₃ being black. But suppose we also learn that identity of indiscernibles holds for these three ravens. Then we lose the confirmation! And we might well learn this. For instance, we might learn that exactly a₁ and a₃ are male, and exactly a₁ and a₂ each have an even number of feathers, and that means that identity of indiscernibles holds.

Moreover, I think most of us have a background belief that our world has such richness of properties that, at least as a contingent matter of fact, the identity of indiscernibles holds for macroscopic objects. If so, then Carnap measure makes induction impossible for macroscopic objects.

Sketch of proof of Theorem 2: Let D be the set of states at which identity of indiscernibles holds. Thus, D is the set of states s with the property that if a and b are distinct, then there is a predicate R such that s(R,a) differs from s(R,b). Observe that if s is any state in D, then |[s]|=n!, where n is the number of names. For, any permutation of the names induces a different state given the identity of indiscernibles, and there are n! permutations. Therefore, P_C({s})=1/(n!|S*|). Hence, P_C({s}) has the same value for every s in D. Therefore, P_C({s}|D)=1/|D|. But, likewise, P_W({s}|D)=1/|D|. The Theorem follows easily from this.

Remark: Theorem 2 gives an intuitive reason to believe Theorem 1. As one increases the number of predicates while keeping fixed the number of names, a greater and greater share of the state space satisfies the identity of indiscernibles.