Aporetics – Page 10 – Maverick Philosopher

The Bundle Theory and the Identity of Indiscernibles

I have been defending the bundle-of-universals theory of concrete particulars (BT) against various weak objections over a series of posts, here, here, here, and here. Now I consider a very powerful objection, one that many will consider decisive. The objection can be cast in the mold of modus tollendo tollens: If BT is true, then the Identity of Indiscernibles is a necessary truth. But the Identity of Indiscernibles is not a necessary truth. Ergo, BT is not true.

1. The Identity of Indiscernibles (IdIn) is the converse of the Indiscernibility of Identicals (InId) and not to be confused with it. InId is well-nigh self-evident, while IdInis not. Roughly, the latter is the principle that if x and y share all properties, then x = y. It is a strictly ontological principle despite the epistemological flavor of 'indiscernible.' As just stated, it is more of a principle-schema than a principle. We will get different principles depending on what we count as a property. To arrive at a plausible nontrivial principle we must first rule out haecceity properties. If, for any x,there is a property of identity-with-x, then no two things could share all properties, and the principle would be trivially true due to the falsehood of the antecedent. Haecceity properties are creatures of darkness in any case as I argue elsewhere.

A plausible, nontrivial, principle results if we allow as properties all and only relational and nonrelational pure properties. A pure property is one that makes no reference to any specific individual. Being married would then be an example of a pure relational property: to be married is to be married to someone, but not to any specified individual. Being married to Xanthippe, however, is an impure relational property. Being obese would be an example of a nonrelational property. Here then is a plausible version of the Identity of Indiscernibles:

Necessarily, for any x, for any y, and for any relational or nonrelational pure property P, if (x has P iff y has P) then x = y.

2. It is obvious, I think, that BT entails IdIn in the above form. Consider a concrete particular, an iron sphere say, at a time. On BT it is nothing but a bundle of universals. This implies that it is not possible that there be a second iron sphere that shares with the first all relational and nonrelational pure properties. This is not possible on BT because on BT a concrete particular is nothing more than a bundle of universals. Thus there is no ontological ingredient in a concrete particular that could serve to differentiate it from another particular having all the same relational and nonrelational pure properties. And if it is not possible that there be two things that differ numerically without differing property-wise, then the Identity of Indiscernibles as above formulated is necessarily true.

I am assuming that BT, if true, is necessarily true. This is a special case of the assumption that the propositions of metaphysics, if true, are necessarily true. If this assumption is granted, then BT entails IdIn.

3. But is IdIn true? Since it is necessarily true if true, all it takes to refute it is a possible counterexample. Imagine a world consisting of two iron spheres and nothing else. (The thought experiment was proposed in a 1952 Mind article by Max Black.) They are the same size, shape, volume, chemical composition and so on. They agree in every nonrelational respect. But they also agree in every relational respect. Thus, each has the property of being ten meters from an iron sphere. What Black's example seems to show is that there can be numerical difference without property-difference. But then IdIn is false, whence it follows that BT is false.

4. This is a powerful objection, but is it fatal? Here are three ways to resist the argument, fit topics for further posts. He who has the will to blog will never be bereft of topics.

a. Maintain that BT is a contingent truth. If so, then BT does not entail IdIn as formulated above.

b. Grant that BT entails IdIn, but deny that scenarios such as Black's are really possible. Admit that they are conceivable, but deny that conceivability entails possibility.

c. An immanent universal can be wholly present at different places at once. So why can't a bundle of universals be wholly present in different places at once? Argue that Black's world can be interpreted, not as two particulars sharing all universals, but as one particular existing in two places at the same time. From that infer that Black's Gedankenexperiment does show that IdIn is false.

Any other paths of resistance?

A Closer Look at Material Composition and Modal Discernibility Arguments

(For David Brightly, whom I hope either to convince or argue to a standoff.)

Suppose God creates ex nihilo a bunch of TinkerToy pieces at time t suitable for assembly into various (toy) artifacts such as a house and a fort. A unique classical mereological sum — call it 'TTS' — comes into existence 'automatically' at the instant of the creation ex nihilo of the TT pieces. (God doesn't have to do anything in addition to creating the TT pieces to bring TTS into existence.) Suppose further that God at t assembles the TT pieces (adding nothing and subtracting nothing) into a house. Call this object 'TTH.' So far we have: the pieces, their sum, and the house. Now suppose that at t* (later than t) God annihilates all of the TT pieces. This of course annihilates TTS and TTH. During the interval from t to t* God maintains TTH in existence.

I set up the problem this way so as to exclude 'historical' and nonmodal considerations and thus to make the challenge tougher for my side. Note that TTH and TTS are spatially coincident, temporally coincident, and such that every nonmodal property of the one is also a nonmodal property of the other. Thus they have the same size, the same shape, the same weight, etc. Surely the pressure is on to say that TTH = TTS? Surely my opponents will come at me with their battle-cry, 'No difference without a difference-maker!' There is no constituent of TTH that is not also a constituent of TTS. So what could distinguish them?

Here is an argument that TTH and TTS are not identical:

1. NecId: If x = y, then necessarily, x = y.

2. If it is possible that ~(x = y), then ~(x = y). (From 1 by Contraposition)

3. If it is possible that TTS is not TTH, then TTS is not TTH. (From 2, by Universal Instantiation)

4. It is possible that TTS is not TTH. (God might have assembled the parts into a fort instead of a house or might have left them unassembled.)

5. TTS is not TTH. (From 3, 4 by Modus Ponens)

The gist of the argument is that if x = y, then they are identical in every possible world in which both of them exist. But there are possible worlds in which TTS and TTH both exist but are not identical. (E.g., a world in which the pieces are assembled into a fort instead of a house.) Therefore, TTS andf TTH are not identical.

If you are inclined to reject the argument, you must tell me which premise you reject. Will it be (1)? Or will it be (4)?

Your move, David.

Varzi, Sums, and Wholes

Achille C. Varzi, "The Extensionality of Parthood and Composition," The Philosophical Quarterly 58 (2008), p. 109:

Suppose we have a house made of Tinkertoy pieces. Then the house qualifies as a sum of those pieces: each piece is part of the house and each part of the house overlaps at least one of the pieces . . . . Are there other things that qualify as the sums of those pieces? UC says there aren't; the house is the only candidate: it is the sum of those pieces.

UC is Uniqueness of Composition:

UC If x and y are sums of the same things, then x = y,

where

(1) x is a sum of the zs =_df The zs are all parts of x and every part of x has a part in common with at least one ofthe zs.

Perhaps commenter John, who knows some mereology and the relevant literature on material composition, can help me understand this. What I don't understand is what entitles Varzi to assume that the Tinkertoy house — 'TTH' to give it a name — is identical to a classical mereological sum. I do not deny that there is a sum of the parts of TTH. And I do not doubt that this sum is unique. Let us name this sum 'TTS.' (I assume that names are Kripkean rigid designators.) What I do not understand is the justification of the assumption, made near the beginning of his paper, of the identity of TTH and TTS. TTH is of course a whole of parts. But it doesn't straightaway follow that TTH is a sum of parts.

Please note that 'sum' is a technical term, one whose meaning is exactly the meaning it derives from the definitions and axioms of classical mereology. 'Whole' is a term of ordinary language whose meaning depends on context. It seems to me that one cannot just assume that a given whole of parts is identical to a mereological sum of those same parts.

I am not denying that it might be useful for some purposes to think of material objects like TTH as sums, but by the same token it might be useful to think of material objects as (mathematical) sets of their parts. But surely it would be a mistake to identify TTH with a set of its parts. For one thing, sets are abstract while material objects are concrete. For another, proper parthood is transitive while set-theoretic elementhood is not transitive.

Of course, sums are not sets. A sum of concreta is itself concrete whereas a set of concreta is itself abstract. My point is that, just as we cannot assume that that TTH is identical to a set, we cannot assume that TTH is identical to a sum.

What is the 'dialectical situation' when it comes to the dispute between those who maintain that TTH = TTS and those who deny this identity?

It seems to me that the burden of proof rests on those who, like Varzi, identify material objects like TTH with sums especially given the arguments against the identity. Here is one argument. (a) Taking TTH apart would destroy it, (b) but would not destroy TTS. Therefore, (c) TTH is not identical to TTS. This argument relies on the wholly unproblematic Indiscernibility of Identicals as a tacit premise: If x = y, then whatever is true of x is true of y, and vice versa. Because something is true of TTH — namely, that taking it apart would destroy it — that is not true of TTS, TTH cannot be identical to TTS.

The simplicity and clarity of modal discernibility arguments like this one cast grave doubt on the opening assumption that TTH is a sum. I am not saying that Varzi and Co. have no response to the argument; they do. My point is that their response comes too late dialectically speaking. If you know what a sum is, you know that the identity is dubious from the outset: the discernibility arguments merely make the dubiousness explicit. Responding to these arguments strikes me as too little too late; what the identity theorist needs to do is justify his intitial assumption as soon as he makes it.

My main question, then, is this. What justifies the initial assumption that material particulars such as Tinkertoy houses are mereological sums? It cannot be that they are wholes of parts, for a whole needn't be a sum. TTH is a whole but it is not a sum. It is not a sum because a sum is a collection that is neutral with respect to the arrangement or interrelation of its parts, whereas it is essential to TTH that its parts be arranged house-wise.

Four-Dimensionalism to the Rescue?

Let us return to that impressive product of porcine ingenuity, Brick House. Brick House, whose completion by the Wise Pig occurred on Friday, is composed entirely of the 10,000 Tuesday Bricks. I grant that there is a sum, call it 'Brick Sum,' that is the classical mereological sum of the Tuesday Bricks. Brick Sum is 'generated' — if you care to put it that way — by Unrestricted Composition, the classical axiom which states that "Whenever there are some things, then there exists a fusion [sum] of those things." (D. Lewis, Parts of Classes, p. 74) I also grant that Brick Sum is unique by Uniqueness of Composition according to which "It never happens that the same things have two different fusions [sums]." (Ibid.) But I deny Lewis' Composition as Identity. Accordingly, Brick Sum cannot be identical to the Tuesday Bricks. After all, it is one while they are many.

Now the question I am debating with commenter John is whether Brick House is identical to Brick Sum. This ought not be confused with the question whether Brick House is identical to the Tuesday Bricks. This second question has an easy negative answer inasmuch as the former is one while the latter are many. Clearly, one thing cannot be many things.

The question, then, is whether Brick House is identical to Brick Sum. Here is a reason to think that they are not identical. Brick Sum exists regardless of the arrangement of its parts: they can be scattered throughout the land; they can be piled up in one place; they can be moving away from each other; they can be arranged to form a wall, or a corral, or a house, or whatever. All of this without prejudice to the existence and the identity of Brick Sum. Now suppose Hezbollah Wolf, a 'porcicide' bomber, enters Brick House and blows it and himself up at time t on Friday evening. At time t* later than t, Brick Sum still exists while Brick House does not. This shows that they cannot be identical; for if they were identical, then the destruction of Brick House would be the destruction of Brick Sum.

This argument, however, rests on an assumption, namely, that Brick Sum exists both at t and at t*. This won't be true if Four Dimensionalism is true. If bricks and houses are occurrents rather than continuants, if they are composed of temporal parts, then we cannot say, strictly and philosophically, that Brick Sum at t still exists at t*. And if we cannot say this, then the above argument fails.

But all is not lost since there remains a modal consideration. Brick House and Brick Sum both exist at time t in the actual world. But there are plenty of possible worlds in which, at t, the latter exists but not the former. Thus it might have been the case at t that the bricks were arranged corral-wise rather than house-wise. So Brick Sum has a property that Brick House lacks, namely, the modal property of being such that its parts could have been arranged in non-house-wise fashion. Therefore, by the Indiscernibility of Identicals, Brick House is not identical to Brick Sum.

So even if the historical discernibility argument fails on Four Dimensionalism, the modal discernibility argument seems to work even assuming Four Dimensionalism.

Please note that my thesis is not that Brick House is a sum that violates Uniqueness of Composition, but that Brick House is not a classical mereological sum. If Brick House were a sum, then it would be Brick Sum. But I have just argued that it cannot be Brick Sum. So it cannot identified with any classical sum. It is a whole of parts all right, but an unmereological whole. What does that mean? It means that it is a whole that cannot be adequately understood using only the resources of classical mereology.

Van Inwagen on Arbitrary Undetached Parts

In order to get clear about Dion-Theon and related identity puzzles we need to get clear about the Doctrine of Arbitrary Undetached Parts (DAUP) and see what bearing it has on the puzzles. Peter van Inwagen provides the following statement of DAUP:

For every material object M, if R is the region of space occupied by M at time t, and if sub-R is any occupiable sub-region of R whatever, there exists a material object that occupies the region sub-R at t. ("The Doctrine of Arbitrary Undetached Parts" in Ontology, Identity, and Modality, CUP, 2001, 75.)

Suppose I am smoking a cigar. DAUP implies that the middle two-thirds of the cigar is just as much a concrete material object as the whole cigar. This middle two-thirds is an undetached part of the cigar, but also an arbitrary undetached part since I could have arbitrarily selected uncountably many other lengths such as the middle three-fourths. Applied to Tibbles the cat, DAUP implies that Tibbles-minus-one-hair is just as full-fledged a material object as Tibbles. Van Inwagen maintains that DAUP is false.

I will reconstruct van Inwagen's argument for the falsity of DAUP as clearly as I can. Consider Descartes and his left leg L. To keep it simple, we make the unCartesian assumption that Descartes is just a live body. DAUP implies that L is a material object as much as Descartes himself. DAUP also implies that there is a material object we can call D-minus. This is Descartes-minus-L. It is obvious that Descartes and D-minus are not the same. (For one thing, they are differently shaped. For another, they are 'differently abled' in PC jargon.) At time t, D-minus and L are undetached nonoverlapping proper parts of Descartes, and both are just as much full-fledged material objects as Descartes himself is.

Now suppose a little later, at t*, L becomes detached from D-minus. In plain English, Descartes at t* loses his leg. (To avoid certain complications, we also assume that the leg is not only removed but also annihilated.) Does D-minus still exist after t*? Van Inwagen thinks it is obvious that D-minus does exist after the operation at t*. DAUP implies that the undetached parts of material objects are themselves material objects. So D-minus prior to t* is a material object. Its becoming detached from L does not affect D-minus or its parts, and if the separation of L from D-minus were to cause D-minus to cease to exist, then, van Inwagen claims, D-minus could not properly be called a material object. Descartes himself also exists after the operation at t*. Surely one can survive the loss of a leg. So after t* both D-minus and Descartes exist. But if they both exist, then they are identical. For otherwise there would be two material objects having exactly the same size, shape, position, mass, velocity, etc., and that is impossible.

In sum, at time t, D-minus and Descartes are not identical, while at the later time t* they are identical. The result is the following inconsistent tetrad:

D-minus before t* = D-minus after t*

D-minus after t* = Descartes after t*

Descartes after t* = Descartes before t*

It is not the case that D-minus before t* = Descartes before t*

The first three propositions entail the negation of the fourth. From this contradiction van Inwagen infers that there never was any such thing as D-minus. If so, then DAUP is false. But as van Inwagen realizes, his refutation of DAUP has a counterintuitive consequence, namely, that L does not exist either: there never was any such thing as Descartes' left leg. For it seems obvious that D-minus and L stand or fall together, to repeat van Inwagen's pun.

That is, D-minus exists if and only if L exists, and D-minus does not exist if and only if L does not exist. D-minus is an arbitrary undetached proper part of Descartes if and only if L is an arbitrary undetached proper part of Descartes. At this point, I think it becomes clear that van Inwagen's solution to the Dion/Theon or Descartes/D-minus puzzle is not compelling. He solves the puzzle by denying that there was ever any such material object as D-minus. But if there was no D-minus, then there was never any such material object as Descartes' left leg. It is obvious, however, that there was such a material object as Descartes' left leg L. So how could it be maintained that there was no such object as Descartes-minus? Van Inwagen makes it clear (p. 82, n. 12) that he does not deny that there are undetached parts. What I take him to be denying is that, for any P and O, where P is an undetached part of material object O, there is a complementary proper part of O, O-minus-P. So perhaps van Inwagen can say that L is a non-arbitrary undetached part of Descartes and that this is consistent with there being no D-minus. If so, he would have to reject the following supplementation principle of mereology which seems intuitively sound:

For any x, y, z, if x is a proper part of y, then there exists a z such that z is a part of y and z does not overlap x , where x overlaps y =df there exists a z such that z is a part of x and z is a part of y.

What the above supplementation principle says is that you cannot have a whole with only one proper part. Every whole having a proper part has a second proper part that supplements or complements the first so as to constitute a whole. Now Descartes' leg is a proper part of Descartes. So the existence of D-minus falls out of the supplementation principle.

It seems, then, that van Inwagen's rejection of DAUP issues in a dilemma. If there is no such object as Descartes minus his left leg, then there is no such object as Descartes' left leg, which is highly counterintuitive, to put it mildly. But if van Inwagen holds onto the left leg, then it seems his must reject the seemingly obvious supplementation principle lately mentioned.

My interim conclusion is that van Inwagen's solution to the Descartes/D-minus puzzle by rejection of DAUP is not compelling.

Van Inwagen Contra Lewis on Composition as Identity

Modifying an example employed by Donald Baxter and David Lewis, suppose I own a parcel of land A consisting of exactly two adjoining lots B and C. It would be an insane boast were I to claim to own three parcels of land, B, C, and A. That would be 'double-counting': I count A as if it is a parcel in addition to B and C, when in fact all the land in A is in B and C taken together. Lewis, rejecting 'double-counting,' will say that A = (B + C). Thus A is identical to what composes it. This is an instance of the thesis of composition as identity.

Or suppose there are some cats. Then, by Unrestricted Composition ("Whenever there are some things, then there exists a fusion [sum] of those things"), there exists a sum that the cats compose. But by Composition as Identity, this sum is identical to what compose it, taken collectively, not distributively. Thus the sum is the cats, and they are it. I agree with van Inwagen that this notion of Composition as Identity is very hard to make sense of, for reasons at the end of the above link. But Peter van Inwagen's argument against Composition as Identity strikes me as equally puzzling. Van Inwagen argues against it as follows:

Suppose that there exists nothing but my big parcel of land and such parts as it may have. And suppose it has no proper parts but the six small parcels. . . . Suppose that we have a bunch of sentences containing quantifiers, and that we want to determine their truth-values: 'ExEyEz(y is a part of x & z is a part of x & y is not the same size as z)'; that sort of thing. How many items in our domain of quantification? Seven, right? That is, there are seven objects, and not six objects or one object, that are possible values of our variables, and that we must take account of when we are determining the truth-value of our sentences. ("Composition as Identity," Philosophical Perspectives 8 (1994), p. 213)

In terms of my original example, Lewis is saying that A is identical to what composes it. Van Inwagen is denying this and saying that A is not identical to what composes it. His reason is that there must be at least three entities in the domain of quantification to make the relevant quantified sentences true. A is therefore a third entity in addition to B and C. It is this that I don't understand. Van Inwagen's argument strikes me as a non sequitur. Or perhaps I just don't understand it. Consider this obviously true quantified sentence:

1. For any x, there is a y such that x = y.

(1) features two distinct bound variables, 'x'and 'y.' But it does not follow that there must be two entities in the domain of quantification for (1) to be true. It might be that the domain consists of exactly one individual a. Applying Existential Instantiation to (1), we get

2. a = a.

Relative to a domain consisting of a alone, (1) and (2) are logically equivalent. From the fact that there are two variables in (1), it does not follow that there are two entities in the domain relative to which (1) is evaluated. Now consider

3. There is an x, y and z such that x is a proper part of z & y is a proper part of z.

(3) contains three distinct variables, but it does not follow that the domain of quantification must contain three distinct entities for (3) to be true. Suppose that Lewis is right, and that A = (B + C). It will then be possible to existentially instantiate (3) using only two entities, thus:

4. B is a proper part of (B + C) & C is a proper part of (B + C).

If van Inwagen thinks that a quantified sentence in n variables can be evaluated only relative to a domain containing n entities (or values), then I refute him using (1) above. If van Inwagen holds that (3) requires three entities for its evaluation, then I say he has simply begged the question against Lewis by assuming that (B + C) is not identical to A. It is important not to confuse the level of representation with the level of reality. That there are two different names for a thing does not imply that there are really two things. ('Hesperus' and 'Phosphorus' both name the same planet, Venus, to coin an example.) Likewise, the fact that there are two distinct bound variables at the level of linguistic representation does not entail that at the level of reality there are two distinct values. There might be or there might not be.

So I cannot see that van Inwagen has given me any reason to think that A is a third entity in addition to B and C. But it doesn't follow that I think that Lewis' thesis is correct. Both are wrong. Here is the problem. 'A = (B + C)' is the logical contradictory of '~ (A = (B + C)).' Thus one will be tempted to plump for one or the other limb of the contradiction. But there are reasons to reject both limbs.

Surely A is more than the mereological sum of B and C. This is because A involves a further ontological ingredient, namely, the connectedness or adjacency of B and C. To put it another way, A is a unity of its parts, not a pure manifold. The Lewis approach leaves out unity. Suppose B is in Arizona and C is in Ohio. Then the mereological sum (B + C) automatically exists, by Unrestricted Composition. But this scattered object is not identical to the object which is B-adjoining-C. On the latter I can build a house whose square footage is greater than that of B or C; on the scattered object I cannot. But A is not a third entity. It is obvious that A is not wholly distinct from B and C inasmuch as A is composed of B and C as its sole nonoverlapping proper parts. Analysis of A discloses nothing other than B and C. But neither is A identical to B + C.

In short, both limbs of the contradiction are unacceptable. How then are we to avoid the contradiction?

Perhaps we can say that A is identical, not to the sum B + C, but to B-adjoining-C, an unmereological whole. But this needs explaining, doesn't it?

Mereological Innocence and Composition as Identity

This is the third in a series. Part I, Part II. What follows is a 10th example of eliminativist/reductivist ambiguity.

One of the axioms of mereology is Unrestricted Composition. Here is David Lewis' formulation (Parts of Classes, Basil Blackwell 1991, p. 74):

Unrestricted Composition: Whenever there are some things, then there exists a fusion of those things.

A fusion is a mereological sum, so I'll use 'sum.' The axiom assures us that, for example, if there are some cats, then there exists a sum of those cats. The cats are many but the sum is one. So it is not unreasonable to think that if there are five cats that compose the sum, the sum is a sixth thing. One could argue as follows: (a) The sum is distinct from each of the cats. (b)There are five cats, each of which exists, and by UC the sum also exists. Therefore, (c) at least six things exist.

But consider this example, adapted from Donald Baxter. You proceed with six bottles of beer to the supermarket 'six items or fewer' checkout line. The attendant protests your use of the line on the ground that you have seven items: six bottles of beer plus one mereological sum. This would be an outrage, of course. The example suggests that the argument to (c) above has gone wrong.

Lewis avoids the mistake — assuming it is one — by pleading that "Mereology is ontologically innocent." (PC 81) That means that a commitment to a cat-sum is not a further commitment over and above the commitment to the cats that compose the sum. The cat-sum just is the cats, and they are it. This is the thesis of Composition as Identity. The xs compose the y by being identical to the y. As Lewis says,

Take them together or take them separately, the cats are the same portion of Reality either way. Commit yourself to their existence all together or one at a time, it's the same commitment either way. If you draw up an inventory of Reality according to your scheme of things, it would be double counting to list the cats and also list their fusion. In general, if you are already committed to some things, you incur no further commitment when you affirm the existence of their fusion. (PC 81-82)

I'm sorry, but this doesn't make much sense. Glance back at Unrestricted Composition. It is not a tautology. It does not say that whenever there are some things, then there are some things. It says that whenever there are some things, then there exists a fusion or sum of those things. Now if the sum of the xs is just the xs, then UC is a tautology. But if UC is not a tautology, then Composition as Identity is false. How can Unrestricted Composition and Composition as Identity both be true?

The problem is already present at the purely syntactic level. 'Y is identical to the xs' is unproblematic if the xs are identical to one another. For then the open sentence collapses into 'y is identical to x.' But if the xs are distinct from each other, then 'y is identical to the xs' is syntactically malformed. How can one thing be identical to many things? If one thing is identical to many things, then it is not one thing but many things. A contradiction ensues: the one thing is one thing and not one thing because it is many things. The gaps in the predicate '. . . is identical to ____' must either be both filled with singular terms or both filled with plural terms.

And now we come back to our main theme, eliminativist/reductivist ambiguity. Lewis wants to say that there is the sum of the xs (by Unrestricted Composition) but that the the sum of the xs is identical to the xs. So he seems to be making a reductionist claim: sums reduce to their members. But I say the thesis is unstable and topples over into eliminativism: there are no mereological sums. For if the sum is just its members, then all that exists is the members so that the sum does not exist!

Fist and Hand, Statue and Lump: The Aporetics of Composition

1. Some maintain that a hand, and that same hand made into a fist, are identical. And there are those who would say the same about a piece of bronze and the statue made out of it, namely, that they are identical at every time at which both exist. This is not an unreasonable thing to say. After all, fist and hand, statue and bronze, are spatially coincident and neither has a physical part the other doesn't have. A fist is just a certain familiar arrangement of hand-parts. There is no part of the fist that is not part of the hand, and vice versa. So at looks as if first and hand are identical. But we need to be clear as to what identity is.

2. Identity is standardly taken to be an equivalence relation (reflexive, symmetrical, transitive) governed by the Indiscernibility of Identicals (InId) and the Necessity of Identity (NecId). The first principle says that, if two items are numerically identical, then they share all properties. The second says that if two items are numerically identical, then this is necessarily the case. Both principles strike me as beyond epistemic reproach. 'Identity' is short for 'numerical identity.'

3. Despite the considerations of #1, it looks as if fist and hand, statue and hunk of bronze, cannot be identical since they differ in their persistence conditions. The hunk of bronze can, while the statue cannot, survive being melted down and recast in some other form. The hand can, while the fist cannot, survive adoption of a different 'posture.' In both cases, something is true of the one that is not true of the other. So even at the times at which the fist is the hand, and the bronze is the statue, the two are not identical: the 'is' is not the 'is' of identity. It is the 'is' of composition and what you have are two things, not one.

What I have just given is a modal discernibility argument. Let me spell it out. Consider a time t at which the hand is in the shape of a fist. At t, the hand, but not the fist, has the modal property of being possibly such as to to be unfisted. So the hand cannot be identical to the fist given that, for any x, y, if x = y, then x, y share all properties.

But there is also this nonmodal discernibility argument. The hunk of bronze existed long before the statue came into existence, and the hunk of bronze exists while the form of a statue. So the hunk of bronze exists at more times that the statues does, which implies the the hunk of bronze is not identical to the statue.

There is also this consideration. Identity is symmetrical. So we can say either fist = hand or hand = fist. But is it the fist or the hand that both are? Intuitively, it is the hand. The hand is the fundamental reality here, not the fist. So how can fist and hand be identical? It seems that fist and hand are numerically distinct, albeit spatially coincident, concrete individuals.

4. The Law of Excluded Middle seems very secure indeed, especially in application to presently existing things. So either the fist is identical to the hand, and there is just one thing, a fisted hand, or the fist is not identical to the hand and there are two spatially coincident things, a fist and a hand. So which is it?

5. If you say that the fist = the hand, then when you make a fist nothing new comes into existence, and when the potter makes a pot out of clay, nothing new comes into existence. And when a mason makes a wall out of stones, nothing new comes into existence. He started with some stones and he ended with some stones. Given that the stones exist, and that the mason's work did not cause anything new to come into existence, must we not say that the single composite entity, the wall, does not exist? (For if it did exist, then there would be an existent in addition to the stones.) But it sounds crazy to say that the wall the mason has just finished constructing does not exist.

6. If, on the other hand, you say that the fist is not identical to the hand, then you can say that the making of a fist causes a new thing to come into existence, the fist. And similarly with the statue and the wall. After the mason stacks n stones into a wall, he has as a result of his efforts n +1 objects, the original n stones and the wall.

But this is also counterintuitive. Consider the potter at his wheel. As the lump of clay spins, the potter shapes the lump into a series of many (continuum-many?) intermediate shapes before he stops with one that satisfies him. Thus we have a series of objects (proto-pots) each of which is a concrete individual numericallt distinct from the clay yet (i) spatially conicident with it, and (ii) sharing with it every momentary property. And that is hard to swallow, is it not?

7. We appear to be at an impasse. We cannot comfortably say that the fist = the hand, nor can we comfortably say that the fist is not identical to the hand. Nor can we comfortably give up LEM. If there are no fists, statues, walls, artifacts generally, then there cannot be any puzzles about their composition. But we cannot comfortably say that there are no such things either.

Do we have here an example of a problem that is both genuine but insoluble?

The Aporetics of Artifacts: Puzzling Over Van Inwagen’s Denial of Artifacts

This post is a sequel to Van Inwagen on the Ship of Theseus. Peter van Inwagen, Material Beings (Cornell UP, 1990), p. 31, writes:

The question 'In virtue of what do these n blocks compose this house of blocks?' is a question about n + 1 objects, one of them radically different from the others. But the question 'What could we do to get these n blocks to compose something? is a question about n rather similar objects. . . . . questions of the former sort turn our minds to various metaphysical and linguistic questions about the "special" n + 1st [read: n + 1th] object and our words for it: What are the identity conditions for houses of blocks?

Why does van Inwagen think that a house of blocks is an object radically different from the blocks that compose it? And why does he think that if there are, say, 1000 blocks, then in the place where the house is, there are 1001 objects? Not only do I find these notions repugnant to my philosophical sense, I suspect that it is their extremism that motivates van Inwagen to recoil from them and embrace something equally absurd, namely, that there are no such things as houses of blocks or inanimate concrete partite entities generally.

In other words, if one begins by assuming that if a house of blocks, for example, is a whole of parts, then it is an object radically different from the objects that compose it, an object numerically additional to the objects that compose it; then, recoiling from these extreme positions, one will be tempted to embrace an equal but opposite extremism according to which there are no such inanimate partite entities as houses of blocks. What then should we say about a house of blocks?

First off, it is not identical to any one of its proper parts. Second, it is not identical to the mereological sum of its parts: the parts exist whether or not the house exists. From this it follows that there is a sense in which the house is 'something more' than its parts. But surely it is not an object "radically different" from, or numerically additional to, its proper parts. If there is a house of 1000 blocks in a place, there are not 1001 objects or entities in that place. After all, the house is composed of the blocks, and of nothing else.

So on the one hand the house is 'something more' than its constituent blocks, while on the other hand it is not a "radically different" object above and beyond them. Think of how absurd it would be for me to demand that you show me your house after you have shown me every part of it. "You've shown me every single part of your house, but where is the bloody house?"

The house, thought not identical to the blocks that compose it, is not wholly diverse from the blocks that compose it . The house is the blocks arranged housewise. The house is not the blocks, and the house is not some further entity "radically different" from the blocks. The house is just the blocks in a certain familiar arrangement. Should we conclude that the house exists or that it does not exist? I say it exists: the house is the blocks arranged housewise, and the existence of the house is the housewise unity of the blocks. Van Inwagen seems to think that there is no house, there are just the blocks. (Of course, he doesn't believe in the blocks either since they too are inanimate partite entities; but to keep the discussion simple, we may assume that the blocks are simples.)

Now if it is allowed that the house exists, it seems clear that the house does not exist in the way the blocks do. But this does not strike me as a good reason for saying that the house does not exist at all. What is wrong with saying that the house is a dependent existent? And what is wrong with saying that about partite entities generally? They exist, but they do not exist in addition to their parts, but as the unity or connectedness of their parts. Saying this, we avoid van Inwagen's absurd thesis that inanimate partite entities do not exist. Of course, this commits me to saying that there are at least two modes of existence, a dependent mode and an independent mode. I suspect van Inwagen would find such a distinction incoherent. But that is a topic for a separate post.

The problem can be set forth as an aporetic pentad:

1. The house is not identical to the blocks that compose it.

2. The house is not wholly diverse from the blocks that compose it; it is not an object numerically additional to the blocks that compose it: given that the house is composed of n blocks, the house itself is not an n + 1th object.

3. The house exists.

4. The constituent blocks exist.

5. 'Exists' is univocal as between wholes and parts: wholes and their parts exist in the same sense.

Each limb has a strong claim on our acceptance. But they cannot all be true. Any four of the propositions, taken together, entails the negation of the remaining one. For example, if the first four are all true, then the fifth must be false. To solve the problem, one of the limbs must be rejected. But which one?

To me it seems obvious that the first four are all true. So I reject (5). Rejecting (5), I can say that the house exists as the connectedness of the blocks. Thus the mode of existence of the whole is different from the mode of existence of its simple parts. But this solution requires us to believe in modes of existence, which is sure to inspire opposition among analytic philosophers. Van Inwagen, if I understand him, denies (2) and (3) while accepting the others.

But van I's solution is just crazy, is it not? Mine is less crazy. But perhaps you, dear reader, have a better suggestion.

Van Inwagen on the Ship of Theseus

Peter van Inwagen's Material Beings (Cornell UP, 1990) is a very strange book, but he is a brilliant man, so one can expect to learn something from it. A central claim is that artifacts such as tables and chairs and ships do not exist. One can appreciate that if there are no ships then the ancient puzzle about identity known as the Ship of Theseus has a very quick (dis)solution.

The Ship of Theseus is a puzzle about diachronic artifact identity. Here is one version. You have a ship, or a rowboat, or any object, composed entirely of wooden planks. You remove one of the planks and replace it with an aluminum plank of the same size. The wooden plank is placed in a warehouse. After this minor replacement, you have a ship and indeed numerically the same ship as the one you started with. It is not a numerically different ship. Now replace a second wooden plank with an aluminum plank, and place the second wooden plank in the warehouse. Again, the numerical identity of the original ship has been preserved. Continue the replacement process until all of the wooden planks have been replaced with aluminum planks. You now have a wholly aluminum ship that is presumably numerically identical to the original wholly wooden ship despite the fact that none of the original matter is to be found in the aluminum ship. After all, the aluminum ship 'grew out of' the original wooden ship by minor changes each of which is identity-preserving.

Now take the wooden planks from the warehouse and assemble them in the form of a ship and in such a way that the planks bear the same relations to one another as the planks in the original wooden ship bore to one another. You now have two ships, a wooden one and an aluminum one. The question is: which of these ships is identical to the original wooden one?

Suppose the two ships collide on the high seas, and suppose the captain of the original ship had taken a solemn vow to go down with his ship. Where does his duty lie? With the wooden ship or with the aluminum one? Is the original ship identical to the resultant aluminum ship? One will be tempted to say 'yes' since the aluminum ship 'grew out' of the original wooden ship by minor transformations each of which was identity-preserving. Or is the original ship identical to the wooden ship that resulted from the re-assembly of the wooden planks? After all, it consists of the original matter arranged in the original way. Since the resultant wooden and aluminum ships are numerically distinct, they cannot both be identical to the original ship.

Van Inwagen makes short work of the puzzle: "There are no ships, and hence there are no puzzles about the identities of ships." (128) One way van Inwagen supports this bizarre solution is by re-telling the story in language that does not make even apparent reference to ships. Here is his retelling:

Once upon a time, there were certain planks that were arranged shipwise. Call then the First Planks. . . . One of the First Planks was removed from the others and placed in a field. Then it was replaced by a new plank; that is, a carpenter caused the new plank and the remaining First Planks to be arranged shipwise, and in just such a way that the new plank was in contact with the same planks that the removed planks had been in contact with, and at exactly the same points. Call the planks that were then arranged shipwise the Second Planks. A plank that was both one of the First Planks and one of the Second Planks was removed from the others and placed in the field and replaced (according to the procedure laid down above), with the consequence that certain planks, the Third Planks, were arranged shipwise. Then a plank that was one of the First Planks and one of the Second Planks and one of the Third Planks . . . . This process was repeated till all the First Planks were in the field. Then the First Planks were caused to be arranged shipwise, and in just such a way that each of them was in contact with the same planks it had been in contact with when the First Planks had last been arranged shipwise, and was in contact with them at just the same points. (128-129)

If I understand what van Inwagen is claiming here, it is that there is nothing in the standard telling of the story, a version of which I presented above, that is not captured in his re-telling. But since there is no mention of any ships in the re-telling, no puzzle about ship-identity can arise. Perhaps van Inwagen's point could be put by saying that the puzzle about identity is an 'artifact' of a certain way of talking that can be paraphased away. Instead of talking about ships, we can talk about shipwise arrangements of planks. The planks do not then compose a ship, he thinks, and so there is no whole of which they are proper parts, and consequently no question about how this whole maintains its diachronic identity under replacement of its parts.

What are we to say about van Inwagen's dissolution of the puzzle? What I find dubious is van Inwagen's claim that ". . . at no time do two or more of these planks compose anything, and no plank is a proper part of anything." (129) This strikes me as plainly false. If the First Planks are arranged shipwise, then there is a distinction beween the First Planks and their shipwise arrangement. The latter is the whole ship and the former are its proper parts. So how can van Inwagen claim that the planks do not compose a ship? Van Inwagen seems to think that if the planks were parts of a whole, and there were n planks, then the whole would be an n + 1 th entity. Rejecting this extreme, he goes to the other extreme: there is no whole of parts. If there were ships, they would be wholes of parts, but there are no artifactual wholes of parts, so there are no ships. The idea seems to be that when we build an artifact like a ship we are not causing something new to come into existence; we are merely re-arranging what already exists. If so, then although a ship's planks exist, the ship does not exist. Consider what van Inwagen says on p. 35:

If I bring two cubes into contact so that the face of one is conterminous with the face of the other, have I thereby brought into existence a solid that is twice as long as it is wide? Or have I merely rearranged the furniture of the earth without adding to it?

Van Inwagen seems to be saying that when it comes to artifacts, there is only rearrangement, no 'addition to existence.' As a general thesis, this strikes me as false. A ship is more than its planks, and van Inwagen seems to concede as much with his talk of a shipwise arrangement of planks; but this shipwise arrangement brings something new into being, namely, a thing that has causal powers that its constituents do not have. For example, a boat made of metal planks properly arranged will float, while the planks themselves will not float.

Sets and the Number of Objects: An Antilogism

Commenter Jan, the Polish physicist, gave me the idea for the following post.

An antilogism is an aporetic triad, an array of exactly three propositions which are individually plausible but collectively inconsistent. For every antilogism, there are three corresponding syllogisms, where a syllogism is a deductive argument with exactly two premises and one conclusion. Here is the antilogism I want to discuss:

1. Possibly, the number of objects is finite.
2. Necessarily, if sets exist, then the number of objects is not finite.
3. Sets exist.

The modality at issue is 'broadly logical' and sets are to be understood in the context of standard (ZFC) set theory. 'Object' here just means entity. An entity is anything that is. (Latin ens, after all, is the present participle of the infinitive esse, to be.)

Corresponding to the above antilogism, there are three syllogisms. The first, call it S1, argues from the conjunction of (1) and (2) to the negation of (3). The second, call it S2, argues from the conjunction of (2) and (3) to the negation of (1). The third, call it S3, argues from (1) and (3) to the negation of (2).

Note that each syllogism is valid, and that the validity of each reflects the logical inconsistency of the the antilogism. Note also that for every antilogism there are three corresponding syllogisms, and for every syllogism there is one corresponding antilogism. A third thing to note is that S3 is uninteresting inasmuch as it is surely unsound. It is unsound because (2) is unproblematically true.

This narrows the field to S1 which argues to the nonexistence of (mathematical) sets and S2 which argues to the impossibility of the number of objects (entities) being finite. Our question is which of these two syllogisms we should accept. Obviously, both are valid, but both cannot be sound. Do we have good reason to prefer one over the other?

Here are our choices. We can say that there is no good reason to prefer S1 over S2 and vice versa; that there is good reason to prefer S1 over S2; or that there is good reason to prefer S2 over S1.

Being an aporetician, I incline toward the first option. Peter Lupu, being less of an aporetician and more of dogmatist, favors the third option. Thus he thinks that the antilogism is best solved by rejecting (1). Peter writes:

(a) If there are infinitely many numbers, then (1) is false. Are there infinitely many numbers? Very few would deny this. How could they, for then they would have to reject most of mathematics. [. . .]

To keep it simple, let's confine ourselves to the natural numbers and the mathematics of natural numbers. (The naturals are the positive integers including 0.) If there are infinitely many naturals, then there are infinitely many objects. If so, then presumably this is necessarily so, whence it follows that (1) is false.

I fail to see, however, why there MUST be infinitely many naturals. I am of course not denying the obvious: for any n one can add 1 to arrive at n + 1. With a sidelong glance in the direction of Anselm of Canterbury: there is no n that fits the description 'that than which no greater can be computed.' In plain English: there is no greatest natural number. But this triviality does not require that all of the results of possible acts of +1 computation actually be 'out there' in Plato's heaven. When I drive along a road, I come upon milemarkers that are already out there before I come upon them. But why must we think of that natural number series like this? I don't bring the road and its milemarkers into being by driving. But what is to stop us from viewing the natural number series along Brouwerian (intuitionistic) lines? One can still maintain that the series is infinite, but the infinity is potential not actual or completed. Peter's first argument, as it stands, is not compelling. (Compare: Everyone will agree that every line segment is infinitely divisible. But it does not follow that every line segment is infinitely divided.)

(b) If propositions exist, then there are infinitely many propositions. Are there propositions? Kosher-nominalists obviously will have to deny that propositions exist. Sentences do not express propositions. But, then, what do they express?

I am on friendly terms with Fregean (not Russellian) propositions myself. And I grant that it is very plausible to say that if there is one proposition then there is an actual infinity of them. Consider for example the proposition *p* expressed by 'Peter has a passion for philosophy.' *P* entails *It is true that p* which entails *It is true that it is true that p,* and so on infinitely. But again, why can't this be a potential infinity?

The following three claims are consistent: (i) Declarative sentences express propositions; (ii)Propositions are abstract; (iii) Propositions are man-made. Karl Popper's World 3 is a world of abstracta. It is a bit like Frege's Third Reich (as I call it), except that the denizens of World 3 are man-made.

I am agreeing with Peter and against the illustrious William that there are (Fregean) propositions, understood as the senses of context-free declarative sentences. I simply do not understand how a declarative sentence-token could be a vehicle of a truth-value. But why can't I say that propositions are mental constructs? (This diverges from Frege, of course.)

(c) Are there sentence types? A nominalist will have to deny the existence of sentence types. But, then, it is difficult to see how any linguistic analysis can be done.

Peter may be conflating two separate questions. The first is whether there are any abstract objects, sentence types for example. The second is whether there is an actual infinitity of them. He neeeds the latter claim as a countrerexample of (1). So again I ask: why couldn't there be a finite number of abstract objects: a finite number of sets, propositions, numbers, sentence types, etc. This would make sense if items of this sort were Popperian World 3 items.

I conclude that, so far, there is no knock-down refutation of (1). But there is also no knock-down refutation of (3) either, as Peter will be eager to concede. So I suggest that the rational course is to view my (or my and Jan's) antilogism as a genuine intellectual knot that so far has not been definitively solved.

Presentism and Existence-Entailing Relations: An Aporetic Tetrad

It is plausibly maintained that all relations are existence-entailing. To illustrate from the dyadic case: if R relates a and b, then both a and b exist. A relation cannot hold unless the things between which or among which it holds all exist. A weaker, and hence even more plausible, claim is that all relations are existence-symmetric: if R relates a and b, then either both relata exist or both do not exist. Both the stronger and the weaker claims rule out the possibility of a relation that relates an existent and a nonexistent. (So if Cerberus is eating my cat, then Cerberus exists. And if I am thinking about Cerberus, then, given that Cerberus does not exist, my thinking does not relate me to Cerberus. This implies that intentionality is not a relation, though it is, as Brentano says, relation-like (ein Relativliches).)

But if presentism is true, and only temporally present items exist, then no relation connects a present with a nonpresent item. This seems hard to accept for the following reason.

I ate lunch an hour ago. So the event of my eating (E) is earlier than the event of my typing (T). How can it be true that E bears the earlier than relation to T, and T bears the later than relation to E, unless both E and T exist? But E is nonpresent. If presentism is true, then E does not exist. And if E does not exist, then E does not stand in the earlier than relation to T. If, on the other hand, there are events that exist but are nonpresent, then presentism is false.

How will the presentist respond? Since E does not exist on his view, while T does, and E is earlier than T, he must either (A) deny that all relations are existence-symmetric, or deny (B) that earlier than is a relation. He must either allow the possibility of genuine relations that connect nonexistents and existents, or deny that T stands in a temporal relation to E.

To fully savor the problem we cast it in the mold of an aporetic tetrad:

1. All relations are either existence-entailing or existence-symmetric.

2. Earlier than is a relation.

3. Presentism: only temporally present items exist.

4. Some events are earlier than others.

Each limb of the tetrad is exceedingly plausible. But they cannot all be true: any three, taken together, entail the negation of the remaining limb. For example, the first three entail the negation of the fourth. To solve the problem, we must reject one of the limbs. Now (4) cannot be rejected because it is a datum.

Will you deny (1) and say that there are relations that are neither existence-entailing nor existence-symmetric? I find this hard to swallow because of the following argument. (a) Nothing can have properties unless it exists. Therefore (b) nothing can have relational properties unless it exists. (c) Every relation gives rise to relational properties: if Rab, then a has the property of standing in R to b, and b has the property of standing in R to a. Therefore, (d) if R relates a and b, then both a and b exist.

Will you deny (2) and say that earlier than is not a relation? What else could it be?

Will you deny presentism and say that that both present and nonpresent items exist? Since it is obvious that present and nonpresent items cannot exist in the present-tense sense of 'exists,' the suggestion has to be that present and nonpresent (past or future) items exist in a tenseless sense of 'exist.' But what exactly does this mean?

The problem is genuine, but there appears to be no good solution, no solution that does not involve its own difficulties.

Frondizi on the Philosophical Attitude

Risieri Fondizi's What is Value? An Introduction to Axiology, tr. S. Lipp (Open Court, 1963) has stood up well since its English debut over forty five years ago. What follows is a noteworthy metaphilosophical observation of Frondizi's:

The philosophical attitude is basically problematic. He who is not capable of grasping the sense of problems and who prefers to seize upon the first solution that presents itself, and which offers him illusory stability, runs the risk of being submerged, together with his so-called solution, in a sea of difficulties. (p. 26)

'Problematic' can mean dubious. But what Frondizi intends is best rendered by 'problem-oriented.' A philosopher is someone who is sensitive to puzzles, problems, and mysteries. Or, as I like to say, the philospher is one who has the aporetic sense. I once heard Roderick Chisholm say that one is not philosophizing until one has a puzzle. That's exactly right. But of course it's an old thought. At Theaetetus 155, Plato tells us that philosophy begins in wonder or perplexity, this being the characteristic feeling of the philosopher. Aristotle echoes the idea at Metaphysics 982b10.

Wilfrid Sellars once likened the philosopher's touch to that of King Midas. Whatever the king touched, turned into gold; whatever the philosopher touches turns into a puzzle. The trouble with this comparison is that it suggests that philosophers create their difficulties. Not so: they discover them, or at least some of them. We could call the ones that are discovered the ground-level problems, distinguishing them from problems that arise as artifacts of theories proposed in solution of the ground-level problems. The ground-level problems are in a certain sense 'out there' independent of our linguistic and conceptual operations. Pace Wittgenstein, they are not engendered by a "bewtchment of our understanding by language" (eine Verhexung unseres Verstandes durch die Sprache). Pace Rorty, they do not arise as artifacts of arbitrarily adopted ways of talking.

The problem of universals, for example, is a perennial problem. It may not interest you, or seem important, but it is there whether you like it or not, and it has repercussions for problems you probably will find important. We attribute properties to things, and sometimes the things to which we attribute the properties actually have them. But what are properties? Are they mental in nature, or perhaps linguistic? Or are properties independent of language and mind? If the latter, are they universals (repeatable entities) or particulars (unrepeatable entities such as sets or tropes)? If properties are universals, can they exist uninstantiated, or can they only exist when instantiated? How are properties related to the things that have them.

These are some of the questions that arise when we think about what is somewhat misleadingly called the problem of universals. 'The problem of properties' is perhaps a better moniker.

The onus probandi is on anyone who claims that this problem (or cluster of problems) is not genuine.

A Modal Aporetic Tetrad

Here is a four-limbed aporetic polyad:

1. The merely possible is not actual.

2. To be actual is to exist.

3. To exist is to be.

4. The merely possible is not nothing.

Each limb is plausible, but they cannot all be true. The first three limbs, taken together, entail the negation of the fourth. Indeed, any three, taken together, entail the negation of the remaining limb, as you may verify for yourself.

Now which limb ought we reject in order to avoid logical inconsistency? (1) is non-negotiable because purely definitional. Everything actual is possible, but not everything possible is actual. 'Merely possible,' by definition, refers to that which is possible but not actual. This leaves us three options.

(2)-Rejection. One might reject the equivalence of the actual and the existent analogously as one might reject the equivalence of the temporally present and the existent. Just as one might maintain that past events exist just as robustly as present events despite their pastness, one might maintain that merely possible items exist just as robustly as actual items. David Lewis' extreme ('mad dog') modal realism is an example of (2)-rejection. On his modally egalitarian scheme there is a plurality of possible worlds all on an ontological par. Each is a maximal mereological sum of concreta. Each of these worlds is actual at itself, but no one of these worlds is actual simpliciter. For each world w, w is actual-at-w, but no world is actual, period. Thus there is no such property as absolute actuality. It is not the case that one of the worlds is privileged over all the others in point of being actual simpliciter. What is true of a world is true of its occupants: I enjoy no ontological privilege over that counterpart of me who is bald now and living in Boston. Actuality is world-relative and 'actual' is accordingly an indexical term like 'now.' When I utter a token of 'now' I refer to the time of my utterance; likewise, on Lewis' theory, when I utter a token of 'actual,' I refer to the world I am in.

Having rejected (2), a Lewis-type philosopher could gloss the other limbs of the tetrad as follows. To say that the merely possible is not actual is to say that merely possible objects (e.g. bald Bill the Bostonian) are denizens of worlds other than this one. To say that to exist is to be is to say that there is no distinction between the existence of an object and its being in some world or other. To say that the merely possible is not nothing is to say that objects which are not denizens of this world are denizens of some other world or worlds.

I am tempted to say that this solution, via rejection of (2), is worse than the problem. For one has to swallow an infinity of equally real possible worlds. Further, my possibly being bald is not some counterpart of mine's being bald in another possible world. (This critique of course needs to be spelled out in detail.)

(3)-Rejection. A second theoretical option is to reject the equivalence of being and existence, of that which is and that which exists. Accordingly, there are things that are but do not exist. They have Being but not Existence. Everything is, but only some things exist. The early Russell, in the Principles of Mathematics from 1903, toyed with this view although he rejected it later in his career. If existents are a proper subset of beings, then one could locate merely possible items in among the beings that do not exist. The merely possible would then have Being but not Existence or Actuality.

This solution leads to an ontological population explosion much as the Lewis theory does.

(4)-Rejection. A third option is to deny (4) by affirming that the merely possible is nothing in reality, that it has no ontological status. One might construe the merely possible as merely epistemic, as being merely parasitic on our ignorance, or as having no status outside our thought. A view along these lines can be found in Spinoza.

Intuitively, though, it seems mistaken to say that there are no genuine, mind-independent possibilities. My writing desk, for example, is one inch from the wall, but it could have been two inches from the wall. It is not just that I can imagine or conceive it being two inches from the wall; it really could be two inches from the wall even though this possible state of affairs was never actual and never will be actual. (Moreover, what I CAN imagine or conceive refers to real but unactual possibilities of imagination and conception; or will you say that these possibilities are themselves derivative from acts of imagining or conceiving? If you do, then a vicious infinite regress is in the offing.))

Now suppose I had provided more rigorous and more convicing rejections of each of the three theoretical options. Suppose that a strong case can be made that all four propositions must be accepted. Then we would have four propositions each of which has a very strong claim on our acceptance, but which are collectively inconsistent. (Assume that the inconsistency is demonstrable.) What might one conclude from that? (A) One possibility is that we ought to abandon the Law of Non-Contradiction. (B) A second is that one of the solutions must be right even though we have good reason to think that every solution is mistaken. (C) A third is that the aporetic tetrad is an insoluble problem, a genuine intellectual knot that cannot be untied.

Note that (A), (B), and (C) form a meta-aporia. Each of them has a claim on our acceptance, but they cannot all be true.

Suppose there are genuine but absolutely insoluble philosophical problems. What would that show, if anything?

The Aporetics of Divine Simplicity

Thomist27 e-mails:

Thank you first of all for a spectacular blog. I discovered Maverick Philosopher a few years ago and have been reading it regularly ever since. Through your blog, I learned that you wrote the SEP's article on divine simplicity, among similar things; I think, then, that you are qualified to answer my questions.

My questions concern divine simplicity and divine knowledge, two nuts that I've lately been making every effort to crack. First, do you think that theism can be salvaged without absolute divine simplicity? I know that there are many theists who don't believe that God is simple, but is such a concept of Deity coherent?

I believe a case can be made, pace Alvin Plantinga and other theistic deniers of divine simplicity, that to deny the absolute ontological simplicity of God is to deny theism itself. For what we mean by 'God' is an absolute reality, something metaphysically ultimate, "that than which no greater can be conceived." (Anselm) Now an absolute reality cannot depend for its existence or nature or value upon anything distinct from itself. It must be from itself alone, or a se. Nothing could count as divine, or worthy of worship, or be an object of our ultimate concern, or be maximally great, if it lacked the property of aseity. But the divine aseity, once it is granted, seems straightaway to entail the divine simplicity, as Aquinas argues in ST. For if God is not dependent on anything else for his existence, nature, and value, then God is not a whole of parts, for a whole of parts depends on its parts to be and to be what it is. So if God is a se, then he is not a composite being, but a simple being. This implies that in God there is no real distinction between: existence and essence, form and matter, act and potency, individual and attribute, attribute and attribute. In sum, if God is God, then God is simple. To deny the simplicity of God is to deny the existence of God. It is therefore possible for an atheist to argue: Nothing can be ontologically simple, therefore, God cannot exist.

A theist who denies divine simplicity might conceivably be taxed with idolatry inasmuch as he sets up something as God that falls short of the exacting requirements of deity. The divine transcendence would seem to require that God cannot be a being among beings, but must in some sense be Being itself . (Deus est ipsum esse subsistens: God is not an existent but self-subsisting Existence itself.) On the other hand, a theist who affirms divine simplicity can be taxed, and has been taxed, with incoherence. As an aporetician first and foremost, I seek to lay bare the problem in all its complexity under suspension of the natural urge for a quick solution.

Second, if my understanding is correct, then according to the doctrine of divine simplicity, God has no intrinsic accidents. How is that compatible with divine freedom? I know it's trite, but I haven't seen a good answer to the question of how God could have properties such as having created mankind or having declined to create elves without their being just as necessary to Him as His benevolence and omnipotence (especially if He is what He does).

This is indeed a problem. On classical theism, God is libertarianly free: although he exists in every metaphysically possible world, he does not create in every such world, and he creates different things in the different worlds in which he does create. Thus the following are accidental properties of God: the property of creating something-or-other, and the property of creating human beings. But surely God cannot be identical to these properties as the simplicity doctrine seems to require. It cannot be inscribed into the very nature of God that he create Socrates given that he freely creates Socrates. Some writers have attempted to solve this problem, but I don't know of a good solution.

Even if there's a solution to that problem, what's to be said about God's knowledge? Isn't His knowledge an intrinsic property of His? But, since the truth of a proposition like the planet Mars exists is contingent, isn't God's knowing it an accidental property, and, furthermore, an intrinsic accidental property?

Well, this too is a problem. If S knows that p, and p is contingent, then S's knowing that p is an accidental (as opposed to essential) property of S. Now if God is omniscient, then he knows every (non-indexical) truth, including every contingent truth. It seems to follow that God has at least as many accidental properties as there are contingent truths. Surely these are not properties with which God could be identical, as the simplicity doctrine seems to require. Now there must be some contingent truths in consequence of the divine freedom; but this is hard to square with the divine simplicity.

And if it is in fact the case that God's knowledge is the cause of things, then how are we to understand His knowledge of the free actions of creatures? I know that God is supposed to be the final cause of these actions, as well as their ultimate efficient cause, but the issue is still unclear to me.

This is also a problem. The simplicity doctrine implies that God is identical to what he knows. It follows that what he knows cannot vary from world to world. In the actual world A, Oswald shoots Kennedy at time t. If that was a libertarianly free action, then there is a world W in which Oswald does not shoot Kennedy at t. Since God exists in very world, and knows what happens in every world, he knows that in A, Oswald shoots Kennedy at t and in W that Oswald does not shoot Kennedy at t. But this contradicts the simplicity doctrine, according to which what God knows does not vary from world to world. The simplicity doctrine thus appears to collide both with divine and human freedom.

I sincerely look forward to your addressing these questions. Thank you in advance for your consideration of these weighty matters.

I have addressed them, but not solved them. Solutions have been proffered, but they give rise to problems of their own — something to be pursued in future posts.