Sunday, August 25, 2013

Thoughts on the Grounding Objection to Molinism

So, I want to get a bit more clear on what the grounding objection to Molinism is saying. As far as I can tell at this moment, the grounding objection seems to go something like this.

The anti-Molinist says that some general statement about the relation between grounding and truth such as the following holds:

(A) If some proposition is true then there is an entity which grounds its truth.

It seems in this context 'grounds the truth of p' just means 'is the truthmaker of p'. The objector to Molinism then proposes:

(B) There could be no entity to ground the truth of CCF's.

Of course from A and B it follows that all CCF's, if they are meaningful, are necessarily false. Hopefully this is all a correct representation of the objection.

Now, one problem is that we seem to be starting with abstract principles and moving to judgments about cases, whereas we should probably go the other way around; for instance, supposing it is true CCF's can't be 'grounded', then in this case we should recognize the truth of some CCF's and conclude there is something wrong with (A), rather than vice versa. For it is more clear to me that I would have gotten chicken had I gone to Panda Express earlier than it is that all truths need to be grounded in the sense we're talking about.

Another problem is that there is no immediate reason to accept either (A) or (B). What are the arguments for them?

Also, there seem to be problems with both premises. Let's start off with (B): Why can't the fact that I would do such and such action in such and such circumstances be the 'ground' of the counterfactual? Note, facts are not propositions. Maybe take facts to be necessarily existing entities.

If for some reason this is not allowed according to (A), then there are still problems for (A) itself. Since it seems to entail the immediate falsity of presentism given that statements about the future are true, (A) would have to be more plausible than presentism, and why think that? And why is (A) true in the face of, say, negative propositions (e.g. the proposition that there does not exist a chair)? Also, is (A) necessarily true? If so, what is the truthmaker of (A)?

Regarding that last question, there seems to be a lurking circularity or something in (A). This is a real problem. Consider the following argument (which I'll admit I came up with rather quickly and thus it may be problematic). Note beforehand: I'm going to define S below as a set; that's inaccurate since it's really supposed to be a sum of entities, but that makes it easier to deal with and there is a sort of function from the relevant statements about the set to the corresponding statements about the sum. Also, I will just assume for ease that S is finite, which is probably not a plausible assumption, but the argument could be made to work in the infinite case if one really wanted it to. Anyway, let us consider the argument:

1. Suppose all true propositions have a truthmaker. [assumption, called (A)]
2. (A) is a true proposition. [premise] Then,
3. (A) has a truthmaker. [by 1 and 2] But,
4. If (A) has a truthmaker, then the truthmaker of (A) would have to be the sum S = {t : For some truth P, t is a truthmaker of P} = {t1, ..., tn, tn+1} [premise]
5. The truthmaker of (A) is the sum S = {t1, ... , tn, tn+1} = {t : For some truth P, t is a truthmaker of P} [by 3 and 4]
6. Suppose S is in S and let S = tn+1. [assumption]
7. For all sums A, if A = {x} U {x1, ..., xn} and A =/= {x1, ... , xn}, then x is a proper part of A. [definition]
8. Then S would be a proper part of S, since S would be {S} U {t1, t2, ... , tn} and S =/= {t1, t2, ... , tn}. [by def. of proper part]
9. But no sum can be a proper part of itself; that is impossible. [premise]
10. So S is not in S. [by 8 and 9]
11. But S must be in S, for S is a truthmaker of some truth (viz. (A)). [by 5 and definition of S]
12. That's a contradiction right there^.
13. So one of 1 - 3 is false, which implies (A) is false. [1-11]

So it's false that all true propositions have a truthmaker.

Also, if (A) entails there are no true CCF's, then we can argue against (A) by giving a very strong argument for the truth of CCF's. Let us consider the following argument, which I will call the 'direct argument' for the truth of CCF's. It seems to have first been endorsed by the great Jesuit theologian Francisco Suarez. Take a CCF like the following:

(*) If Alfredo were to defend Molinism, Reginald would have accepted it.

This is of the form A => B. We can verify either A => B or its counterpart A => ~B as follows: Have Alfredo defend Molinism, and see if Reginald accepts it or not. If he accepts it, then someone would have been correct in asserting (*) at some time e beforehand. If Reginald does not accept it, then someone asserting the counterpart CCF of the form A => ~B at time e would have been correct instead. Either way, at least one CCF was true at e, so at least some CCF's are true at some time. But whether the antecedent A is true in the actual world or not shouldn't make a difference metaphysically speaking; for why does the truth about this counterfactual depend on the actual world's in fact being actual? Also, there is nothing special about this particular CCF nor its being said at e. So all CCF's or their counterparts are true. And that entails (A) is false.

So, while it's not even clear to me CCF's can't be grounded as per (B), we have a lot of reasons to think (A) is incorrect anyway and not much reason to think it is true (it seems to me whatever intuitive force (A) has can probably be captured just as well by weaker principles). Hence, we should reject the proposition that all propositions must have a truthmaker, and thus the grounding objection to Molinism fails.

Sunday, August 18, 2013

My Theology Blog

For any readers of this site, I have created a new Catholic blog called The Scriptorium, which you can find at I will continue to post here at The Analytic Scholastic as well.

It may seem to complicate things to keep two blogs going at once. However, I think it actually complicates things less to do so. I post here less frequently, and when I do I tend to post rather longer, technical philosophical thoughts which I'm having, thoughts which aren't always directly related to religion or Catholicism. On the one hand, for those who want to delve deep into the issues, this is a good thing. But it has the downside of not being as accessible to everyone. The Scriptorium on the other hand won't be as much about technical philosophy, especially areas of philosophy which are very unrelated to religion. While maintaining rigor and clarity it should be quite a bit more accessible than this blog. So I feel it is better to not get the two mixed up.

Anyway, if you are interested in theology, philosophy of religion, or Catholicism in general, please do check out The Scriptorium. I think you'll like it!

Thursday, July 11, 2013

Immaterial Animals?

In a previous post, I gave an argument for thinking substance dualism is true as opposed to hylomorphic dualism. To re-cap, the argument goes as follows. I will number the premises slightly differently to avoid confusion later:

(1*) Some disembodied conscious subject psychologically continuous with me continues to exist after death.
(2*) I am this subject.
(3*) The animal associated with me does not continue to exist after death.
(4*) So I am not identical to the animal associated with me.

I explain and defend the argument more fully in the previous post, and explain why I think substance dualism follows.

In conversation elsewhere, I discussed this argument with Prof. Robert Koons and some other friends (Koons' new blog, which I recommend, can be found here). Most of them suggested that we should deny the third premise (3*), and say that the animal actually continues to exist in an immaterial state after death. Prima facie this reply seems obviously incorrect to me. Denying (3*) seems to me to be an instance of going to unreasonable lengths in order to deny a conclusion. I sincerely do not mean that to be rude or arrogant, I just really don't think there is good reason for denying it. Still, I want to repeat some of the comments I made in reply to this proposal. I have two worries about this proposal: First, that it stretches the concept of an animal to the point of absurdity, and second that it is impossible given some basic Aristotelian metaphysics.

The first worry is that this seems to be stretching the concept of an animal quite a bit. I feel I begin to lose a grip on what the essence of an animal is if not something which is material, which has biological organs, biologically functions in a certain way, interacts with a certain environment, is alive, etc. So if hylomorphic dualism entails the possibility of immaterial animals and animals with none of these features, this may actually count as an argument against the plausibility of hylomorphic dualism.

Moreover, suppose at some time t all forms of life in the universe cease to exist but our souls subsist. Then, at least given one scholastic criterion of God's omnipotence i.e. that God has the power to bring about whatever involves no explicit contradiction, God could bring about that there are animals who are never any of the things I listed (material, having biological organs, etc.); for he could create ex nihilo a different world just like this one except the first moment of this world is t. So it would seem hylomorphic dualism not only entails the possibility of immaterial animals, but it also entails the possibility of animals who are never at any time material, living, having biological organs, etc.

This seems to me to count against hylomorphic dualism; at the very least animals must have these material features at some time (most likely when they began to exist). The sense of "must" here is not only statistical, as in "they usually begin in a material state", but rather metaphysical in the sense of, "necessarily (in all metaphysically possible worlds), they begin to exist in a material state." Maybe one could reply that the situation I describe is actually impossible, because somehow it involves a contradiction to suppose an animal is created in an immaterial state; but I do not see any prima facie contradiction in this situation, at least if one thinks it is possible for them to exist immaterially. So until this contradiction is made explicit I do not have a good reason to think there is one.

The second objection I have is based on Aristotelian metaphysics. Traditionally in the Aristotelian "genus-et-differentiam" tree animals will fall under the broad category of material substances, and since genus and specific difference give you the essence of something (in our case, rational + animal), and since animal falls under 'material substance', being material is part of the essence of all animals. But then I take it if you have a certain feature as part of your essence, then necessarily if you exist then you have that feature (thought I don't believe the converse always holds). Hence, it is not possible to be immaterial and still be an animal. This argument can be formalized as follows:

1. The essence of 'man' is 'rational animal'.
2. 'Animal' falls under the genus 'material substance'.
3. If one of the components of the essence of x falls under a genus, then that genus is part of the essence of x.
4. So, 'material substance' is part of the essence of 'man'.
5. But if P is part of the essence of x, then necessarily if x exists then x is P.
6. So, necessarily if man exists then man is material.

That can be made more precise but it should suffice. Let me explain premise 3 since it's the only one that I think might be a bit unclear. To take an example, you might say that 'chair' has as its essence 'three-or-four-legged furniture piece' or something like that. In this case then 'furniture piece' is a component of the essence of the chair. 'Furniture piece' falls under the genus 'substance'; so then premise 3 would in this case imply that 'substance' is part of the essence of the chair. Simple enough.

Now, premise 1 I take to be part of hylomorphic dualism. Premise 2 is the traditional Aristotelian picture (see e.g. Aristotle's De Anima Bk. II 412a2-b6 plus St. Thomas's commentary, as well as Oderberg's 'Real Essentialism' pg. 99). Besides, it would be very weird for an Aristotelian to deny it, since then you will have some some animals falling under 'material substance', others falling under 'immaterial substance', and hence the term 'animal' will be at best an analogical term between us and lower animals. Which seems clearly to not be what is going on anywhere in Aristotle. The rest of the premises I take to either be self-evident or to clearly be true in the Aristotelian framework. But I was quite surprised to find some people denying premise 5, so I will defend it.

Someone might object to premise 5 because they think there are counter-examples. For instance, you might say that the feature of being bipedal is part of the essence of humanity, and yet clearly people who are missing their legs can exist and yet be human. However, I do not think it is part of the essence of humans to be bipedal; for one, if anything, it "flows" from the essence but is not part of it. Essence is to be identified properly speaking with the definition by genus and specific difference (or at least the non-linguistic correlate of this definition), and the genuses above are said to be "virtually contained" in the essence (this is the term Oderberg uses anyway; I prefer to just say they're part of the essence). And being bipedal doesn't show up anywhere in the essence (whereas given the definition "rational animal", "material substance" does show up in the essence). So being bipedal is not part of the essence of humans.

But what about a feature which is presumably essential, like that of being rational. Rationality is part of our essence, yet clearly there are many people who are both irrational and in fact cannot even immediately exercise their rationality. The problem with this proposed counter-example is that it is a misunderstanding of what is meant by "rationality." By rationality is meant having a tendency to exercise our intellect. So, if we analyze the term "rationality", what is essential to being human is having a tendency to exercise our rational capacities, not actually exercising our rational capacities (where tendency is taken in a teleological sense rather than statistical sense). I think Aristotle even says some things like this in the De Anima or somewhere, though to be fair I don't have my copy at the moment so I can't confirm this.

These are some objections against premise 5; but what can be said in its favor?  To answer this we need to get clear on what is meant by "essence." On the traditional understanding, or at least any understanding relevant to this discussion, I would contend that premise 5 is definitely true.

When I talk about essence, by "the essence of x" I mean what it is to be x. This is the understanding of Aristotle (the Greek phrase translated as 'essence' in Aristotle literally means 'what it was to be'; see e.g. here where Greek can be loaded on the right). On this definition, it seems to me to clearly follow that for x to exist it must have its essential properties. For in order to be x you must have the essence of x; but in order for x to exist it must be x; so in order for x to exist it must have the essence of x.

Now, maybe one will reply that you can have the essence of x, yet only have the essence of x partially, and thus premise 5 will not be true since you can be missing part of the essence. The problem with this reply is that if you only have the essence of x partially, and the essence of x is what it is to be x, then you can only partially be x, i.e. you can only be x in some non-literal, lesser sense. So if something only possesses the essence of x partially, it is not x (but only partially x). For x to exist (rather than partial-x, say) it must fully be itself (there's no "half-identity" or something), and so to be x one must have the essence of x fully, and thus have all the parts of x's essence. So on the traditional reading of essence coming from Aristotle, 5 is still true.

Maybe though one could give a modified definition of essence, and say something like the essence of x is that which grounds what it is to be x. However, even on this somewhat vague definition, I think it follows that if it is part of your essence to be P then you must be P if you exist. For if the essence of x grounds what it is to be x, then what it is to be x depends on the essence of x (since if A grounds B then B depends on A). Hence, if the essence of x includes being P, then x's being what it is to be x depends on x's being P. But if x's being what it is depends on x's being P, then x's being what it is entails x's being P; i.e. x must be P if x exists. So If the essence of x grounds what it is to be x, then it follows x must be P if x exists. That's a long and somewhat wordy argument, but hopefully it is clear enough. I am pretty confident it could be made more precise and detailed.

All of this aside I think premise 5 is self-evidently true and doesn't really need to be defended, especially given an Aristotelian framework. But even countenancing all of these objections it still seems 5 must be true.

For what it's worth, I don't think either Aristotle or Aquinas would deny any of what I've said here; for instance, I am pretty sure I have read Aquinas agreeing that it is no longer the person who exists after death but only their soul (in one of his commentaries on the Scriptures; maybe someone can find the reference). I only bring this up not as a matter of appeal to authority, but to give evidence that in the Aristotelian framework it is completely natural to agree with my premise at the beginning which says the animal ceases to exist at death. And thus, if one holds to basic Aristotelian metaphysics, one should agree with my conclusion, that we are not animals.

Wednesday, July 3, 2013

Essence and Ontological Dependence

This is my term paper from my independent study last quarter on ontological dependence. I will say beforehand that I did not have enough time to make it great, and there is a lot more I could have said. However, I believe it contains a relatively good summary of Kit Fine's position, and I think the stuff toward the end about causation is somewhat original (albeit sketchy). So hopefully someone will find it interesting and useful.

I.  Introduction – Examples and What We Want

In many areas of philosophy, as well as common discourse, it is normal to say that one thing depends on another. Moreoever, one of these uses of the word 'depends' is a distinctly ontological sense, as opposed to, say, a notion of epistemological dependence or logical dependence. I will use the term 'dependence' throughout this essay to stand for this particularly ontological notion, unless otherwise stated. So for instance, we might say that a composite depends on its constituents. Or we might say a smile depends on the mouth of which it is a smile. Or that a hole depends on the thing which it is a hole in. This is a philosophical datum and the only reason one would deny it or feign incomprehension seems to be hard-headedness.

Nevertheless, while most people are prepared to affirm some dependence claims, it is not entirely clear what this notion of dependence means, nor is it clear what sort of criterion we can give to identify when one thing depends on another. Certainly there are features of relations which we can use to rule out their being relations of dependence. For instance, plausibly, x depends on y only if y does not depend on x. So we know that the relation is asymmetric. But this still does not help us to say what dependence is, nor does it give us any necessary or sufficient conditions for a relation of dependence holding.

I will try to explain some approaches to this idea of dependence. Some of them will be more fruitful than others, and I'll try to point out some of the issues which accompany each of them. We will see how they handle cases like the ones listed above, and whether they satisfy the necessary conditions for being a good theory of ontological dependence. We will see that the best explanation of dependence is one which ties it to the idea of essence. Finally, we will examine whether this notion relates in any special way to other notions like causation.

II. The Essential Background

It is plausible to think that there is some connection between the idea of ontological dependence and that of essence. The essence of something can be thought of as the nature of a thing, or simply what it is to be that type of thing (this latter phrase is in fact the literal translation of Aristotle's term for 'essence'). Properties then can be said to be part of the essence of something, and these we will call the essential properties of a thing. Any good reductive definition of essence then is going to be one which tries to capture this notion. One can of course define the word 'essence' however one wants, but the hope is that the definition will capture most of the intuitive meaning.

One way of explaining essence is to say the essence of something is the set of all that thing's essential properties, and then reduce the idea of an essential property to a modal notion. Let us offer the following modal definition of essential properties:

Property P is an essential property of x just in case necessarily if x exists then x is P.

The term 'necessarily' is used here in the most broadly logical or metaphysical sense, rather than, say, a merely physical or epistemic sense. This definition seems to capture some of our intuitions about essence on the face of it. For instance, it seems somewhat plausible that to say I'm essentially human means it's not possible for me to exist and yet not be human.

What then follows about the idea of ontological dependence? Well, the relation of dependence is one between entities. And when we say that something depends on something else, we take this to mean the one couldn't exist without the other. So the most natural way to define ontological dependence in terms of our modal conception of essence is as the following modal definition of dependence:

X ontologically depends on Y just in case 'existing only if Y exists' is an essential property of X.

It may seem a bit unfair and a misnomer to simply call this the modal definition of dependence, since no modal terms are explicitly mentioned in the definition at all. Nevertheless, it is easy to remember it by this name and it helps us to remember that essence is going to be analyzed in modal terms.

Now, by our definition of an essential property, it will follow by the rules of logic that X depends on Y just in case necessarily X exists only if Y exists. As we see then, if this account works it will be a successful definition of ontological dependence entirely in terms of the notions of existence and metaphysical modality. Unfortunately, the problem is that this definition does not work. For, in short, it does not capture the asymmetry which holds between a dependent and the thing upon which it depends. Let's take a look at some examples. We will bring these examples back later when considering the plausibility of other accounts, so it is good to keep them in mind.

Consider the case of Socrates and his singleton set, i.e the set containing only him. Sets are abstract objects, and thus they exist necessarily. But if the consequent of an implication is necessarily true, then the implication is necessarily true. So necessarily, Socrates exists only if his singleton set exists. From the modal definition of dependence it follows that Socrates depends on his singleton. But this seems wrong; it is the set which depends on him, if anything. So we see one problem with the modal definition of dependence is that it often gets things backward and it does not preserve the asymmetry of dependence.

Suppose we also have two necessarily existing objects. They can both be abstract objects even, such as the property of being red and the number 2. Necessarily, if 2 exists, then the property of being red exists. So it would follow on the modal definition that the number 2 depends on the property of being red. It appears then the modal definition of dependence also creates dependence where it does not even exist.

Consider also the case of me; I exist only if I exist. In fact, this is a logical truth, so necessarily I exist only if I exist. It follows then that I depend on myself. This could be taken as an objection in itself if one thinks that to avoid circularity ontological dependence must be antisymmetric, i.e if X depends on Y then Y does not depend on X. But at the very least it seems it should not follow as a logical truth that I depend on myself.

Other counter-examples can be adduced, but these help to bring out some of the core problems with the modal account of dependence. This leaves us with two options: Either (i) we can reject the definition of dependence in terms of essence altogether, or else (ii) we can identify a problem with the modal definition of essence and modify it accordingly.

There are some reasons against the first proposal and some reasons in favor of the second proposal. Against the first proposal, it seems that in some important sense when, for instance, we say my smile depends on my mouth, we are saying that my smile just couldn't be the way it is were it not for my mouth i.e it could not have its essence were it not for my mouth. In favor of the second proposal Kit Fine, who originally brought up many of these counter-examples, considers the modal definition of essence itself to be problematic, and moreover thinks he has an alternative conception of essence which suffers from none of these problems and is also able to accurately account for dependence. If one can reduce dependence to essence, it is better to do that than to introduce a new primitive notion of dependence. So instead of abandoning the tie between dependence and essence we should examine whether this Finean picture of essence is helpful.

Before going into Fine's own theory of essence we should lay out his problem with the modal conception of essence. His counter-examples against the modal definition of an essential property are pretty much the same ones brought against the modal definition of ontological dependence. To take the example of singleton Socrates again, it seems that it doesn't have anything to do with what Socrates is that he be a member of singleton Socrates and thus it does not seem to be part of his essence, even though necessarily if he exists he is a member of this set. So the modal conception cannot be an accurate representation of what we normally mean by claims about essence. Fine proposes then an understanding of essence which is not in terms of modality. It is to Fine's own picture that we now turn.

III. A More Fine-Grained Account

Before I begin I feel I should make a few remarks about Fine exegesis. In different papers on this topic Fine says slightly different things, and so the reader should keep in mind that when presenting what I call the Finean account I will be presenting what seems to me to be the most coherent synthesis of all these different strands, generally favoring what he says in later papers as opposed to earlier ones.

Now, to get clear, like any theory of essence, Fine's is trying to capture what is meant when we say the essence of something is just what it is to be that thing. Fine starts with the notion of the operator "true in virtue of the identity of the F's." This is to be treated as a sentential operator, represented by □F which is always indexed relative to some predicate. We will call this 'the essentialist operator'. So, for instance, □FQ should literally be translated as saying it is true in virtue of the F's that Q. Now, whether or not this idea can be reduced, Fine for his purposes takes this operator as a primitive notion, which is not to be defined in simpler terms. Still, it can be explained a little bit more to give us an intuitive notion. Roughly, □FQ means that Q is true in virtue of the things which are F being what they are by their very nature. □FQ may also be able to be explained as Q being true in virtue of the state of affairs of the F's each being self-identical. We might also be able to say □FQ means there is no other fact than the F's being themselves which explains Q. These seem to be the best interpretations I can give to this operator.

Given that we have some notion of this "true in virtue of the identity of" operator, the rest follows. Similarly to the modal notion, we give the following Finean definition of an essential property:

Property F is an essential property of x just in case if x exists then it is true in virtue of the identity of x that x is F.

This requires a bit of explanation. I said earlier that the essentialist operator is always relative to a predicate, whereas here in natural language it is relative to a variable. This can easily be dealt with if we use as our predicate one which denotes the property of being identical to x, and thus only denotes x. Call this property I. Then to say it is true in virtue of the identity of x that P is the case, we can say □IP.

Given that this Finean definition makes sense, let us define the essence of x as the set of all propositions true in virtue of the identity of x (where □I is the operator 'true in virtue of the identity of x'):

The essence of x =df the set of all propositions P such that □IP

We will define dependence in terms of essence. Before moving onto dependence though, which is pretty straightforward given a correct characterization of essence, we must point out the distinction Fine makes between constitutive and consequential essence, and a certain restriction of essence necessary for a proper account of dependence. Fine understands constitutive essence as follows:

Proposition P belongs to the constitutive essence of x just in case P is not logically entailed by some more basic proposition in the essence of x.

So the propositions in the constitutive essence of something are the most basic propositions of that thing's essence. Consequential essence on the other hand is defined oppositely:

Proposition P belongs to the consequential essence of x just in case P is entailed by propositions in the constitutive essence of x.

Consequential essence then is the set of all the essential truths about a thing which can be derived logically from the thing's constitutive essence.

The problem then is that we have to restrict the consequential essence of an object appropriately if we are to not get the sorts of counter-examples the modal account suffered from. For the logical truths can be derived from any proposition, and for any object x it is a logical truth that x = x, thus entailing any object at all will be in the consequential essence. To the end of avoiding this we define the notion of a generalization of a proposition for proposition P(y) with y a constituent of that proposition (i.e a term in the formula expressing the proposition):

The generalization of a proposition P(y) =df the proposition that for all v, P(v)

This lays the ground for the definition of generalizing out:

Object y can be generalized out of a collection of propositions C just in case for all P(y), if C contains P(y) then C contains the generalization of P(y).

Now we can finally give our sought-after Finean definition of ontological dependence:

x ontologically depends on y just in case y is a constituent in a proposition in the essence of x and y cannot be generalized out of the consequential essence of x.

This definition which includes the clause about generalizing out prevents things being dependent on any object whatsoever. As far as the examples we brought against the modal account go, since this definition does not make any explicit reference to metaphysical necessity, there is no guarantee that because something exists necessarily that it will end up being a constituent in a proposition in the essence of an object. So the singleton Socrates example can be avoided for instance, along with any other counter-example which is derived strictly from the properties of implication and modality.

But how does this work with our every day examples? Take the case of the smile depending on the mouth of which it is a smile. It is plausible to believe that the smile can only be itself in virtue of the mouth's being there; so it is true in virtue of the smile's identity that the mouth exists if the smile does, and thus the mouth is part of a proposition in the smile's essence. So it follows straightforwardly that the smile depends on the mouth. Just as we wanted. Fine's picture then appears to be able to bear much fruit.

IV. Case One: Duns Scotus and Causation

It is nice evidence in its favor that Fine's account of dependence is able to handle the cases we want as we want it to. Still, dependence is not only supposed to  play a role in every day needs, but is also supposed to do a very large amount of work in abstract metaphysical thinking. Given its central role, we should examine whether it can provide any illumination or bears any relation to a more meaty notion, like causation.

One of the best people to turn to for this is the late-13th century thinker, Duns Scotus. Fine's account is best described as being a neo-Aristotelian account of essence. In this respect then Fine's essentialism is probably one of the most congenial for interaction with medieval thought and Scotus in particular of any contemporary account. After all, most medieval philosophers took a broadly Aristotelian metaphysics for granted, including an Aristotelian notion of essence.

Now if essence is going to be helpful in understanding the notion of causation, as the name suggests it will most likely be helpful when we come to Scotus's idea of an essentially ordered causal series. In Scotus's thought, all causal series can be divided into two broad categories, accidental and essential. Essentially ordered causal series can be separated by the following three criteria:

(1) In essentially ordered causes, the posterior depend on the prior to exert their causal influence.
(2) In essentially ordered causes, the prior are more perfect (important) than the posterior.
(3) In essentially ordered causes, all the causes are simultaneously required to produce the effect.

Accidentally ordered causal series are those series which do not satisfy these criteria. Scotus takes it that in any essentially ordered causal series, the relation of causation will be transitive, so that if x causes y and y causes z then x causes z.

These criteria require some expounding. As regards claim (1), it is pretty clear what is meant. It is saying the later causes in the series could not exert their causal influence were it not for the prior causes in the series, since they depend for their ability to exert their causal influence on the prior causes. As far as (2) goes, the best I can make of this is that the higher a thing is in a causal series, i.e the more things it is causally prior to, the more important is its causal influence for producing the lower causal effects. While probably more can be said of this idea of "importance", clearly we have some understanding of this. For instance, while I might have gotten a little bit of help from the TA, the instructor was much more important in causing me to pass my math exam. This makes sense. With respect to point (3), this is simply saying that if at a time t an effect in an essentially ordered causal series exists, then all the causes in the series also exist at t and exert their causal influence at t to produce the effect.

With that said, the distinction is mostly clear. So, now that we have some understanding we can give a couple examples to illustrate the distinction.  One famous (if imperfect) example of an essentially ordered causal series is a man moving a ball by means of a stick with his arm. The effect here is the movement of the ball. Arguably, in this causal series it is in virtue, and indeed primarily in virtue, of the arm's motion that the stick moves the ball at all; so criterion (1) and (2) are satisfied. And the arm must be exerting its causal influence at the same time that the ball pushes the stick in order for the ball to be moving; so criterion (3) is satisfied. Hence, this counts as an essentially ordered causal series.

An example of an accidentally ordered series is a series of fathers causing the existence of their sons. While it is sufficient for being an accidentally ordered series that the series fail to satisfy any of (1) through (3), this one appears to fail all of them. (2) seems to be clearly not satisfied in this case; if anything, it is the posterior rather than the prior in the series which are more important; I depend more on my dad's causal influence than my grandfather's in order to begin existing. And clearly my dad can beget me even if my grandfather has already ceased to exist, contra (1) and (3). So this is a perfect example of an accidentally ordered causal series.

Before getting to the point there is one more notion to get out of the way, that of an 'accidental unity' as it is called by the medievals. While some people may get scared by the 'spooky' scholastic terminology, the notion is relatively simple. An accidental unity is just the composite of an object and some feature (accident) which that object has. Now, as far as what it counts as a unity or composite in virtue of, it is in virtue of the relation of the feature's inhering in the object. An example would be the object white-Socrates. This is not just reducible to Socrates, for Socrates can fail to be white, but clearly white-Socrates cannot fail to be white. For it is a composite of Socrates and his whiteness.

Now, with all that said, how can a Finean definition of essence contribute to our understanding of causation here? While I'm not so sure about (2), I would suggest that it is definitely applicable in further analyzing criteria (1) and (3) of an essentially ordered causal series. I will examine the point about (1) first, since I think it actually helps us to understand why (3) is true as well.

It seems likely on the face of it that (1) can be further analyzed with our notion of ontological dependence, especially considering that the very word 'dependence' appears in it. But let us go back to the example of the arm, stick, and ball. In this series, all the causes in the series are acting so as to produce the motion of the ball. So the effect being produced is the motion of the ball. [It is interesting to note that this effect is an accidental unity: the composite of the ball plus the feature of its being in motion (though this need not always be the case; substances taken alone can often be effects as well).]

Now, let us ask what is actually causing the motion of the ball, in the most strict sense of being a sufficient rather than merely partial cause of the ball's motion: Is it properly speaking the stick and the arm, or is it the causing-stick and causing-arm, i.e the accidental unity of the stick plus its feature F of causing this ball's motion in this causal series (and similarly for the causing-arm)? It seems to be the latter, for if you do not have the stick plus this feature F, then you will not have any motion at all. [I should note: It is consistent with what I've said that F can possibly be reduced further to a conglomeration of even more basic features.]

But then, given the causes are properly speaking the causing-stick and the causing-arm it becomes quite easy to analyze (1) further: For we can say the causing-stick depends on the causing-arm in precisely the sense of our Finean definition earlier, viz the causing-arm appears as a constituent in an essential truth about the causing-stick. For let us take our feature F in full: It is the feature of the stick's causing this ball's motion in this causal series. Now the stick only has this feature because of the force imparted to it by the arm. So the accidental unity of the causing-stick has its very identity in virtue of the action of the causing-arm. But this statement is true in virtue of the identity of the causing-stick; so the causing-arm is a constituent in an essential proposition of the causing-stick, and so the causing-stick depends on the causing-arm (and thus, derivatively, on the arm).

If that is all clear, then (3) makes sense as well in a pretty straightforward way. For on the Finean account of essence, while not all necessary truths about a thing are essential truths, it is plausible to think that all essential truths are necessary truths about the thing. That is to say, if it is true in virtue of the identity of x that P, then necessarily if x exists then P is true. Now, going back to (3) obviously the effect depends on its most immediate cause in order to be produced; but then if what I've said about (1) is right, the most immediate cause depends ontologically—in the most strict Finean sense--on all the prior causes. But given that the immediate cause depends on the prior causes, it is true in virtue of the identity of the immediate cause that it exists only if the prior causes exist. So, given what I've said about essential truths entailing necessary truths, it will follow that necessarily if the immediate cause exists then so do the prior causes. But if x's existence necessarily entails y's existence in this sense, then at all times at which x exists, y exists. So if the immediate cause of the effect exists, then all the causes of the effect exist at the same time. So, given that we are dealing with an essentially ordered series and have (1), criterion (3) will follow.

As a side point, I think it is interesting to note that (1) appears to follow from (3); this may be part of the reason why in the cases of accidental series we can often refute (1) by just refuting (3). If (1) entails (3), which itself entails the existence of the prior causes, then having the prior causes not exist will show not only (3) fails but that (1) fails as well; thus, for instance, it is enough to point out that my grandfather doesn't exist to show that in the causal series of fathers the posterior do not depend on the prior for their causal efficacy.

All said and done then, Fine's notion appears helpful here. Of course, the claims I've made may need to be modified somewhat on further consideration, but I think they are on the right track. Very roughly, the point I'm trying to get across is that the causes in essentially ordered series are accidental unities, that these unities depend in the full Finean sense on the prior ones, and so (1) and (3) will follow. It seems very plausible to me that much more can be said about this than I have said here, and I have a strong inchoate idea about how far this could go in other directions and how much more precise it could be made. Yet even if one does not find this plausible as an explanation of what is actually going on in these cases of causation, it still might be a way to explain what (1) and (3) actually mean in the hands of medieval philosophers.

So, at the very least, it does seem that the Finean notion of dependence can potentially play a role in helping us to further analyze and understand important metaphysical notions like causation. In virtue of this, I would suggest that a Finean essentialist definition of dependence is very important and requires further investigation. Other important areas that seem fruitful for investigation, just to list a couple, might be the area of grounding, and whether these notions of ground are the same as ontological dependence or not, as well as the idea of reduction and whether reduction claims are a species of dependence claims.

Sources Consulted
-Aristotle, and W. D. Ross. Aristotle's Metaphysics. Oxford: Clarendon Press, 1924.
-Fine, Kit. "Essence and Modality." Philosophical Perspectives 8 (Logic and Language) (1994): 1-16.
-Fine, Kit. "Ontological Dependence." Proceedings of the Aristotelian Society 95 (1995):  269-290.
-Fine, Kit. "The Logic Of Essence." Journal of Philosophical Logic 24.3 (1995): 241-273.
-Fine, Kit. "Senses of Essence." In Sinnott-Armstrong, W., Raffman, D. and Asher, N. (eds.): Modality, Morality, and Belief: Essays in Honor of Ruth Barcan Marcus. New York: Cambridge University Press (1995): 53-73.
-Gorman, Michael. "Ontological Priority and John Duns Scotus." The Philosophical Quarterly 43.173 (1993): 460-471.
-Koslicki, Kathrin. "Varieties of Ontological Dependence." In Correia, F. and Schnieder, B (eds.) (2012): 186-213.
-Weingart, R.G. "The Logic of Essentially Ordered Causes." Notre Dame Journal of Formal Logic 12.4 (1971): 406-422.
-Zalta, E. N. "Essence And Modality." Mind 115.459 (2006): 659-694.

Monday, June 24, 2013

Narrative History, Substance Dualism, and Scholasticism

In my last post, I explained why I find the evidence in favor of substance dualism convincing over hylomorphic dualism. Since what I'm about to say is somewhat disconnected from the last post, I felt I should save it for later. Nevertheless, I thought I should say something about the historical narrative often painted around substance dualism when brought up in connection with hylomorphic dualism. Before beginning though, I want to say that I consider hylomorphic dualists more of an ally against materialism than enemies; that I truly sincerely seek the truth here and also seek to be faithful to my Catholic Christian beliefs; and finally that I do not intend for any of my comments to be rude or disrespectful to the people I am disagreeing with, as I take their position seriously and try my best to understand it on its own terms.

Now, basically, you sometimes hear it claimed that substance dualism led to materialism, that substance dualism brought about a metaphysical revolution at the expense of scholasticism, that substance dualism leaves behind the consensus of the greatest philosophers, that substance dualism helped bring about the downfall of western civilization, etc. Usually the trouble is supposed to have started with Descartes' formulation of substance dualism. Now, many philosophers would probably just dismiss these questions as irrelevant, but I honestly take these points a bit more seriously than most, at least to the extent that I take quite seriously the arguments and thoughts of many of the people who were not substance dualists--like Aristotle and Aquinas.

First, how I think we should handle these sorts of claims in general:

-While I'll say some things in defense of Descartes, note that I defined substance dualism quite minimally earlier, as the theory that the soul is an immaterial substance distinct from the body and we are identical to the soul. Hence, it is extremely important to note that not everything which can be ascribed to Cartesian metaphysics can thereby be ascribed to substance dualism.
-We have to be careful to avoid slippery slope fallacies, of the form "This belief had or has such and such undesirable effects, therefore it's false."
-When writing our narratives, we should be careful about painting too broad a picture or making very tenuous connections between ideas or figures and historical facts.
-The best narrative will be good history, and when it comes to the history of philosophy that is best accomplished by looking at the ideas the philosophers actually held.
-When seeking the truth we should focus primarily on arguments, not persons or narratives.

Second, what I think is correct in these sorts of claims:

-Many people who were greatly influenced by Descartes abandoned scholastic metaphysics for no good reason whatsoever.
-After Descartes scholastic philosophy did not have a very large role in early modern philosophy.
-Cartesian metaphysics is different in important ways from many of the preceding metaphysical systems.
-Belief in substance dualism does in fact have certain ethical implications, though I don't think they are as radical as some people make them out to be.

Third, where I think these sorts of claims are wrong:

-Substance dualism in no way entails materialism. It is a theory of how the mind is not material. Now if you say that the idea of materialism could not have developed were it not for substance dualism, I reply that first, Leucippus and Democritus existed before Descartes, and second, materialism could not have developed were it not for many things, including the belief in matter, yet this does not impugn in any way our belief in matter.
-Substance dualism is not based on such radically different metaphysical grounds that things which are intelligible to hylomorphic dualists are simply inaccessible to substance dualists. I, as a substance dualist, admit all the famous scholastic distinctions and at least think I understand most of them. That includes the four causes, the distinction between essence and existence, matter and form, substance and accident, act and potency, nature and suppositum, etc.
-Descartes was not such a novel break from the past. There have been many great philosophers throughout history prior to Descartes who have held substance dualist views. Think, for instance, of Plato, Plotinus, and St. Augustine, as well as Avicenna with his Floating Man argument.
-Descartes' substance dualism was not such a radical break from scholasticism. Though Aquinas almost certainly was the exemplar of hylomorphic dualism as I've defined it, scholasticism does not equal Aquinas (even if he was certainly one of the greatest scholastics). In Ockham for instance we see the body as being able to exist apart from the soul, at least if God wills it. In Buridan the soul is viewed more like a force pervading the whole body. Later in Suarez the soul and the body are each conceived as separable substances, albeit "incomplete" ones when taken alone. So in fact the path from them to Descartes is rather continuous, and we can see this insofar as Descartes continues to use scholastic terminology and makes use of the distinctions he received from Suarez. Again, all these people accepted the broadly Aristotelian metaphysical framework of scholasticism, and yet they were not all hylomorphic dualists as I've defined it above. So I think one can remain broadly scholastic without being a hylomorphic dualist (I'll say more about this last point--on the consistency of my generally scholastic views--in a later post).
-As regards the downfall of western civilization, Descartes was a devout and orthodox Catholic, and probably even converted the Queen of Sweden. So it is likely he did not do this terrible thing. More importantly though, as of now I have found nothing in substance dualism as I've defined it which is repugnant to the Catholic faith or heretical or implies something heretical, so this charge is simply unfounded.

Friday, June 21, 2013

Why I Am a Substance Dualist

I. Introduction

In this post I'm going to explain why I'm a substance dualist rather than a hylomorphic dualist. So the title might be an instance of false advertising, insofar as I'm not really going to explain in detail why I'm not a materialist of any sort. I will say that I honestly think there is almost no evidence for reductive materialism/physicalism/naturalism whatsoever, and that there is very strong evidence for at least some minimal form of dualism. (See, just for a couple of examples, this and this.) I won't say much more than this, other than that I find the sort of "classic" dualist objections from qualia, intentionality, consciousness, personal identity, etc. among many many others quite convincing, and I honestly do not feel there has been any good reply. I don't mean to sound dogmatic, it's just that, based on the evidence, at this moment I don't consider materialism to be a serious contender for the truth. If anyone is genuinely curious about this or feels otherwise though, please feel free to ask.

II. Animalist & Substance Dualist Background

Before explaining my view, I should explain the views in question. The definitions should be clear enough, but they can be made more detailed and precise.

Substance dualism I take to be the theory that the soul is a substance distinct from the body, the soul is immaterial, and that we are identical to a soul. Hylomorphic dualism I understand to be the theory that we are identical to rational animals, which are composites of matter and form, and that our soul (which is the form) has immaterial components. Animalism is the view that human persons are identical to human animals. Clearly then hylomorphic dualism is a form of animalism. Since in characteristic scholastic fashion I basically believe all material substances are hylomorphic composites, and I'm a dualist, as far as I'm concerned establishing animalism is both necessary and sufficient for establishing hylomorphic dualism. So for me the dispute really comes down to deciding between substance dualism and animalism.

In modern times, the biggest proponent of animalism is probably Eric Olson. For those uninitiated into Eric Olson's papers, the argument he gives in its favor is basically as follows:

(a) There is an animal where you are.
(b) It is thinking and you are thinking
(c) There is only one thinking thing where you are; so,
(d) You are an animal.

(a) seems impeccable. Just look at your body: It is clearly alive and in the shape of an organism, it descends from an evolutionary history, and moreover all biologists think it belongs to the species 'homo sapiens'. (b) I am a bit more unsure of; maybe you will say that thinking and rationality cannot develop out of matter, so that no organism could have thoughts. However, clearly even lower animals have some sort of conscious perceptual experience, and thus at the very least you and the animal are both having conscious awareness. You could revise (c) to say there is only one conscious thing where you are (where consciousness means having some sort of experience), and the conclusion would still follow.

Since (c) is the only premise which appears not unquestionable, the cost of denying this argument's conclusion is to say there is more than one conscious subject wherever you are. This seems counter-intuitive to say the least. First of all, all sorts of new obligations follow which we generally would deny exist; for instance, whenever you are fasting, you are subjecting the animal which follows you around to fasting and such against its will. More generally whenever you subject yourself to pain, you subject the animal to it as well. Whenever you make love to a person, not only you but another animal are as well. Etc.

I'll be honest, I have found this to be the most compelling argument in favor of some sort of animalism, which says that rather than me being identical to a soul I'm identical to an animal. (But more on this in a moment.)

However, with all that said, I find this argument quite compelling for the thesis that I am not identical to an animal:

(1) Some disembodied conscious subject psychologically continuous with me continues to exist after death.
(2) I am this subject.
(3) The animal associated with me does not continue to exist after death.
(4) So I am not identical to the animal associated with me.

(1) I take to be a datum of Christian theology (at the very least it's Catholic teaching). (2) I take to be extremely hard to deny; suppose I am about to die, and before I die I begin to say in my head "IIIIIIIIIIII think, therefore I exist". It is extremely counter-intuitive to think that if I am holding the "IIIIIIIIIIII" syllable as I die, that before my death it refers to me, then after death it either ceases to refer or refers to something other than me. In effect, denial of (2) seems to be a denial of Descartes's "cogito"; for you could be thinking, but then a moment later cease to exist--while thinking the same thought! If one doesn't like this specific example, one could probably easily come up with something similar enough to get the point across, the main point being that it is very counter-intuitive to believe that I could be thinking, and meanwhile this same conscious subject turns into a person different from me. Since (3) is equally if not more obvious than (2), from all these three premises (4) follows.

Given (4), the best way to make sense of all these facts seems to me to be substance dualism.

III. Substance Dualist Rejoinder

For a while I've sort of been at a fork in the road, not really sure which way to go; to me, both arguments have seemed very good on the face of it. However, on re-examining it, I think that Olson's argument will not appeal to substance dualists, because of his premise (b) that both the animal is thinking and you are thinking. For the animalist, the motivation behind this premise is that it seems animals can think, and thus the animal associated with you should be able to as well. However, I think the substance dualist can plausibly deny that animals think per se, if 'animal' is just taken to be the biological organism. Rather, substance dualists can plausibly say that animals have souls too; since they are also conscious, the same sorts of arguments for the immateriality of our minds would apply to theirs just as well. Hence, it is the animal's soul which thinks, properly speaking. So all Olson can really justifiedly say is that both you and an animal's soul are thinking, from which it follows you are identical to an animal's soul; but the substance dualist will have no problem agreeing that we are identical to some animal's soul.

Note what saying animals have souls does not mean: It does not entail that humans are basically just like lower animals. Human souls and lower animal souls may be very different, insofar as one possesses intellect and the other does not. It also does not entail that animals will survive death, though it may entail the possibility of their doing so. Finally, it does not entail anything really more weird than hylomorphic dualism does: For even on hylomorphic dualism it is common to say that animals (and even plants) have souls, the difference being that humans possess a rational soul including intellectual capacities, whereas lower animals have only the capacity for sensation. The substance dualist seems to me to be saying something very similar.

IV. Conclusion

There is a lot more that could be said about substance dualism: How exactly does it fit into an Aristotelian metaphysical framework? What are the deeper metaphysical, epistemological, and even ethical implications? Does it fit in with Christian theology in other areas, like Christology or the Resurrection? Doesn't it just make us out to be angels? How does the soul interact with the body? What is the nature of the union between the soul and the body? While I think these are all good questions and replies can be given to all of them, I don't want to try to answer everything in this post. Answering all these questions would probably turn into a book-length project. All I want to do is explain why I've chosen substance dualism over hylomorphic dualism: While I am open to other arguments as well as hearing other options, as of now I have found the arguments in favor of substance dualism convincing, and the arguments in favor of animalism, which I take to be the key consideration, wanting.

Wednesday, June 19, 2013

Two Inadequate Arguments for a Finite Past

In this post I will consider two arguments which have at times been brought up in connection with the Kalam Cosmological Argument (KCA), which I will call the "subtraction argument" and the "argument from traversing an infinite," the former of which I have heard from Dr. William Lane Craig. The KCA goes as follows:

1. Whatever begins to exist has a cause.
2. The universe began to exist.
3. So the universe has a cause.

The arguments in question are designed to defend the second premise, which is presumably implied by the past's being finite. I should note that I think the second premise is true and there is strong evidence in favor of its truth. Alexander Pruss has given an excellent argument here, to which I have heard no compelling reply. I also think there is very strong scientific indication of the premise's truth, which Craig has adequately demonstrated. I just don't think these two arguments demonstrate its truth.

The "argument from traversing an infinite" goes something like this:

1*. If the past were infinite, one would have to cross an infinite temporal distance to get to the present moment.
2*. If one had to cross an infinite temporal distance to get to the present moment, then one could not get to the present moment.
3*. So if the past were infinite, then one could not get to the present moment.
4*. But we are at the present moment.
5*. So the past is not infinite.

The argument requires some unpacking. First of all, to say the past is finite is to say there was a beginning of time, and to say the past is infinite is to say there was no beginning of time. Second, 'temporal distance' means the length of time between one moment and another. There is a perfectly good way to define finite temporal distance. If we take our measure of time as a second, we can assign the current time the number 0, the time one second ago -1, the time two seconds ago -2, and so forth. To find the temporal distance from one time t1 to another t2, we take the number assigned to t1 and the number assigned to t2, and take the absolute value of the difference between the two. For instance, take the time 1000 seconds ago. To find the temporal distance from that time to the present time you take the absolute value of -1000 minus 0, which is of course 1000 seconds. Pretty simple.

However, problems begin to arise when we start to talk about an "infinite temporal distance." This phrase is ambiguous, and depending on which interpretation of this phrase we take it will either cause problems for premise 1* or for premise 2*. First, the phrase could mean something analogous to the way finite temporal distance has been defined above. However, infinity is not a real number, so you simply cannot define an infinite temporal distance the same way as above. There is no number "-infinity" from which you can subtract, say, -5. So if this is what is meant, then premise 1* appears to be false, since no real sense can be given to an infinite distance in this way.

On the other hand, crossing an infinite temporal distance could just mean that the set of all the numbers assigned to the seconds is infinitely large. This makes perfectly good sense of the phrase, but then in that case it is not clear why premise 2* is true. As Thomas Aquinas points out, there being an "infinite temporal distance" in this sense is perfectly consistent with all the temporal distances from the past to the present being finite, where "temporal distance" is defined as it was earlier:

"Passage is always understood as being from term to term. Whatever bygone day we choose, from it to the present day there is a finite number of days which can be passed through. The objection, however, is founded on the idea that, given two extremes, there is an infinite number of mean terms." [ST Ia q.46 a.2]

So for instance, the distance from the present to one second ago is 1 second, the distance from the present to two seconds ago is 2 seconds, etc. and so on forever and ever back into time. Hence, no matter how far you go back in time, the distance in the way I've defined above from any given past moment to the present will be finite, and thus you will only have to cross a finite number of seconds to get to the present moment. But of course any finite number of seconds can at least in principle be crossed; hence, premise 2* is false.

So much for the "traversing an infinite" argument then. The "subtraction argument" goes something like this:

1'. If the past were infinite, then an actual infinity would be possible.
2'. If an actual infinity were possible, then one could perform subtraction on infinities.
3'. But if one can perform subtraction on infinities, then one will get contradictory results.
4'. So if the past were infinite, then one would get contradictory results.
5'. So the past is not infinite.

This seems to be one of the arguments William Lane Craig gave in his debate against Peter Millican. Let me first point out an ambiguity in the phrase "actual infinity," after which I'll assess the argument under each interpretation. Here are two possible meanings of the phrase "actual infinity":

(ACT1) An actual infinity exists just in case for some time, at that time there exist distinct concrete objects such that the size of the set containing all and only them is infinite.

(ACT2) An actual infinity exists just in case there is some set containing only distinct concrete objects whose size is infinite.

Some elaboration is in order. First off, both definitions presume when speaking of actual infinities that we are dealing with concrete objects. While my understanding is that Craig does not believe there are infinitely many numbers (he is a nominalist), presumably his argument doesn't presuppose this view; Craig only wants to rule out the possibility of infinitely many concrete objects. Now as for the definitions themselves, the difference between (ACT1) and (ACT2) is that in (ACT1) you only have an actual infinity when all the concrete objects exist at the same time. In (ACT2) you could have finitely many objects at t2, finitely many at t1, and so forth, yet if you take a set containing concrete objects from different times, and the times go back to infinity, you will still have an actual infinity. So both of these definitions make fine sense. However, the assessment of the argument will depend on which interpretation we take.

Let's deal with the first definition, (ACT1). Given our definition of actual infinity in (ACT1), premise 1' does not appear to be true, or at least not obviously true. It is consistent with holding that the past is infinite that at each time there are only finitely many concrete objects. And if you believe only objects in the present moment exist, then 1' is definitely not true. The fact that there were objects at each time in the eternal past in no way implies an infinite collection of simultaneously-existing objects.

My main concern is with 2' and 3' though. Take 2', since it is also ambiguous to a certain extent. The problem is it is not immediately clear what is meant by "perform subtraction on infinities." Craig acknowledges that the ordinary operation of subtraction is not defined for "infinity". If he did mean this, i.e. the ordinary operation of subtraction, then clearly 3' would be true but 2' would have no support at all. So it cannot mean the ordinary operation of subtraction.

Now, Craig gives us an example to support 2' and 3', and this may help us understand what he means. Suppose we have an infinite number of coins. Then we can take away all the coins except three of them. And in this sense we can be said to perform subtraction on infinity, i.e. taking away some number of things from an infinitely large collection. This definition of performing subtraction on infinities makes sense.  But then why is 3' true? Craig says, considering our infinite number of coins, that you can take away infinitely many coins and be left with 3 coins, and thus infinity minus infinity = 3; but you can also take away infinitely many coins and be left with 2 coins, and thus infinity minus infinity = 2; hence, 2 = 3, which is our contradiction.

The problem with this argument is that it runs on an equivocation: We agreed that we are not using "performing subtraction" or "minus" in the normal sense of the arithmetical operation, since this just makes no sense. So "infinity minus infinity = 3" must simply be shorthand for saying "taking away infinitely many objects from an infinite collection leaves us with 3", and similarly with "infinity minus infinity = 2." But then if "2 = 3" means that 2 is identical to 3, then it certainly does not follow that 2 = 3; all that follows is that you can take away an infinite number of things and be left with 3, and also take away an infinite number of things and be left with 2, and this is certainly not a contradiction! It only looks like a contradiction when we are illicitly inferring "2 = 3", as if the phrase "infinity minus infinity = 2" were using "minus" and "=" in the same way as "5 minus 3 = 2." It would be like if I had infinitely many pennies and dimes, and I said, "infinity minus infinity = a penny, infinity minus infinity = a dime, so a penny = a dime." Clearly I am making an illicit inference here, and for the same reason Craig's argument makes an illicit inference as well.

Now, this whole time I have been working under the assumption that throughout the argument "actual infinity" is meant in the sense of (ACT1). But under interpretation (ACT2) the situation is even worse, since it is not clear 2' is true. It seems that in order to "subtract" infinitely many coins in the sense defined above, all of them must exist at the same time. But if "actual infinity" is taken in the sense of (ACT2), then it is not required that all of the infinite number of coins exist at the same time, and thus 2' has no support. And of course, with the exception of premise 1', all the same criticisms I have just given apply equally well under (ACT2). So, interpreted charitably, the argument seems to be a failure, with the primary problem being in premise 3'.

I should note one more time that, in spite of all my criticisms of these two arguments, I think there are good reasons for thinking the KCA is sound. I just don't think these are among them.