Thomistic Moderate Realism Reduces to Armstrongianism or Platonism

I've always had difficulty understanding Aquinas's "moderate realist" view on universals, at least as that view is expounded by his interpreters. It seems that they want to have their cake and eat it too: Thomists both believes that universals are "objective" and "extramental" in some sense. But they also are not extreme realists, i.e., (modern) Platonists. They also seem to say things that make it sound as if universals are merely conceptions; if taken at face value, that reduces to conceptualism or nominalism of some sort.

Here's one way of bringing out the problem.

Assuming there are universals, then:

• 1. Either (a) universals exist extramentally or (b) they do not exist extramentally.
• 2. If (b), then nominalism or conceptualism, QED.
• 3. If (a), then either (i) they only exist in the objects that have them, or (ii) they sometimes exist outside of the objects that have them.
• 4. If (i), then that is Armstrong's view.
• 5. If (ii) then that is Platonism.
• 6. So if (a) is true, then either Armstrong's view or Platonism is true, and so moderate realism either reduces to Armstrong's view or Platonism.

6 already seems to disambiguate the Thomistic view in a way that makes it unacceptable, and does not let it have all of the desirable qualities it is supposed to have.

• 7. If Armstrong's view holds, then universals depend on the objects that have them, and therefore cease to exist if the objects do.
• 8. But if the universals cease to exist, then statements about non-existent things, like "dinosaurs are big creatures", do not require universals to be true, and so universals are superfluous.
• 9. So if Armstrong's view holds, then universals are superfluous.

That only leaves Platonism, and there are huge problems with Platonism.

To be fair, I am leaving much unsaid here, and I am not making any distinctions within "Platonism." But ultimately I think this is basically correct; the way Platonism is construed in modern times basically just is the view that there are non-mental, objective, necessarily existent, universals.

Naturalness vs. "Arbitrariness" and "Simplicity" in Mereology

There are three answers one can give to the question: "When does composition occur?"

(1) Always.
(2) Sometimes.
(3) Never.

The first view is something like David Lewis's view: Any plurality of objects composes a third. Hence, for example, there is an object consisting of Barack Obama, my left leg, an orange, and half of the beach in Santa Monica. (We could call it the BLOB.) This is the view encompassed in classical mereology.

The third view is something like Peter Van Inwagen's view in 'Material Beings'. This view holds that (with maybe a few very specific exceptions), there are not, literally, any composite objects. There are just "simples" -- atoms in the void, physically proximate to each other and arranged in various ways.

The second view encompasses all other possibilities. One of these possibilities is "common sense" ontology, or something like it. One such view might hold that things like physical organisms, tables and chairs, rocks, planets, stars, maybe even galaxies, etc. are composite objects. But not just any plurality of things constitutes an object on this view; for instance, there is definitely no object such as the BLOB.

One argument (I think due to Van Inwagen) says that (2) can be ruled out rather easily because it is arbitrary and/or overly complicated. Hence, we must choose between (1) and (3).

However, it seems to me that people who hold to (2) might argue that their ontology only encompasses what is natural; just as there is a distinction between natural properties (like 'having mass') and gerrymandered properties (like 'being Barack Obama-or-my leg-or-an orange-or-half of Santa Monica Beach'), there may well be a distinction between natural composites and gerrymandered composites. And just as one might choose to privilege the natural properties by saying they are the only ones that exist (as D.M. Armstrong does), so one might choose to privilege the natural composites by saying they are the only ones that exist.

Obviously more needs to be said than this and this view would need to be fleshed out. But I'm more interested in the methodological question, and all I need granted is that it is a distinction one could coherently use so as to avoid (1) or (3).

Now, people like Van Inwagen might (probably, would) respond to this view by claiming that it is arbitrary, that it multiplies distinctions, that the notion of "naturalness" is mysterious and vague, and so on.

But I think it is worth noting here an "arbitrariness" in this objection: Claiming that some entity is more natural than another (or, by extension, that one's theory is more natural) is no more mysterious than claims that that (2) is arbitrary and complex, and that (1) and (3) are non-arbitrary and more simple. Defining the sense in which (1) and (3) are "non-arbitrary" and "simpler" is no easier than defining the sense in which (2) is "more natural."

Frankly, simplicity and arbitrariness, as used in this way, seem to me to be just as bad off as the other notions that anti-hyper-intensionalists use as criteria of theory choice; they are themselves hyperintensional notions in fact. That's not to say that they are bad off -- I do think there is an intuitive sense in which theories can be "simpler" and "less arbitrary" than other theories. But it is arbitrary to use "arbitrariness" and "simplicity" as criteria for selecting between metaphysical theories, and then pretend you don't know what it means when one says that his theory is more "natural" than others or, relatedly, posits entities that are "more natural."

Russell on Existence in TPLA III: The Argument from Transferability

In the last post we explained what Russell's views on existence are and how they entail a "higher-order" theory of existence according to which existence is not a feature of individuals but of some "higher-order" things, viz. propositional functions. Since this is, initially, a very counter-intuitive proposal ("Socrates exists" is meaningless on Russell's view!), Russell ought to have some arguments to defend his view. This is going to be a long series of posts, so we'll discuss several of Russell's arguments from The Philosophy of Logical Atomism, but in the next two posts we'll discuss one of his arguments in particular: What I will call the "Transferability Argument." But before that I'll briefly mention Russell's motivation for having a theory of existence like his in the first place.

In the first place, the whole notion of existence comes up in connection with what we might call “negative existential” statements. A negative existential statement is a statement saying that something does not exist: For instance, that Socrates does not exist, or that dogs do not exist. These present an initial puzzle. On the one hand, if they are true, then it seems “Socrates” and “dogs” do not refer to anything, and so it’s not clear what could make the sentences true. On the other hand, they seem to be saying that something, “Socrates” or “dogs,” has the feature of “not existing.”

Now, this doesn’t immediately support Russell’s view on existence, but it does give one impetus to develop some sort of view that would address the question of negative existentials. It is interesting to see how Russell’s view deals with the problem. In the first place, ‘Socrates does not exist’ is simply meaningless according to Russell, since it doesn’t make sense to attribute existence to an individual, and so neither does it make sense to deny existence of an individual. On the other hand, since existence is a property of propositional functions, “dogs do not exist,” is easy to deal with: it is the same as saying ‘x is a dog’ is impossible. This involves no shady references to non-existent dogs or anything of that sort. One need only say that ‘x is a dog’ is never true.

With that said, it is not enough to point out that Russell’s view gives an answer to this question. Russell’s view is still prima facie implausible, and there might also be other positions available. Hence, Russell needs to give some direct arguments specifically for his view and arguments against alternatives. We will discuss just one of the arguments that Russell gives, which I call “the Transferability Argument.” The argument is quite subtle in fact, and it is rather complicated. But I think it is worth thinking through because it incorporates several interesting assumptions from logic and the philosophy of language.

Before delving into it, I want to define what we will call a ‘transferable predicate’. Russell does not use this terminology himself, but he uses the concept, and his argument is easier to state with this terminology. Now, a predicate F is transferable in my sense just in case (i) F can  be meaningfully applied to some kind G, and (ii) for any kind G that F applies to, ‘G’s are F’ is true only if every individual x that is a G is also F. For instance, the predicate ‘green’ is transferable: It applies to a generic kind term like ‘men’, since we can say ‘men are green’, and ‘men are green’ is true only if each man is himself green. The predicate ‘green’ “transfers” to the individual men. The predicate ‘numerous’ on the other hand is non-transferable: While we can say ‘men are numerous’, it does not imply any particular man is himself numerous. Indeed, this last statement is meaningless.

With that said, Russell’s Argument from Transferability can be reconstructed as follows:

• (1’) ‘Unicorns exist’ is false, but meaningful. [Premise]
• (2’) If there is an individual sense of ‘exists’, then ‘exists’ is transferable. [Premise]
• (3’) If ‘exists’ is transferable, then ‘Unicorns exist’ implies ‘a exists’, for some proper name ‘a’ of some particular unicorn. [Premise]
• (4’) So, if there is an individual sense of ‘exists’, then ‘Unicorns exist’ implies ‘a exists’, for some proper name ‘a’ of some particular unicorn. [By 2’ and 3’]
• (5’) If ‘a’ is a proper name then ‘a is F’ is meaningful only if ‘a’ refers. [Premise]
• (6’) So ‘a exists’ is meaningful only if ‘a’ refers. [5’, Universal Instantiation]
• (7’) Suppose there is an individual sense of ‘exists’. [Supposition for Reductio]
• (8’) Then ‘Unicorns exist’ implies ‘a exists’ for some proper name ‘a’ of some particular unicorn. [By 4’ and 7’]
• (9’) If ‘unicorns exist’ is false, then ‘a’ does not refer. [Premise]
• (10’) So ‘a’ does not refer. [By 1’ and 9’]
• (11’) So ‘a exists’ is meaningless. [By 10’ and 6’]
• (12’) No meaningful statement can imply a meaningless statement. [Premise]
• (13’) So, ‘unicorns exist’ is meaningless. [By 8’, 11’, and 12’]

But this contradicts our assumption in (1’). Hence, we must reject our assumption in 7’:

• (14’) There is no individual sense of ‘exists’. [By 7’ – 13’ and Reductio ad Absurdum]

This is an extremely interesting and rich argument. It is the best reconstruction I can give of Russell’s argument. The argument is clearly valid. It has a total of six premises: 1’, 2’, 3’ 5’, 9’, and 12’. I think it is useful to isolate the premises so that we can see precisely the principles at work here:

• (1’) ‘Unicorns exist’ is false, but meaningful.
• (2’) If there is an individual sense of ‘exists’, then ‘exists’ is transferable.
• (3’) If ‘exists’ is transferable, then ‘Unicorns exist’ implies ‘a exists’, for some proper name ‘a’ of some particular unicorn.
• (5’) If ‘a’ is a proper name then ‘a is F’ is meaningful only if ‘a’ refers.
• (9’) If ‘unicorns exist’ is false, then ‘a’ does not refer.
• (12’) No meaningful statement can imply a meaningless statement.

The first premise is uncontroversial I assume. The third premise seems to follow from the definition of ‘transferable’.  12’ also seems straightforward: If, by hypothesis, p is meaningful, then it does not imply anything meaningless, and this is just what 12’ says. That leaves 2’, 5’, and 9’ as the crucial premises.

I take it that the motivation behind 2’ is that if ‘existence’ is just another predicate of individuals, like ‘green’, say, then it should be transferable in precisely the way they are. After all, how could it be that ‘frogs are green’ is true but that ‘green’ does not transfer to all of the individual frogs? But if ‘exists’ is just like ‘green’ then it should behave in the same way.

5’ seems to be motivated by the fact that we are supposing ‘a’ to be a proper name in Russell’s sense. Recall that, according to Russell, a logically proper name is a word whose meaning just is a particular object; in other words, the proper name ‘a’ is meaningful only if, and because, ‘a’ refers. So if ‘a’ is meaningless, then the whole sentence ‘a is F’ will be meaningless too.

Finally, 9’ is motivated by the fact that if ‘unicorns exist’ is false then there simply aren’t any unicorns for ‘a’ to refer to, and so ‘a’ cannot have a reference.

On the face of it, there is some reasonableness about all of these premises. However, I think there are worries for all of them. In the next post I will try to raise some of those worries.

Geach on Good and Evil: Some Counterexamples?

In Peter Geach’s paper “Good and Evil” Geach draws a distinction between attributive adjectives and predicative adjectives. An adjective A, as in the phrase “is an A B,” is predicative just in case the phrase can be broken down into “is a B” and “is A.” Otherwise A is attributive. For instance, “This is a sweet pastry” can be broken down into “This is a pastry” and “This is sweet.” So “sweet” is a predicative adjective. On the other hand “This is a small elephant” cannot be broken down into “This is an elephant” and “This is small,” so “small” is an attributive adjective.

Geach wants to argue for the thesis that “’good’ and ‘bad’ are always attributive, not predicative, adjectives.” I would like to bring up three intelligible, legitimate uses of the word “good” that are not obviously attributive, and suggest that they may provide counter-examples to Geach’s thesis. The first is what I will call “comparative goodness.” The second is what I will call “relative goodness.” The third is what I will call “global goodness.”

In the first case, consider the claim that it is better to be a Socrates dissatisfied than a pig satisfied. Or consider the claim that God is the greatest conceivable being and its apparent implication that it is even better than being a dissatisfied Socrates to be God. Both of these claims seem, on the face of it, intelligible. They may be open to dispute, but if they are open to dispute then that presupposes they are intelligible.

Now, ‘better’ and ‘greater’ are not, by themselves, the same word as ‘good’. But these claims presuppose that there is some scale of being according to which a satisfied pig is somewhere below dissatisfied Socrates, dissatisfied Socrates is presumably below satisfied Socrates, and satisfied Socrates is somewhere below God. Now, if we suppose, just for argument’s sake, that the satisfied pig is at the very bottom of the spectrum, and that God is at the very top, it seems intelligible to say that being satisfied Socrates – this being somewhere in the middle – is good. This is what I have called the “comparative sense” of goodness. But in this sentence, “Being satisfied Socrates is good,” there is not even another adjective for “good” to modify. Hence, this appears to be a case where “good” is used in a predicative sense.

The second case, which I have called “relative goodness,” is the type of goodness we have when something is good for something else. It is what we express by saying “X is good for Y.” For example, “This spaghetti is good for me,” or “my dog Paco is good for me.” It is interesting to note that both of these sentences might be true even if this spaghetti is neither good food nor good spaghetti and even if Paco is not a good dog. This spaghetti might be terribly undercooked. It might be incredibly cheap. But the relief it gives me may be enough to make it good for me. And Paco may be a crooked, maimed, and disobedient beast – hardly a good dog – but the licking he gives me at the end of the day makes him good for me still.

So things that are good in this sense can’t be likened to cases where I eat spaghetti and say “Oh, this is good!” In that case the adjective that “good” is supposed to modify is implicit (e.g., “Oh, this is good [spaghetti]!”). Whereas here it is not clear at all what the implicitly modified adjective would be; nor would it be clear what the meaning of such a construction might be (what does it mean to say “This spaghetti is good [spaghetti] for me”?); and as we’ve just shown, it is false that, in general, the F that is good for me is a good F (the spaghetti that is good for me need not be good spaghetti). So it doesn’t seem that “good” modifies any adjective in this case; so relative goodness seems predicative.

The third case is what I call “global goodness.” This is the type of goodness we attribute to whole facts or propositions. For instance: “It is good that God created the universe,” or “It is bad that animals suffer needlessly in factory farms.” Of course, these constructions could be turned into subject-predicate form too (that-p is good), in which case it will be clear that global goodness is prima facie predicative. Note that these are not just roundabout ways of saying “It is morally bad;” for even if nobody were responsible for the suffering of beasts – say, if they were harmed by some natural disaster – it would still be bad that they suffered (an objective tragedy, if you will).

But in these cases, it is not clear what the implicit adjective would be that “good” is supposed to modify. Should we say that-p is a bad proposition? This doesn’t seem to make much sense. Or that it is a bad state of affairs? This doesn’t seem to make much sense either: It’s not clear what counts as a good state of affairs or a bad one. States of affairs just are what they are. (If a state of affairs consists of an object having a property, as some philosophers say, then is a good state of affairs one where the object really has the property? Or maybe it must have the property well?). Besides, even if one did not believe there were states of affairs at all (an open question in metaphysics), one could still affirm that it is bad that animals needlessly suffer. This isn’t the case with genuinely attributive uses of “good,” since one cannot consistently believe “he is a good robber” and at the same time believe there are no robbers. Hence, this cannot be an attributive use of goodness. So this sense of “goodness,” global goodness, seems to be predicative: We can say, in a seemingly intelligible manner, “It is good that-p” or “that-p is good,” good simpliciter.

Russell on Existence in TPLA II: Russell's Second-Order View of Existence

In my last post I introduced the topic of Russell on existence. Now I'll deliver. Let’s see what Russell thinks.

First, it is helpful to understand some of Russell’s technical vocabulary. In particular, for our purposes, we should consider his notion of a name, of a definite description, of a proposition, and of a propositional function.

For Russell, a logically proper name (or, for short, just a name) is a word whose meaning is a particular, i.e., an individual object or entity. For instance, intuitively, the name “Socrates” directly denotes the particular object, Socrates. Or the name “Paco” directly denotes my Chihuahua, Paco. Now, this is simplifying a little bit, since Russell has a whole theory of what a particular is and which words actually are proper names, but this isn’t really essential to his account of existence. One could hold to views about existence that are basically the same as Russell’s even if one modified his account of particulars and the extent of the proper names.

What is important, however, is that proper names be contrasted with definite descriptions. A definite description is some phrase that is meant to describe a particular, unique individual. For instance “the Chihuahua that I have had since 6th grade” is a definite description. (As it turns out, it does successfully denote something: my dog Paco.) “The dragon flying above my head” is a definite description too, though to my knowledge it is one that does not refer to anything.

Note: Names are not definite descriptions and definite descriptions are not names. The meaning of a name is just the object it refers to; the meaning of a definite description includes all of the predicates mentioned in the description (for instance, in the last example, "dragon," "flying", and "above my head" are all part of the meaning of the description).

It is important to bring up this contrast between definite descriptions and proper names because Russell gives a separate account of existence statements for each. What we are interested in when talking about “individual existence” statements is existence statements whose subject term is a proper name. This is the type of existence statement Russell will say is meaningless.

A proposition for Russell is, in essence, something that can be asserted, or something that can be true or false. For instance, that it is raining is a proposition, or that Paco is black is a proposition. Once again, this is simplifying a bit, but the particular details of Russell’s views on propositions are not essential here.

Finally, there is the notion of a propositional function. Russell says that a propositional function is “any expression containing an undetermined constituent, or several undetermined constituents, and becoming a proposition as soon as the undetermined constituents are determined.” Examples include ‘x is a man’ or ‘n is a number’ or ‘(x+y)(x-y)=x2-y2’. So, if we were to “fill in the blanks” so to speak we would have a full proposition. For instance, replacing ‘x’ with ‘Paco’ gives the proposition that Paco is a man. Replacing 'n' with '2' gives the proposition that 2 is a number.

Russell's propositional functions can be necessary, possible, or impossible. Russell defines this as follows. A propositional function is:

• Necessary, when it is always true;
• Possible, when it is sometimes true;
• Impossible, when it is never true.

Russell technically says that we have to take at least one of these locutions – “always true,” “sometimes true,” etc. – as undefined. But intuitively, “always true” means that every instance of the propositional function is true. For example, ‘x is x’ is a propositional function that is “necessary’ in Russell’s sense, since it is “always true,” whereas ‘x is a man’ is a propositional function that is possible but not necessary. These locutions are clearly not meant in a temporal sense.

(Interesting side-note: Obviously Russell's definition of "possible" and the like is not at all the definition we would immediately think of when we hear these words. What's interesting is that it's not clear whether he even meant to capture what we do with possible worlds semantics. He makes it explicit that he thinks previous thinking about modality is confused and problematic in some way, but it's not clear whether his discussion of modality is trying to capture some sort of traditional modal phenomenon as opposed to just making stipulations, nor whether his attitude toward traditional notions of modality is one of revision or rejection. Another interesting question: Is there any way to modernize Russell here? Is he on to anything at all? Anyway, enough of this digression...)

That brings us finally to Russell’s theory of existence. Russell’s official view is that “existence is a predicate of a propositional function.” In particular, if F is a type or kind of entity, then to say that F’s exist is just a shorthand way of saying that the propositional function ‘x is F’ is possible:

• (EXIST): F’s exist iff ‘x is F’ is possible (in Russell’s sense above).

For instance, dogs exist iff ‘x is a dog’ is possible. Or men exist iff ‘x is a man’ is possible.

This makes Russell's view a "second-order" or "second-level" view of existence. If we think of individual objects or entities as the "first level" and we think of things that apply to individuals -- propositional functions -- as the second level, then existence is a property of things at the second level, since it is a property of propositional functions. Hence Russell's view has been variously described as a "second-order", "higher-order", or "higher-level" view of existence.

So, according to Russell, “It is of propositional functions that you can assert or deny existence.” On the other hand, to say of a particular thing in the world that it is exists or not is “strictly nonsense.” After all, it doesn’t make sense to say of a particular object a that it is “possible” or “sometimes true.” Hence, individual existence statements are meaningless.

This is of course rather shocking on the face of it. We seem to make true individual existence claims all the time. But on Russell’s view, “John exists” isn’t simply false. It isn’t even a loose way of speaking. It’s simply nonsense. Moreover, the seemingly indubitable inference from “I think” to “I exist” is not invalid on this view; it isn’t even an argument, since arguments have to have propositions as their conclusions, and “I exist” isn’t even a comprehensible thought. What one might have thought incorrigible turns out to be unintelligible.

Nonetheless, as repugnant to common sense as this might seem at first, common sense is not infallible. And to be fair, we have only laid out Russell’s views and have not presented his arguments for them. In the next post I'll consider some of the reasons why, exactly, Russell might have come to this conclusion.

[Part III is here!]

Russell on Existence in TPLA I: Why Care?

In the next few posts I'm going to talk about Bertrand Russell's views on existence as one finds them in his The Philosophy of Logical Atomism (TPLA for short). (Note: I have already written the posts, so I will actually deliver!)

In TPLA Bertrand Russell offers a brief but intriguing account of the notion of existence. Russell holds forcefully to the view that existence cannot be said to apply to individual objects – at least, not without descending into nonsense. According to Russell we cannot meaningfully say this or that particular thing exists; instead, only types or kinds of things can be said to exist.

I will first try to make clear what, exactly, Russell’s views on the matter of existence are, at least insofar as he talks about it in TPLA, and I will clarify his technical terminology along the way. I will then attempt to lay out what are, so far as I can tell, Russell’s arguments for his views, as well as some of the problems concerning existence that motivate him to have a view in the first place. After questioning the soundness of Russell’s arguments I will lay out an alternative view that deals with some of the problems of existence he has identified. This alternative view of existence is more similar to that held by the majority of people before him, including the medieval Scholastics. Ironically, it turns out that this view is actually similar to some of what Russell says about propositions and propositional functions (as well as his own earlier view before TPLA).

Before I begin, however, I’d like to explain why I think it is worthwhile to think about Russell’s views on existence and to bother to critique them. After all, Russell gave the Lectures for TPLA nearly a hundred years ago, and hardly anyone would agree with the precise details of his account of existence (let alone his general metaphysic).

It is probably true that Russell’s precise views are not generally accepted, and arguably parts of his technical “machinery” are archaic. Nevertheless, Russell’s spirit lives on. Russell’s claim that existence is not a predicate of individuals and that existence should ultimately be defined in a “higher-order” way, in terms of quantification, has been widely accepted by many prominent philosophers.  Indeed, the slogan that “existence is what is expressed by ‘existential quantification’” is standard orthodoxy nowadays. (See, for instance, Frege’s  Foundations of Arithmetic sec. 53, Quine’s “On What There Is,” C.J.F. Williams’ What is Existence?, and Peter van Inwagen’s “Being, Existence and Ontological Commitment,” just for a few examples.)

So, aside from the fact that Russell was a great thinker, and the general guideline that it is worthwhile to interact with great thinkers, we should think about his views on existence because views like his are held in one form or another even today.

[Part II is here.]

Two New Papers: Aristotelian Structuralism/Metaphysics of Ineffability

I just wanted to call to the attention of any readers that I've put up two papers from last semester! One is on the philosophy of mathematics, and the other is on the metaphysics of ineffability. I made a couple of posts about these issues (e.g., here and here) and these are my more considered thoughts after a semester of reflection.

The first paper is a general overview of an Aristotelian version of structuralism. That paper is here. In that paper I try to lay out as clearly as possible what the view is, and lay out some of the arguments and examples that support the view. Some might find the discussion and argument concerning what I've called "mathematical treating-as" to be interesting. To be honest, I find the view quite compelling.

I wasn't able to come up with a uniform semantics for this view in time to make it perfectly polished, though I do know what I want to say about this now (hopefully more about this in future posts/papers). I am pretty confident now that a uniform semantics for Aristotelian-type structuralism can be given.

The second paper (here) is on the metaphysics of ineffability. I talked about this problem before and was puzzled then. I remain puzzled now. But I feel that I've got a good grasp about what types of ineffability there are and what types of arguments can be given for each. My paper basically identifies several types of ineffability, and defends a substantive version of ineffability against an "idealist" type argument that was conceived by my professor, Thomas Hofweber. I felt quite pleased with this paper by the end of it; it's a very interesting topic and the paper is filled with lots of arguments and examples (hopefully some of them are good!).

If anyone has any thoughts on all of these feel free to comment or shoot me an e-mail!

Ineffability: A Serious Threat to Ambitious Metaphysics

Consider a squirrel on a tree. There are things we can represent that are simply not within the capacity of the squirrel's mental structure to represent. For instance, we can represent complex mathematical truths about the shapes and physical relations obtaining in the squirrel's environment. Despite the fact that all these facts are "happening" right in his face, our little squirrel hasn't the slightest clue about them. And it's not just that he's ignorant about them like someone who doesn't know math or physics; it's that it's completely beyond a mind like his to even represent things like facts and propositions. They are completely ineffable to the squirrel.

That all seems fair enough. But here's where things start to become worrisome: Why shouldn't we think that we are in relation to some possible being in the same way that the squirrel is related to us? (Or, if you believe in God, why not think that we are related to some actual being in the same way that the squirrel is related to us?) In other words, why not think that there are some aspects of reality that are completely ineffable to us, even if maybe they are effable with respect to some other, more advanced being? Why not think that there could be some species of creature who could comprehend things that are completely beyond our mind's capacity to even represent?

This is not a new argument or anything. See for instance Ch. VI of Thomas Nagel's book 'The View From Nowhere'. Moreover, I'm sure many of us have thought of this possibility before. But do we really consider the implications of this argument? I know that I've thought of this before, but haven't drawn anything of significance from this. However, I'm realizing now that this is a very important and deep question.

Note that this view seems to go hand in hand with the idea of metaphysical realism, which holds that there is an objectively existing world that is in some sense totally (or at least mostly) mind-independent. If you hold that view, it seems rather strange to think that somehow all of reality, of necessity, must be in principle comprehensible to us. And if reality were in its totality to be comprehensible to us it would be a rather strange coincidence. Moreover, there seems to be nothing particularly special about us. We are on a continumm with non-sentient creatures, insects, and squirrels on one side and with angels and God on the other (and probably a lot of things in between). Hence, it seems quite likely, on realism, that we are in a situation similar to that of the squirrel: There are aspects of reality which are simply beyond the representational capacities of our minds.

But then there is trouble. Interestingly, those who have high hopes for metaphysics tend to be metaphysical realists, but that very same metaphysical realism tends to undermine the high hopes for metaphysics. For example, suppose we characterize metaphysics as the study of the most fundamental or general aspects of reality. Suppose moreover that we are metaphysical realists. Then, probably, there are aspects of reality which are simply beyond the representational capacities of our minds. But in that case, for all we know, the most fundamental or general aspects of reality are within the sphere of things that are completely ineffable to us. So there is reason to doubt the possibility of having any substantial metaphysical knowledge.

In fact, maybe by a similar though distinct argument we can get a stronger conclusion. We have some reason to think that as minds become more advanced on the "great chain of being" that I've described, they become able to represent more (and more) fundamental aspects of reality than those before them. Higher beings have concepts that are more fundamental than those of the minds on lower levels. (Technically, they have concepts of things that are more fundamental.) And they probably have more of them. For instance, some lower animals can probably represent things like 'cause' and 'object' and even 'agent', but it seems doubtful whether bees could do the same (or to the same degree). But in that case, granted we are probably pretty far from the high end of the continuum, probably the most fundamental aspects of reality are only representable by beings on the higher end. So, probably, we cannot represent the most fundamental aspects of reality. So, probably, ambitious metaphysics is hopeless.

This might seem like a fun philosophical puzzle, but actually it is rather important, because if I sit down and ask myself whether I really think metaphysical realism is true, I am with utter and literal sincerity inclined to say, "Yes." And if I sit down and ask myself whether I really think some aspects of reality are ineffable for the reasons described, I am with utter and literal sincerity inclined to say, "Yes." And, to bring the trilemma to completion, I have high hopes for metaphysics and sincerely think it is essential to truly understanding the world.

What to do then? Does the argument against substantial metaphysics work? What are the implications for metaphysics and other areas of philosophy depending on which way one goes? How might different views solve the issues here?  These are interesting questions. Since I've been thinking about this stuff for a class I'm taking, I'll probably have a chance to write a term paper on it. I have inklings about where we might go, but I have no clear answer at the moment.

Shameless Hyperintensionalism in Ethics (And Other Areas)

A hyperintensional position in a sentence is one where substitution of necessarily co-extensional statements does not preserve truth value. So for instance, 'believes' is a hyperintensional position. Alfredo believes triangles all have three sides doesn't necessarily imply Alfredo believes that triangles all have angles adding up to 180 degrees (and vice versa). After all, I might not know this yet.

What I will call a metaphysical hyperintensional position (and which I will just call 'hyperintensional' hereafter) is, intuitively, one where the hyperintensionality doesn't arise because of some mental attitude. This can be defined more precisely, but for my purposes some examples will suffice.

For instance, the operator 'essentially' is hyperintensional, at least on one understanding of 'essentially.' To use a historical example, Socrates is essentially a rational animal; he is not essentially risible, though necessarily if something has the one property it has the other. Or to use the more contemporary example, Socrates is not essentially a member of his singleton set -- this doesn't have to do with what he is, at the most fundamental level, in himself -- even though he has the property of being so necessarily. Grounding, intrinsicality, naturalness, reduction; these all seem to be hyperintensional as well.

Many metaphysicians are skeptical of these concepts. (Though of course many are not! Which is why they are such a hot topic of discussion lately.) I think there is usually some ambiguity in what this "skepticism" amounts to, but it is often made explicit in terms of the good old, "I don't know what that means." This skepticism looms especially large among those of a certain breed of metaphysician, whose generation either made advances over extensional concepts by employing intensional ones, or learned from those who did. (On all of this, see Daniel Nolan's very enjoyable paper.)

One thing I find worth noting is that people in ethics use these concepts shamelessly, both in first-order normative ethics and in meta-ethics. Meta-ethical claims are consistently stated in terms of "in virtue of" (just witness the Euthyphro Dilemma) or "grounding." There is a consistent flow of talk about what is essential to an action (as opposed to what is just necessarily true of it), as well as the intrinsic features of an action. Ethicists seek real definitions of what's right and wrong, seek to categorize things into natural kinds, seek to reduce properties to other ones, and seek to explain less fundamental facts in terms of more fundamental principles; in general, ethicists have no qualms about using metaphysical hyperintensional concepts, while many metaphysicians claim skepticism even about their intelligibility.

Philosophers of mind initially seem to be a bit more careful in this regard, at least those working in the metaphysics of mind. Witness the debates about supervenience for instance. However, it seems many philosophers of mind are realizing that they have really been trying to raise issues that can only be adequately stated using hyperintensional language. And many times even those who are keen to phrase things in terms of supervenience will explicitly distinguish this purely modal notion from what's really doing the work (explanation, grounding, reduction, and so forth).

Many perfectly legitimate metaphysical debates themselves are best phrased in terms of hyperintensional concepts. And if we look in areas other than metaphysics, a very similar pattern seems to arise; think of philosophy of science, philosophy of math, philosophy of religion, free will and moral responsibility, etc.

One of the clearest examples though is in ethics. It might be worth it for me to demonstrate my empirical claims with specific examples (though frankly it's harder for me to think of papers in ethics which don't make any use of these concepts than those which do). I might try to do this at some point. But I think it's just worth pointing out for now: The sense I get from everything I've read in ethics over the years is that ethicists have no problem whatsoever using any of the hyperintensional notions that metaphysicians will sometimes claim to have no understanding of. Obviously that is not universally true, and some positions in ethics are in fact more "deflationary" than others. But arguably this is not the norm.

Considering that ethics is often one of the closest areas of philosophy to "real world" problems this is important, since it lends some weight toward thinking that these are not arcane, idiosyncratic notions, but would be considered completely intelligible to most people (and they are; anyone taking intro to ethics will "get" the Euthyphro Dilemma; in fact, their understanding of this problem will probably be much clearer than any problem involving modality for example). Also, if you grant a certain level of epistemic autonomy to ethics -- it doesn't need to wait on the approval of metaphysicians to be considered legitimate -- then it would be wrong to pronounce, based on an ill-defined "skepticism," that these concepts are unintelligible. The interaction between metaphysics and ethics should really be one of reflective equilibrium. And in that case, the initial reaction to hyperintensional concepts should be one of cautious approval rather than default skepticism.

Modal Realism and the Serviceability Argument

Here's a quote from David Lewis: "Why believe in a plurality of worlds? -- Because the hypothesis is serviceable, and that is a reason to think that it is true."

Question for David Lewis and other modal realists: Lots of worlds are serviceable, not just the metaphysically possible ones. Many times when we do semantics, discuss language, give thought experiments, etc., worlds which are strictly logically possible but not metaphysically possible are helpful. For example, one of the ways that intensional semantics deals with oblique transitive verbs, control verbs, etc. is by invoking worlds where, for instance, water is not H2O, or where Hesperus is not Phosphorus. Presumably these are not metaphysically possible worlds, but rather 'logically' possible worlds. (Sometimes metaphysically possible worlds are called 'broadly' logically possible worlds; by 'logically' possible worlds I mean what are sometimes called 'strictly' logically possible worlds.)

Do these exist too, in exactly the same way as the metaphysically possible ones? If yes, then we run into problems. After all, isn't it only the metaphysically possible worlds which can exist? If not, then what is the distinction between metaphysical possibility and mere logical possibility supposed to mean? In fact, if merely logically possible worlds exist just like the metaphysically possible ones then there is no distinction. But there is, of course, a distinction.

At the very least, aren't the metaphysically possible worlds the only ones which could be actual? But if 'actual' is indexical as Lewis thinks, and the logically possible worlds exist on a par with the metaphysically possible ones, then any of these worlds could be actual.

Personally, I think there's just as good reason to admit the existence of logically impossible worlds as there is to admit the existence of possible worlds (though I don't think there's much reason to admit the existence of either).  If we really needed possible worlds, I think we'd need impossible ones too. But if logically impossible worlds are serviceable too then that makes things even worse for the modal realist. After all, what would it mean to say that a logical contradiction actually holds true in a concrete world just like ours? Clearly there are no such concrete worlds, since whatever concretely exists must at least be possible. But even if one resists the need for impossible worlds, the metaphysically possible worlds are a proper subset of the strictly logically possible ones, and it should be clear that these latter are "serviceable" too.

In sum, if Lewis's argument works for the existence of concrete metaphysically possible worlds, then it works for the existence of metaphysically impossible worlds too. But these can't exist concretely; that's the whole point of making some metaphysically possible and others not. Hence, Lewis's argument does not work. This can be taken as either reason to abandon the 'serviceability' criterion of existence, or as reason for rejecting concrete possible worlds. I'm inclined to reject both.

"Whatever is Moved is Moved by Another"

In this post I am going to try to defend Aquinas's First Way, specifically against the attacks brought against it by my friend Alex. Alex has written a fine explanation and critique of Aquinas's first and most famous Way, the argument from motion. The paper can be found here. Unlike many attacks on Aquinas's argument, Alex's reading of Aquinas is sympathetic and charitable, and thus at the same time his criticisms are incisive and well-taken. Anyone who wants to fully understand my post should read Alex's paper first; nevertheless, I will summarize some of his most important results.

To be specific, I'm going to defend Aquinas's premise that whatever is moved is moved by another, which we shall call (MOV). I do not claim that Aquinas ever made the defense I am making. In fact, I think the argument I give is in some respects new. But when all is said and done what I am concerned with is whether Aquinas's premise is defensible and true.

First of all, Alex points out that the term 'motion' in scholastic philosophy really means change. And to say that an object is changing with respect to some feature P is to say that it is going from being potentially P to actually P (more on this terminology here). I will take this for granted in everything I say about change. Now, in summary, Alex's main objection to Aquinas's defense of (MOV) is that either it is (a) valid but palpably unsound or (b) all its premises are true yet it is invalid, i.e. does not prove the premise (cf. his paper for details). However, Alex does think that Aquinas can defend the following more modest premise, which David Oderberg attributes to Aquinas:

(ACT) If something changes from being potentially F to being actually F then there must be some actual being that initiates this change.

The problem is that the more modest and highly defensible premise (ACT) is not equivalent to (MOV), leaving (MOV) undefended and the First Way ultimately uncompelling.

Before I present my argument in favor of Aquinas's (MOV), we need some definitions. First, the definition of an external object, (EXT):

(EXT) x is an object external to y just in case x is not y and x is not a part of y. [def.]

Let's also define what I will call the notion of change per se. (This is my own terminology.) Intuitively, something changes something else per se if it is the most immediate and fundamental efficacious cause of the change, [or the only sufficient cause such that you can't get any 'closer' to the change]. For instance, my hand pushes a stick which pushes a rock; the idea is that the stick, specifically its tip, is what changes the location of the ball per se. Here is a somewhat more formal definition of changing per se, which we will call (CPS):

(CPS) x is a cause per se of a change in something y with respect to feature P by action E just in case (i) x changes y with respect to feature P by action E, (ii) if at the same time as action E there is an action F of some parts of x, and these parts also change y with respect to feature P by action F, then the action F taken alone is not sufficient for changing y with respect to P, and (iii) x's action E taken alone is causally sufficient for changing y with respect to P [def.]

This definition can be made more precise, but the concept should be somewhat clear. The idea behind what I've called change per se is that whatever changes something else per se is the thing that changes y in the most immediate sense and a sense more proper than other things. So, for instance, take the following objects: Me, my arm, my arm's atoms, and a stick. When my arm changes the location of the stick, I can be said to change the location of the stick; however, I cannot be said to change its location per se, since, arguably, if somehow my arm persisted in its motion without the rest of my body (maybe by a miracle it was detached and could float, pushing things around), it would still be sufficient for the stick's changing with respect to its location (contra (ii)). On the other hand, arguably, my arm, or at least some part of it, changes the stick's location per se by its motion, since clearly it can be said to be changing the stick's location, thus satisfying (i). It arguably satisfies (iii) for the reasons stated, and it arguably satisfies (ii) since intuitively if you removed most of the arm but left a chunk of it or a few of its atoms, and they did the same thing as they did when my whole arm's motion occurred, then they would not be able to bring about the stick's change of location.

Now maybe you will disagree with my example and say that given my definition the arm does not change the stick's location per se. But the example is simply to illustrate what I'm trying to get at. If you deny the example is an example of change per se then you should understand what I mean. Also, I would not be surprised if my definition requires chisholming; nevertheless, I think it is on the right track, and helps get my point across. What is most important is just that we have some intuitive understanding of what I mean by something's changing something else per se.

Now we need the following premises. I will translate them into predicate logic, and from my translations it should be clear which formulas correspond to which English phrases.

(1) For all x, if x is changed with respect to P by something y then there is some actual thing z which changes x with respect to P.

Translation 1: ∀x[∃yCxy→∃z(Az∧Cxz)]

(2) For all x and y, if y is actual and x is changed per se with respect to P by y, then y is either an object external to x or y is a proper part of x.

Translation 2: ∀x∀y[(Ay∧Dxy)→(Eyx∨Pyx)]

(3) For all x, if x is changed with respect to P by something actual y, then there is a z which is actual and changing x per se with respect to P.

Translation 3: ∀x[∃y(Ay∧Cxy)→∃z(Az∧Dxz)]

(4) For all x and y, if y is changing x per se with respect to P, then y is changing x with respect to P

Translation 4: ∀x∀y(Dxy→Cxy)

(5) For all x and y, if x is external to y, then x is not identical to y.

Translation 5: ∀x∀y(Exy→x≠y)

(6) For all x and y, if x is a proper part of y, then x is not identical to y.

Translation 6: ∀x∀y(Pxy→x≠y)

Let's examine whether these premises are plausible or not. 4, 5 and 6 can easily be shown to follow from the definitions of 'change per se', 'external object', and 'proper part' respectively, so I will not talk about them any more. 1 is basically just a more precise statement of (ACT), so I won't say too much in its defense, but the premise is eminently plausible: Upon a small amount of reflection it is simply obvious that what is merely potential cannot have any power to bring about something actual. Merely potential chemical reactions do not bring about any actual chemical reactions. So the only thing which can bring about something is something which actually exists, and doesn't merely potentially exist. 

The crucial premises then are 2 and 3. 3 is quite plausible on the face of it. For surely if something is changed at all, then there is something which changes it in the most immediate sense i.e. changes it per se. There must be some most immediate explanation or cause of a change right? If there isn't, then the change can never come about. This seems intuitive enough.

(The intuition is this: There seems to be some sort of infinity problem here, though the problem isn't with an infinite regress but rather with what we can call an infinite "progress" of causes. If there is no immediate cause, there has to always be another cause that's "closer" to the effect, but never one that actually "gets" to the effect. If it isn't clear what I mean I can elaborate.)

What about 2? The idea behind 2 is that some things can truly be said to bring about changes in themselves in some sense, but they can't be said to bring about per se changes in themselves; properly speaking, it is the parts which are bringing about the change in the whole. For instance, dogs can move themselves only because their legs do. So, the only thing which can bring about a per se change in something is either something external to it or else one of its parts.

Suppose to the contrary that the cause x of the per se change in y with respect to P is not one of y's parts and is not something external to y. Then since clearly whatever is not a proper part of y and is not external to y is identical to y, it follows  x = y. So y brings about a per se change in y. Now either (a) some of the parts of y bring about the change in y or (b) none do (either way, definitely no parts bring it about per se, as per our assumption).

Assume (a). If none do, then y's parts remain completely the same, yet there is a change in y. But surely y taken alone is not sufficient for explaining the change in y, and thus y does not cause a per se change in itself! After all, how could y change itself with no external influence and no action of any of its parts at all? It would have to be a spontaneous causa sui! So on the supposition that the parts do not act in any way so as to bring about the change in y, it follows y cannot be a per se cause of a change in itself. Since we assumed however that y does cause a per se change in itself, it follows we must reject this supposition and conclude that some of the parts do in fact bring about a change in y. In other words, we must reject (a) and assume (b).

Assume (b). Suppose on the other hand that some of the parts do help cause the change in y. By the definition of per se change, the action of these proper parts of y is not sufficient for explaining the change in y; but nevertheless the action of y taken apart from any external cause is. This seems to make little sense; y still appears to be acting as a causa-sui, since it is still causing a change in itself at least in part independently of the action of its parts. Since this is impossible--nothing can be a self-cause except by the action of its parts--we must conclude that the parts do not help cause the change in y. Thus (b) is false.

Since both (a) and (b) are false, and either (a) or (b) must be true given our assumption that y causes a per se change in itself, we must reject our assumption that y caused a per se change in itself. But if that is the case, then given that there is no external cause of y then x (the cause of the change in y) must be a proper part of y, as we set out to prove.

So much for premises 1 and 2 then. Now, given that all the above premises 1-6 are true we can prove:

(7) For all x, if x is changed with respect to P by some y, then x is changed by some z non-identical to itself.

Translation 7: ∀x[∃yCxy→∃z(Dxz∧x≠z)]

I won't explain the proof here; instead, for anyone who doubts me, I have attached a formal proof below. From 7 and 4 of course it can be shown quite easily that whatever is changed with respect to P is changed with respect to P by some non-identical z: That is to say, whatever is changed is changed by another. Hence, given my 1-6, Aquinas's premise is secure.

Proof of 7:

[Note: If you can't see the proof, right click and either open in a new tab or else save the image and zoom in with some image viewer. I did the proof rather quickly so it is not the most elegant and could be done in fewer steps, but it gets the job done.]

An Issue With Metaphysical Reduction

Take a fact F. In general, what does it mean to say that fact F metaphysically reduces to fact F'? Note I am speaking of metaphysical reduction as opposed to conceptual reduction. First of all, the latter has to do with concepts and propositions rather than facts. For example, when we say that being a bachelor just means being an unmarried male, or when we say the proposition that Alfredo is a grandfather just means that Alfredo is the father of a parent, these count as examples of conceptual reduction. These explications of meanings are just the result of fully specifying the nature of our concepts as they stand. These are very simple examples, but the more complex instances of conceptual reduction in philosophy follow the same general idea as these ones.

Metaphysical reduction on the other hand has to do with facts in the world and how they stand in relation to each other. I take it that the following necessary condition imposes a restriction on the relation of metaphysical reduction:

• (R) If fact F metaphysically reduces to fact F' then (i) fact F holds in virtue of fact F' holding and (ii) the holding of fact F is nothing over and above the holding of fact F'.

As an example, physicalists often say that all mental facts are reducible to physical facts. I take it that this at least means that the mental facts hold in virtue of the physical facts and that they are nothing over and above the physical facts.

Now, (i) and (ii) seem to me to be in tension with each other. In fact, on the most straightforward reading of (ii) their simultaneously holding leads to a contradiction. Hence, we must find some other way to explain (ii), since it does not seem like a primitive relation. This is rather difficult. Let me explain.

By (i), reducibility must be an asymmetrical relation. This means that if F reduces to F' then F' does not reduce to F. For suppose F reduces to F'. Then F holds in virtue of F'. But the 'holding in virtue of' relation is asymmetrical, since otherwise there would be circular chains of ontological dependence. So if F holds in virtue of F', then F' does not hold in virtue of F, and thus by (R), F' is not reducible to F.

The problem is that the most straightforward reading of (ii) is that the holding of fact F is identical with the holding of fact F'. After all, suppose F and F' are not identical and we are dealing with a world of just F and F' (here I'm abbreviating, and I should really be saying the holding of F and the holding of F'). Then there is a perfectly clear sense in which F is something over and above F', viz. there are more things in the world than F! For if F =/= F', then for some x, x =/= F'. So there is something out there in the world which is extra-mentally distinct from F'. That seems to be a legitimate sense in which F is something over and above F'. So if F is not something over and above F' then F = F'.

But of course, if that were the case, then the 'in virtue of' relation here would not be asymmetrical, since if F = F' and F holds in virtue of the holding of F', then by substitution of equals F' holds in virtue of the holding of F. So reducibility would not, in fact, be asymmetrical. And that is a contradiction, since we earlier established it was.

One option is to say that the 'in virtue of' relation is not asymmetrical. But that seems deeply problematic insofar as it doesn't allow us to capture the reducibility we want to pick out. After all, every materialist will accept that all mental facts reduce to physical facts, but no materialist would ever dare say the physical facts reduce to the mental facts! (Personally I find the latter suggestion more plausible than the former, but regardless it is not something the materialist would ever claim.)

Instead, we have to find a sense in which one could say fact F is nothing over and above F' even though F is not identical to F'. And I'm not sure how to explain this. No idea if this works or not, or whether it is at all helpful, but here's a thought: Let us denote by 'a full truthmaker of P' a truthmaker of P which is not a constituent or part of some other truthmaker of P. Let Q be the proposition expressing the holding of F. Maybe we can say F is nothing over and above F' if the set of all full truthmakers of the proposition Q contains only F'. That would make (i) superfluous it seems. Or at least from pretty uncontentious premises (i) would follow as a consequence. This theory is a little weird though, since the question arises as to what, metaphysically speaking, explains why Q would be distinct from the proposition expressing the holding of F'.

With that said, I don't know if that's on the right track. And even if it gets the extension of the relation right it might not even produce a deeper understanding. The point being, I don't myself know how to explain (ii). Like I said though, it doesn't seem like this is a primitive or undefinable relation. I wonder then what we can say about it.

The Universe is Contingent (And Therefore Needs an Explanation)

One common fallacy is the fallacy of composition, where one argues from the fact that each part of a thing has a certain feature to the conclusion that the whole thing has that feature. For instance, one could argue that every brick of the house is cube-shaped, therefore the house is cube-shaped. Or one could argue that each part of one's brain is unconscious, therefore the whole brain is unconscious. These inferences are fallacious.

However, I think it is worth noting that not all inferences from properties of parts to properties of the whole are invalid. If each part of a wall is made entirely of stone, then the whole wall is made entirely of stone. Similarly, if each part of the ball is entirely red, then the whole ball is entirely red. And so on.

Contingency seems to be like this, at least in this case. So here's an argument that the universe must be contingent:

Quantifier Variance and the Semantics of Quantifiers

In my previous post I explained the basic idea behind quantifier variance. Now I want to criticize it. In particular, I said I want to point out some problems with the quantifier variantist's simultaneously affirming the following two statements:

(i) the different quantifiers behave the same logically; and

(ii) the different quantifiers have different meanings.

Let's do a little basic semantics. Let's define the truth function τ[ψ]^U,g relative to models U and g for the cases of quantified formulas ψ as follows. The following definitions are true for all models M, all variable assignments s, all variables x, and all formulas φ. If a formula is not assigned to T it is assigned to F:


τ : {<ψ,U,g>|ψ is a formula, U a model, g a var. assign.} → {T,F}

• (τ-∀): τ[∀xφ]^M,s = T ⇔ for all variable assignments s′, if for all variables v, s(v) ≠ s′(v) ⇒ v = x, then τ[φ]^M,s′ = T 
• (τ-∃): τ[∃xφ]^M,s = T ⇔ for some variable assignment s′, for all variables v, s(v) ≠ s′(v) ⇒ v = x, and τ[φ]^M,s′ = T 

Basics of Quantifier Variance

When I say that there are tables is it unambiguous what I'm saying? Quantifier variantists say no. Or at least they would say that in certain contexts it is not. In particular, the sentence is ambiguous when we are engaging in metaphysical debate about the existence of the table, as in the following case.

Consider the debate between what I will call compositionalism and anti-compositionalism. Compositionalism is the thesis that there are composite material objects, while anti-compositionalism is the thesis that there are not. Take the case of a world with just a table and its parts, and suppose we are considering a form of compositionalism which says there are tables. Assume further that there are exactly n atoms which, according to this form of compositionalism, are proper parts of the table. Note that we are using a philosophical definition of 'atom', according to which an atom is a material object which has no proper parts. Anti-compositionalism says there is no table; there are just the n atoms. 

In essence, compositionalism says (A) there are n+1 distinct things (viz. the n atoms, plus the table), while anti-compositionalism says (B) there are n things and there are no more than n things. Note that (A) and (B) can be adequately translated into a quantified language which only contains variables, quantifiers, sentential connectives, and the identity sign with the usual interpretation. For example, (A) would be translated as follows:

∃x₁∃x₂...∃x_n((x₁≠x₂ ∧ ... ∧ x₁≠x_n+1) ∧ (x₂≠x₃ ∧ ... ∧ x₂≠x_n+1) ∧ ... ∧ (x_n≠x_n+1))

Pure Actuality

Many scholastic theologians, most notably Aquinas, make the claim that God is "pure actuality." This is supposed to do a lot of philosophical and theological "work"; it is by showing that there exists a being which is pure actuality that Aquinas is able to deduce many of the divine attributes. However, it is not immediately clear what this even means if one is not familiar with the metaphysical context of medieval philosophy.

A charitable interpreter who has read some medieval philosophy may be able to see how scholastics use this claim and identify certain inferences from this claim as being valid and others not. But it'd be nice if we had a more precise characterization of what it means to say God is 'pure actuality', so that we can see if all that Aquinas says follows actually does follow from this claim. Moreover, once we have a precise characterization of what Aquinas is even asserting, we can begin to more clearly assess the plausibility of the claim itself and whether Aquinas has established it. I propose the following definition:

• x is pure actuality if and only if for all (intrinsic) P, if x is P then x is actually P.

For completeness and wider scope of application, I also propose the following definitions of a thing's being 'composed of' or 'having' actuality and potentiality:

• x is composed of potentiality if and only if for some (intrinsic) P, x is P and x is potentially P
• x is composed of actuality if and only if for some (intrinsic) P, x is P and x is actually P.

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.

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.

Reply to William Lane Craig on Divine Simplicity

Dr. William Lane Craig has made a response to my previous post where I argued that his own view of divine sovereignty entails the truth of divine simplicity. Now, Craig is actually correct about one thing: My argument does not by itself entail that God is identical to all his parts. This only follows from the conclusion of my argument if you grant that God really has a will, intellect, etc. Craig does not grant this, since he doesn't think talk about things having parts is metaphysically substantive.

There are a lot of things to say about Craig's response here. Maybe the first is to simply note that he is denying that anything really, in the metaphysically deepest sense, has any parts. This is surely an unacceptable conclusion. Personally I would think it's better to simply deny God has any parts rather than to deny anything has parts. Absent this option, if I didn't believe in divine simplicity I would even modify my account of divine sovereignty just to save parthood. For otherwise I honestly don't know how Craig explains kidneys, brains, legs and their relations to the people who have them. This is just a datum of experience, that there are at least some parts.

Craig tries to use an argument by Peter van Inwagen to back up his thesis. However, the problem is that Van Inwagen's argument only demonstrates the falsity of the doctrine of arbitrary undetached parts, which is the idea that any region of a body can be taken to be a proper part. His argument can only go through if we are dealing with 'parts' like Dottie* which are constituted by enough matter in such a form that a person can survive by becoming identical to them. It's not obvious though that I could ever become identical to, say, my heart. So his argument would not go through with those sorts of proper parts.

Now, I'm inclined to reject the doctrine of arbitrary undetached parts anyway so I'm happy to accept the soundness of the argument. But it just doesn't demonstrate that there are no proper parts. And if it did entail that, then--like Peter Geach did with Tibbles the Cat--I would just take the argument to establish the relativity of identity rather than the complete lack of proper parthood. More importantly, it's not even obviously sound. We might just deny the premise that Dottie becomes identical to Dottie*, since Dottie seems to be an animal (or a soul) and Dottie* seems to be a 'lump'. In virtue of their falling under different sortals these two objects have different identity conditions associated with them, and thus by Leibniz's law they are non-identical. They are merely constituted by the same matter.

There's also something to be said about Craig's underlying Carnapian sympathies. There is intense debate about taking this sort of view about language and metaphysical methodology (cf. the Chalmers volume on metametaphysics), and suffice it to say for now that I'm not too sympathetic. I will criticize this neo-Carnapian line of thought later, but this post should be enough to see why Craig's response is inadequate.

Is 'Existence' Univocal Because 'All' Is Univocal?

In this post, Bill Vallicella presents another argument by Peter van Inwagen for the univocity of existence and questions it (he posted and refuted the first argument here). I think Vallicella has a point. Still, I might grant that the Quinean like Van Inwagen can translate a singular existential statement so as to have the same form as a general existential statement and argue the conclusion still does not follow. Quine, in his famous paper "On What There Is," proposes that we treat for instance the relation '__ = Pegasus' as a single-place predicate '=Pegasus'; one can call this 'pegasizing' or formally 'P'. Then 'Pegasus exists' will just be 'something pegasizes', which will just be translated to, '(Ex)(Px)'. Even ignoring the problematic aspects here I would pose a different objection. I would accuse Van Inwagen's argument of being a 'non sequitur'. Vallicella states the argument thus:

(1) 'Every' is univocal.

(2) 'Exist(s)' and 'every' are interdefinable: 'Fs exist' is equivalent to 'It is not the case that everything is not an F.'

Therefore

(3) 'Exist(s)' is univocal.

Clearly, as is, this argument is not valid. To make it valid we need some further premise. I'm not sure what sort of plausible premise Van Inwagen is using to get to his conclusion, but maybe it is something like

(2.5) If two terms are interdefinable then each of the terms' uses share the same sense relation.

(Just a clarificatory point: Sense relations are things like 'univocity' or 'equivocity', and Van Inwagen thinks that all the uses of 'exists' are univocal.) How are we to understand 'interdefinable' here? Surely not as meaning that for each 'exists' statement there is a semantically identical 'every' statement, i.e. one with the exact same meaning, for that would be utterly question-begging. We must construe it then as something like 'for each 'exists' statement there is a logically equivalent 'every' statement'. The problem is that (2.5) is not obviously true on this interpretation. I'll explain.

I think we can admit that 'some' and 'every' are univocal, that these two are interdefinable in the sense that logically equivalent statements can be expressed in terms of each, but still say that 'some' doesn't fully capture the meaning of 'exists', and thus neither does 'every'. Of course, every 'some' statement is logically equivalent to another 'there exists' statement, but that does not imply they are semantically identical.

On the idea that 'exists' is analogical, the natural language quantifier 'there exists' has many senses, but all beings can be said to exist in one of those senses; thus the range of this quantifier includes all beings (regardless as to which sense of 'being' can be said of them). And since there are no non-existent beings, the range of the quantifier 'some' is over all beings. So the two quantifiers range over the same domain of discourse; and since for any 'some' statement there is a logically equivalent 'there exists' statement, it follows that we can translate logically equivalent statements involving either of them with the same symbol in predicate-logic, '(Ex)'. This is also why they are each logically equivalent to at least one 'all' statement. But it simply doesn't follow that they all share the same sense relation (univocal, equivocal, etc.). It is true that our 'some' quantifier ranges over only and all beings, but it ranges over them regardless as to which of the many analogous senses of 'being' can be said of them. So it's consistent with both 'some' and 'all' being univocal that 'being' or 'exists' are not.