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.

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.]

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.

Essence and Counterpossibles

In my last post I made the point that the predicate position in real definitions is hyperintensional. So, even if two predicates have the same intension, i.e. necessarily apply to all the same things, they might not be able to be substituted for each other in the real definition while preserving truth value. This means that, among the necessary properties of a thing, we have to distinguish those which are essential to the thing from those which are not.

Maybe one way to do this is by using counterfactuals with impossible antecedents, also known as counterpossibles. 

The general idea is this. Counterpossibles, according to a certain semantics, are also hyperintensional. You can insert intensionally equivalent antecedents into the same counterfactual, but only some of these counterfactuals will be true while others will be false. So maybe we can use a particular counterpossible schema (as we will see, that in (1*)) to discriminate between properties that are essential and those that are non-essential. 

More specifically, we can take two intensionally equivalent properties F and G, insert F into the antecedent, insert G into the antecedent, and the counterfactual may have a different truth value depending on which of F or G is substituted. Thus, the essential vs. non-essential distinction will be able to be defined in terms of the truth or falsity of instances of a certain counterfactual schema. To make the point more vivid, the counterfactual schema will be like a box we can insert intensionally equivalent properties such as F or G into. If we put in G for instance and the box outputs TRUE then G is essential; if it outputs FALSE then G is not essential. Since counterpossibles are hyperintensional, the 'box' won't always give the same output for properties with the same intension.

So: Consider two intensionally equivalent properties F and G. This means that, necessarily, if anything has F in a world then it also has G in the same world, and vice versa. If F and G are intensionally equivalent then the properties λz[□Fz] and λz[□Gz] are also intensionally equivalent, and thus the formulas λz[□Fz](x) and λz[□Gz](x) are intensionally equivalent. Now, consider a counterfactual of the form:

• (C) If φ had been the case then A.

Since we are going with an interpretation allowing for non-trivially true counterpossibles, we can substitute in for φ either of two necessarily false propositions, P or Q, but it won't automatically follow that (C) will come out true under both substitutions.

So suppose it is true that a is necessarily F and necessarily G.

Let P be '¬λz[□Fz](a)', let Q be '¬λz[□Gz](a)' and let A be '¬λz[∃yy=z](a)'.

Again, it doesn't follow automatically from the semantics of counterpossibles that substituting P in for φ will give you the same truth value as substituting Q for φ, despite the fact that these two formulas are intensionally equivalent. What we can do then to distinguish whether F is essential or G is essential (or neither) is substitute in P for φ and Q for φ in (C). If (C) comes out true, then the property is essential; if it comes out false, then it is not.

With this hypothesis in mind, here is a very rough first stab. For any x:

• (1) For any P, if P is a de re necessary property of x, then P is an essential property of x if and only if had x lacked P then x would not exist.

More formally, let |x| be λz[z=x]: 

• (1*) For any P: If λz[□Pz](x) then: □|x|Px iff (¬λz[Pz](x) □→¬λz[∃yy=z](x)])

Again, keep in mind that the counterfactuals here are to have a semantics where they can be read as non-trivially true counterpossibles, and thus not according to the standard Lewis-Stalnaker semantics. Also, the 'essentialist box', □F, comes from Kit Fine's 'Logic of Essence'. Roughly, □FA means that in virtue of the essence of the F's, A holds. So if |x| is the property of being identical to x, then □|x|Px turns out to mean that in virtue of the essence of x P holds of x. Fine parses out the logic this way for its semantical and logical elegance.

Consider one example given by Kit Fine: Suppose we have the property λz[z∈{z}]. Intuitively, this is the property satisfied by something whenever it is in its singleton. Assuming that, necessarily, I am a member of my singleton, then this is a de re necessary property of me, i.e. λz[□z∈{z}](a) However, it seems to not be an essential property of me.

So, is it true that the following holds?

• (A) Had Alfredo lacked λz[z∈{z}], Alfredo would not exist.
• (A*) ¬λz[z∈{z}](a) □→ ¬λz[∃yy=z](a)

It seems that (A) does not hold. For it's irrelevant to my nature whether there are any abstract objects, such as sets, at all. After all, it seems that if nominalism were true, I would still exist. This lends some weight toward thinking that, if I had lacked λz[z∈{z}], I would still exist. But then by the criterion in (1), since the counterpossible does not hold for the property λz[z∈{z}], it must follow that λz[z∈{z}] is not essential to me. 

Given a de re necessary property of x, P, it might also be that the truth of the appropriate counterpossible is just a necessary condition for a property's being essential. In other words:

• (NEC) If P is essential to x, then were x to lack P x would not exist. 
• (NEC*) If □|x|Px, then (¬λz[Pz](x) □→¬λz[∃yy=z](x)]).

Or it might be a sufficient condition as well, i.e.:

• (SUFF) Given that if x were to lack P x would not exist, then P is essential to x.
• (SUFF*) If (¬λz[Pz](x) □→¬λz[∃yy=z](x)]), then □|x|Px.

Keeping in mind of course the sense of the term 'essence' in mind, and the relevant semantics for counterpossibles, (NEC) seems definitely true, and probably uncontentious. I suppose the interesting question is whether (SUFF) is true. I think the examples lend some support to the idea, such as the case of the singleton set given above.

In the case of (SUFF) it is particularly important that we use the right semantics for counterpossibles. If (SUFF) is true then this is very useful when talking to those who don't recognize the sense of essence at stake here: If they already know how to evaluate the counterpossible in the antecedent according to a 'non-trivial' semantics, then we simply say, "Plug in the property for the antecedent; if the counterfactual holds non-trivially, the property is essential. Now you know what I mean." This might also be a nice way to interpret people who give multiple definitions of 'essential' and 'accidental' properties, such as Aristotle. Aristotle gives a modal definition of essential properties which can sound like it might contradict other definitions of his; but (1) is 'modal' too, and it might be a way to interpret Aristotle that makes him consistent.

Probably potential counter-examples to the hypothesis come to mind. I know that I already see some issues. But it might be useful to see how far this hypothesis can go. And maybe if the hypothesis doesn't hold in general (I bet it doesn't) it might at least help us pick out an important class of the essential properties. After all, it seems in part that the reason we recognize λz[z∈{z}] as non-essential to me is because in some (non-trivial) sense had I lacked it λz[z∈{z}] I would still exist. Had nominalism been true, I'd have still been real (save if you're a Platonist/Pythagorean of a certain sort, in which case maybe you'd have good, non-trivial reason to deny that the counterfactual holds).

It might seem overly complicated to do this quasi-formally as I have, but one thing I'd like to do is look more at the formal semantics of essence such as Fine's for instance (hence the essentialist 'box' operator from Fine's papers), and see how this relates to the formal semantics of counterpossibles (whatever that happens to be). Given the inter-dependence of the two notions, maybe the correct semantics for counterpossibles will help us find the correct semantics for essence, and vice versa. Maybe the notion of essence will help us give a more principled similarity metric for counterpossibles in certain contexts. Also, maybe the notions of essence and counterpossible will relate closely to other notions, such as grounding, dependence and explanation. It might be an interesting project to see how far formal methods can help us here in finding relations between these concepts.

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.

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))

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.

Naturalness in the World

I'm currently taking a course on a newly concocted discipline in analytic philosophy called metametaphysics. For Aristotle or Aquinas the topics grouped under this heading would probably just fall under the science of metaphysics. We are primarily reading from the recent Chalmers volume and Ted Sider's new book Writing the Book of the World. In the latter book quite a bit turns on a distinction which Sider calls variously 'naturalness', 'fundamentality', 'structure', and 'carving at the joints'. My professors claim to not understand what Sider is talking about. Admittedly, Sider could be quite a bit more clear. However, I think that an Aristotelian would want to agree at least in some respects with his general point insofar as we'd admit some boundaries in reality to be privileged over others.

Consider an example Sider gives which is represented in the figure above (I'll paraphrase these examples a little bit). Suppose there is a world filled with red and blue liquid. There are many true ways that we can divide this world up when we describe it. We can divide it into those parts satisfying the normal understanding of our predicates 'red' and 'blue' as in the first figure. We could also divide it into those parts satisfying different predicates, call them 'bled' and 'rue', which correspond respectively to the portions left and right of the diagonal in the second figure. Both of these ways of describing the world are true. There are indeed bled and rue portions of the world just as much as there are red and blue ones; or, to put it another way, there are divisions of the world along the lines of both the first and the second figure. Yet the first way of speaking seems in some way to be natural while the second seems bizarre. Sider wants to assert that the first way of dividing up the world, i.e. dividing it up along the lines of our 'red' and 'blue' predicates, is better because it describes those features of reality which are in some way privileged; to put it in his terms, it describes those features which are natural/ fundamental/carve at the joints/are part of the structure of reality.

It seems like Sider gets a lot of his account from David Lewis's work. It is at least similar to the natural vs. non-natural distinction Lewis makes in his paper "New Work for a Theory of Universals." Consider two properties: being green and being grue, where grue is defined here as the property of being green and observed before 3000 A.D. or blue and not examined before 3000 A.D. This property which we have called grue appears "gerrymandered" in a way the property green is not. In Lewis's term grue is a "less natural" property than green is. Sider agrees on this point with Lewis. It should be noted that what makes grue a less natural property is not the syntactic complexity of its definition--after all, we can just give it a simple name like grue--but rather that it is in some way privileged over these other properties as being a more fundamental feature of reality. (Lewis may actually disagree here but I think his view is highly implausible and Sider does not seem to endorse it.)

Take a final example. Consider two classes of things: the electrons, and the electron-or-cows (EoC's), the latter class consisting of everything which has the property of being an electron or a cow. The things in the first group seem to go together quite well in a way which the objects in the latter group do not. The electrons do not go together better simply because they share many properties. For one, the EoC's share many properties. In fact, the EoC's share infinitely many properties: they each have the property of being an EoC or four feet long, the property of being an EoC or five feet long, the property of being an EoC or six feet long...and so on.  So it's not simply the number of properties shared which distinguishes the two. It's the fact that a grouping of things into electrons and cows gets the way reality fundamentally is, whereas a grouping of things into electron-or-cows does not.

So these are just a few examples where we can contrast fundamental/natural features of reality in opposition to gerrymandered or non-natural ones. It is necessary to use such illustrations since fundamentality is taken to be a primitive distinction which is likely not definable in more basic terms. There are a lot of questions that can be asked here: is fundamentality/naturalness/carving at the joints/structure of reality/etc. itself fundamental? Is fundamentality supposed to be a property? Is fundamentality a feature of our thoughts and concepts, of entities, or both? What is fundamental? How can we know? Is the same notion at work in each case? These are all good questions, some of which are discussed in the book.

However my primary concern is this: Is Sider really getting at some objective distinction or not? What I mean by 'objective' is whether his distinction between the fundamental/non-fundamental really corresponds to something in the external world. Personally I think he is onto something and makes an interesting case regardless as to the connections with Aristotelianism. However, I think an Aristotelian would want to admit the distinction even if he might disagree about what the fundamental things are. I will try to explain why in later posts.

If anyone is reading, I'd especially like to hear your thoughts on (1) whether you think Sider's distinction makes any sense and (2) whether the examples illustrate the point.