Modernizing (and Medievalizing) Analyticity

A project I've been hoping to do eventually is to re-work the concept-containment notion of analyticity. Something has always seemed to me intuitive about it, at least in the "Bachelors are unmarried" sorts of cases. (From the 19th century onward, for various reasons, the analytic truths seem to have become equated with the logical truths; this seems wrong to me.)

I'd also like to see whether the notion of analyticity comes up at all in medieval philosophy, and whether any of the tools of medieval philosophy could be of use for this project (or, what would be just as interesting, whether they would find the concept-containment notion of analyticity hopelessly confused).

There are several problems for the concept-containment approach, but a big one is this:

1. Forms of Analytic Judgments: Kant (sometimes) defined analyticity in terms of conceptual containment, roughly as: A judgment of the form 'A are B' is analytic iff the concept B is contained in the concept A.

The worry is that there are many judgments that do not seem to have this form, but where they seem to also be true in virtue of 'meaning' or 'concepts' (or something close). Some examples, taken from SEP:

(11) If Bob is married to Sue, then Sue is married to Bob.
(12) Anyone who's an ancestor of an ancestor of Bob is an ancestor of Bob.
(13) If x is bigger than y, and y is bigger than z, then x is bigger than z.
(14) If something is red, then it's colored.

Other examples, from Jerrold Katz:

(15) Mary walks with those with whom she herself walks.
(16) Mary walks with those with whom she herself strolls.
(17) Poor people have less money than rich people.
(18) Rich people have more money than poor people.

It's not totally clear how all of these examples can be analytic (assuming they all are) if we take analytic truth to mean that the predicate-concept is contained in the subject-concept.

I don't have a worked-out answer to this yet, but I suspect that some of the work in cognitive linguistics might be helpful. In fact, Katz' own work might be helpful here, though from what I understand Katz is a Platonist (rather than a "conceptualist") about meaning, and so it would have to be properly adapted.

In addition to these strategies, where medieval philosophy might be useful here is in seeing how we might, in fact, be able to reformulate all "basic" sentences and then parse them out so that they technically obey the constraint of "subject-copula-predicate" form.

As Terence Parsons points out, medieval scholastic Latin is unique in that it is a natural language and yet there is no distinction between a sentence's ordinary surface grammar and its logical form.

Now, if one had a close enough association between words and mental concepts, one might then be able to get a nice conceptualist-type semantics going. And it seems certain medieval thinkers did have precisely this sort of close association between mental concepts and meaning, viz., in the theory of "subordination" (I'm thinking of Buridan and Ockham right now).

This suggests that we could translate basic ordinary-language sentences into medieval subject-copula-predicate sentences, and the concept-containment idea could become more useful again.

I'm not sure it will be as simple as that or that this will solve everything, but I suspect it will make things easier.

There are some other issues for a conceptual containment theory of analyticity too:

2. Conceptual Containment: There are really at least two problems here: (a) a theory of concepts, and (b) a notion of containment. Can we give a plausible and clear theory of both? (And, what would be even better: Can we give a theory of both that is mathematizable and subject to rigor and computation once we are given a case?) And how neutral can we be here with respect to different theories of concepts and containment?

How can medieval philosophy help here? Medieval philosophy certainly contains much discussion of concepts, and that should certainly be useful.

As for the notion of containment, I can't help but think of Scotus' theory of "repugnance" and "non-repugnance" by which he assesses the modal status of basic propositions -- the kinds of propositions concept-containment seems to be after (see page 162, here). What's interesting is that Scotus seems to define repugnance as a relation holding between terms. That seems to make it a semantic relation rather than a conceptual relation for Scotus; moreover, the relation's holding is said to be grounded in "notae," which are in some sense objective features of the external world.

So this isn't exactly concept-containment analyticity; but still, in terms of its formal/logical properties, non-repugnance/repugnance behaves similarly to concept-containment/exclusion. (This seems to be an instance of a more "externalist," metaphysical picture of containment and exclusion relations; I get the sense that this sort of externalism is the norm in medieval philosophy, especially pre-Nominalism, though even after that as well.) Moreover, like repugnance-relations, concept-containment relations are supposed to ground the modal status of propositions, and moreover, they both seem to deal with the same sorts of propositions. So I can't help but think Scotus will be helpful here, even if he probably wouldn't have a view of analyticity like Kant's.

3. Rigor: How would a rigorous conceptual semantics go? Can we make it as formal, precise and mathematical as the non-conceptualist semantics have been? Some work has already been done on this sort of thing, but there's more to be said. Again, I wonder if the medieval logic might help us here, since Parsons and others have shown it to be entirely rigorous and fit for mathematical treatment. Medieval logic at its height was generally far more sophisticated than anything after it -- certainly more sophisticated than anything Kant did on logic -- and so I can only imagine it will make this project easier.

4. Externalism: Although the conceptual-containment approach seem plausible in certain cases, so does semantic externalism. Gillian Russell, a professor here at UNC, has done some brilliant work on updating analyticity to take these post-Kripkean insights into account. However, she tends toward the more externalist side of things, and I'd like to see whether we can salvage more of the connections between meaning, analyticity and concepts than she does, while still giving a reasonable account of the externalist insights (something I worry cognitive semantics a la Peter Gardenfors hasn't quite done -- see 4.1 here, for instance).

One thing I worry about in trying to find analyticity in medieval philosophy is that medieval philosophy seems to tend much more toward externalism about semantic content (though this is only a hunch I get -- I can't point to anything specific). But maybe I'm wrong and there is room for analyticity in medieval philosophy; and even if not, the project may still be worthwhile, since maybe we will find new reasons to either abandon or revise our conception of analyticity in interesting ways we've never thought of.

So I plan to do some research on medieval logic and semantics in the next few weeks, and maybe this will help my concept-containment project. Moreover, I think it is an interesting historical project in its own right to see whether something like analyticity can be found (or reformulated) in terms of medieval semantic categories.

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.]

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.

One More Reason Why Russellian Descriptivism is False

Russellian descriptivism about proper names can be summarized rather roughly in terms of the following four theses. For any object O, where 'O' is a proper name of O:

(1) There is a unique definite description D associated with the term 'O'.
(2) The speaker believes that D is uniquely true of O.
(3) D uniquely picks out O.
(4) The proposition expressed by "If O exists, then O is D" is knowable by the speaker a priori.

So, very roughly, in the case of Barack Obama one might associate with this name the definite description 'the president of the United States from 2008 to 2016' and believe that this picks out Barack Obama uniquely; this is how one refers to Barack Obama, by means of the associated description.

There are many excellent reasons to believe a theory like this is false. Kripke, Donnellan, and others have pretty much put it to rest. But one more reason is that it entails another false thesis known as the identity of indiscernibles (or at least a somewhat modified and equally false version of this thesis). The identity of indiscernibles basically states that if two things share all the same features then they are the same object. [On the falsity of this thesis, check out Max Black's famous paper.]

The basic line of reasoning which leads me to say descriptivism of the described sort entails this principle is, informally, as follows (taking Barack Obama again as our example): By clause (3), 'the president of the United States from 2008 to 2016' uniquely picks out Obama; and by Russell's analysis of definite descriptions it follows from this that anything having the feature *being the president of the United States from 2008 to 2016* will be Barack Obama. Now, suppose two things x and y both have all the same features, and x has this feature. Then so does y. But since anything which has this feature is Obama, y is Obama. We can apply similar reasoning in any other case, since by (3) everything will have a feature P which uniquely picks it out, so if two things x and y have the same features then both of them will have P, and thus both will be identical to x.

There are two qualifications to be made. First, there is a sense in which the identity of indiscernibles can be trivially true. If we include among the features of a thing x the property *being identical to x* then it is obviously and trivially true that two things having all the same features will be identical. Even excluding features such as these the argument still goes through though, since presumably on a descriptivist theory the relevant description for, say, Barack Obama is not going to be 'the thing which has the feature of being Barack Obama'. Descriptivist theories aim to not be blatantly circular.

Second, the descriptivist might say that his thesis only applies to those referents which are nameable, whereas the identity of indiscernibles applies to all things regardless as to whether they can have names applied to them. However, the identity of indiscernibles is still false even when restricted to nameable things. You can give names to Max Black's two spheres and this provides a counter-example to the restricted form of the identity of indiscernibles.

For those not satisfied with my rather informal "proof" I've made a more formalized proof (with cheap symbols and lines, since OpenOffice does not appear to have any quantifiers). It should be pretty easy to interpret but if not please tell me. The first premise says for all x, if x is nameable, then there is a feature P such that for all y, if y is P then y = x, and x is P. The conclusion states that for any two nameable objects x and y, the identity of indiscernibles holds. Hence, if I have made no mistakes, descriptivism entails a restricted but still false identity of indiscernibles.