Greg Cavin on Bayes' Theorem and Miracles

I wrote most of this post several months ago when my friend Calum Miller came to southern California for a semester abroad. Unfortunately, I simply never got around to finishing it up. Hence, this post comes about five or six months late. However, I still think it's worth posting, in case someone watches the video or comes upon the type of fallacy that I suspect goes into the argument. Here's the post:

A couple weeks ago I went to a debate between my friend Calum Miller and philosopher Greg Cavin on the Resurrection of Jesus. The video can be found here. Cavin's opening speech on Bayes starts at (6:00). He gets into his arguments again at (13:40). In this post I'll discuss a small part of Cavin's opening speech.

At the beginning, Cavin claims that he will show that it is "virtually 100% certain that no miracles ever occur."

Greg Cavin formulates the argument in terms of an assessment of a comparison between probabilities. While Cavin goes into a ton of mathematical detail that I suspect could be simplified to get to the main point, a little bit of it is probably necessary. He formulates the argument in terms of the Odds Form of Bayes' Theorem.

In general, the Odds Form of Bayes' Theorem is as follows. For any events A, B, and D:

P(A|D)/P(B|D) = P(D|A)/P(D|B) * P(A)/P(B)

Cavin comes up with a partition of probability space which is exhaustive and exclusive. In other words, at least one of the following hypotheses holds and if one holds then the others do not.

• M: At least one miracle has, had, or will occur in the universe.
• L: The laws of the sciences as these are currently formulated in standard reference works, without any supernatural non-interference proviso, are true and are laws of nature in their restricted domains.
• (¬M & ¬L): Neither M nor L hold. 

With respect to L, what it is saying is that if you have a law of science 'S,' then a statement of the law will just be of the form: "For all times, all places, S," rather than "Except for the intervention of some supernatural force, for all times, all places, S." In other words, laws of science lack the underlined "proviso."

M and L are taken to be incompatible because if a miracle occurs, i.e. if M is true, then that entails the failure of at least one "un-provisoed" law of science at some time and place, whereas L entails all "un-provisoed" laws of science hold at all times and places. Cavin defines the evidence E with respect to which we will evaluate these probabilites as follows (27:00):

• E: The total evidence, which is a combination of T & C, where T and C are understood as follows:
• T: All of the traces (call them Ti) of miracles. These are all of the pieces of evidence people could take to provide evidence for a miracle.
• C: All of the confirmation instances (call them Ci) of the laws of science. These are all of the pieces of evidence people could take to provide evidence for the various scientific laws.

The partitioning of probability space.

Applying Bayes' Theorem to the argument at hand, this is the Ratio of Posterior Probabilities of L vs M:

P(L|E)/P(M|E) = P(E|L)/P(E|M) * P(L)/P(M)

In other words, the left hand side compares the likelihood of L given the evidence with the likelihood of M given the evidence.

Now, a crucial part of Cavin's argument is in calculating the ratio P(E|L)/P(E|M). This is done by calculating probabilities of all of the Ti given L and M and calculating the probabilities of all the Ci given L and M. If these are lower on M than they are on L, then P(E|L) will be higher than P(E|M). His official argument here is from (31:00) to (34:00), but I asked him a question later that gets to the same point.

After the talk, I asked Cavin why he thought M could not explain C and T as well as L could. In other words, why are, say, the confirmation instances of science less likely given that miracles have occurred than if L holds? He said, "Well, if I told you, 'This is a desk,'  what would that explain? Not much. How can you make any predictions from that? So, likewise, how can the proposition that at least one miracle holds explain anything? It could hardly have any predictive or explanatory power." Of course, that seems true. If the only sentence you knew to be true were "At least one miracle occurs," then you wouldn't be able to predict much, just as you couldn't predict much from just knowing "This is a desk." Hence, the argument goes, P(E|M) is very low.

However, it's a little bit misleading to put things this way. P(E|M), strictly speaking, isn't defined in terms of how much you can predict from the single proposition that at least one miracle occurs. This is clear after considering some very basic probability theory.

First, note that we can always define P(M) as P(M∩A) + P(M∩¬A) for any event A. This can clearly be seen by the following diagram:

P(M) = P(M∩A) + P(M∩¬A)

A is marked out in dark blue.

¬A is marked out in light blue.

Suppose that 'A' denotes some hypothesis, maybe the hypothesis 'The laws of nature almost always but not always hold.' Then the probability that some miracle happens is equal to the probability that some miracle happens and A holds plus the probability that some miracle happens and A does not hold. Again, P(M) = P(M∩A) + P(M∩¬A).

From this we can infer: P(E|M) = P(E|M∩A) + P(E|M∩¬A). Now, you might still think that this is lower than P(E|L) for various reasons. But you certainly couldn't infer it from the type of argument I sketched above. That would be much too easy.

Maybe I am misrepresenting what Cavin said. I hope I'm not. But if I am, let's just say that if someone were to argue in the way I represented Cavin as arguing, then they would be committing a fallacy.

There were many other interesting issues that came up during the debate, such as the likelihood of the laws of nature holding most of the time given theism, and these deserve attention. But for now I think it's worth noting that Cavin's argument doesn't go through as easily as it might have seemed.

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.

Review: 'An Aristotelian Realist Philosophy of Mathematics' by James Franklin

I recently finished reading James Franklin's marvelous book, An Aristotelian Realist Philosophy of Mathematics, and I want to advertise it here. This is a great book. It is empirically informed by a wide knowledge of both actual mathematical practice and contemporary mathematics itself, along with other relevant areas of study such as perceptual psychology, neuroscience, and engineering. It also engages with much of the cutting edge in contemporary philosophy of mathematics, especially in the later chapters. This is some of the best of what Aristotelianism has to offer. I really hope people will read it.

Franklin aims to give an account of mathematics as the science of quantity and of structure. Franklin gives particularly clear definitions of both quantity and structure--something often lacking among contemporary structuralists in my opinion--and this in itself is a very important advance. According to his account, mathematics studies structural universals and quantities. These universals and quantities are the type of thing that can be found in the real world and can be literally had by concrete objects. Of course, not all mathematical structures are had by some concrete object, but it is essential to his account that they could be, i.e. that they are metaphysically possible.

While quantity seems to me to play a less central part in his project, his clear account of structure allows him to take his views a long way. Franklin understands a property to be purely structural just in case it can be defined completely in terms of 'part', 'whole', 'same', 'different', and purely logical vocabulary. The relations of 'part' and 'whole' will probably come into play in geometry, as well as set theory, graph theory, topology, analysis, etc. So, for instance, on this definition, the property of being a Euclidean space could probably be defined purely structurally; see for instance Hilbert's axioms. Also, the Peano axioms seem to describe purely structural relations, since they only invoke logical vocabulary and identity (other than the names for the relations being defined, of course). Franklin gives many more examples, so I refer the reader to his book for a treatment of further cases.

Franklin contrasts his approach with Platonism and nominalism in contemporary philosophy of mathematics. Unlike Platonism, the universals studied by mathematics can be literally instantiated by concrete things in the real world. What mathematics does is study these possibly instantiated structures. Mathematics does not study abstract, particular individuals. Number systems, for instance, would not be cashed out as consisting of abstract individuals (numbers), but as either systems of quantities or as structures which can be instantiated by concrete things. (Franklin's account of number, in fact, cashes out numbers as being relations which are literally instantiated in the world by material heaps and 'unit-making' universals.)

Against nominalism on the other hand, Franklin assumes that there are, in fact, mathematical universals that can be literally shared by different things. Again, Franklin also assumes that there are, in addition to those universals instantiated in the real world, universals which are not instantiated but are at least possibly instantiated.

By his choice of example he shows how contemporary philosophers of mathematics often miss the most central cases of mathematics. Contemporary philosophy of mathematics often has a Platonist bias, focusing on those cases that are less essential for use in real world applications (such as huge sets, large infinities, etc.). This is to the detriment of the most central and basic cases, which are the simple, often discrete and finite structures widely used in real-world applied sciences, and which are less amenable to Platonist interpretation.

He gives a far more plausible account of mathematical knowledge and empirical mathematical application than that offered by most Platonists. He also argues that contemporary philosophy of mathematics tends to not pay enough to attention to how mathematics is actually done, and therefore misses those aspects of mathematical practice that make more sense on an Aristotelian view. He shows a much closer parallel between actual mathematical practice and actual empirical scientific practice than is often recognized (for instance, by the unquestionable use of induction, plausible reasoning, and explanation in mathematics; he rightly notes that (in)formal proof is often only the last step in the equation). Franklin goes on to apply the Aristotelian conception of mathematics to many other philosophical issues, such as mathematical necessity, infinity, approximation, and ontology.

With that said, there are several parts of the theory that could be potentially problematic and call for more investigation. Just to shotgun a few of them out:

• The reliance on a classical mereology of heaps and arbitrary sums (this is important for his definitions of whole numbers and sets).
• The reliance on (immanent) universals, problematic from a trope nominalist perspective such as my own, and which might use a bit more explanation.
• The commitment to uninstantiated universals (an idea classically denied by most Aristotelians, including Aristotle himself, and one which moves Franklin's account toward a "semi-Platonism" as he calls it).
• His commitment to all mathematical structures being metaphysically possible (this is interesting to me; I bet Franklin's account could be seamlessly extended given a proper account of impossibility, impossible objects, impossible universals, and impossible worlds, and I bet this isn't essential to his view).
• Giving a general, unified semantics for mathematical language (it's less than clear from the book how this is to be done; for instance, with the complex and negative numbers, Franklin gives what appear to be examples, or maybe geometrical/economical interpretations. But what would he say are the straight up truth-conditions for, say, -2 + 3i = 2(-1 + 3/2i)? Or (-2)(-3) = 6? Or of more general laws governing number systems?).
• Showing more precisely and in individual cases how a more wide range of mathematical concepts are definable either purely structurally or quantitatively (ideally, it'd be nice if we could get to the point of giving a general paraphrase scheme or a general procedure--Franklin's account of set theory being purely structural is suggestive, so maybe we could show how any set-theoretical entity or relation could be defined structurally, and thereby show all mathematics to be interpretable structurally; either this or the last question I hope to work on for my term paper this semester).
• The apparently ad hoc fictionalist account of zero and the empty set combined with a realist account of everything else (I can see fictionalists asking why we need the realistic ontology in some cases but not others).
• Related to this last point, some unclarity/implausibility in the theory of ontology and ontological commitment at play, as well as some unclarity about the ontological status of mathematics (if it were made more clear when or why we are committed to some things but not others, and in what way, it'd probably be easier to answer questions such as the last one).

I don't have enough time to spell all these worries out, though if anyone is curious I can explain what I mean, and maybe after reading the book some of these worries will be clear. And I don't think these are damning or insuperable criticisms either; I think they are problems to be investigated, but Franklin's account seems to me to be certainly on the right track.

One last potential criticism that I feel kind of bad about making: I feel like the book doesn't really engage much with what's been said in contemporary neo-Aristotelian metaphysics and ontology. I feel bad about saying that because of the huge swaths of literature the book does, in fact, engage with (the number of works referenced is amazing; one wonders how somebody can read so much). But in certain respects (the mereology for instance, or the role of states of affairs), it seems like the book draws on some concepts with which many current Aristotelians might take issue. And like I said, the book's understanding of ontological commitment could have been a bit more clear; here, engagement with contemporary Aristotelian metaphysics (among others) might have been helpful as well.

Overall though this is an excellent book, and maybe even a game-changer, at least for me. It contains many more interesting ideas and arguments to grapple with than I've been able to discuss here. Whether one buys into it or not, Franklin admirably demonstrates the fruits of an Aristotelian approach, at least on one understanding of that term. He makes use of a wide variety of examples, from a wide variety of real world sciences (including, but very much not limited to, pure mathematics). By doing this he demonstrates how important it is to pay attention to actual empirical results and practice when doing any sort of metaphysical or epistemological investigation into the philosophical status of mathematics. And this seems to me to be one of the most important marks of the general Aristotelian attitude.

Link: Pope Francis: Marriage is Indissoluble

Pope Francis states, in very clear terms, the traditional Catholic teaching on the indissolubility of marriage, here.

Quote: Foot Making Fun of Expressivists

In her book 'Natural Goodness' Philippa Foot criticizes (lightly mocks) expressivist accounts of moral evaluation because they seem to make evaluation of human action completely disconnected from our evaluation of other biological aspects of human well-being, as well as the evaluation of the goodness of other kinds of animals and plants.

"For it is obvious that no expressivist account will do in those other domains: we cannot think that the use of the word 'good' is to express a 'pro-attitude' in what we say about the roots of nettles or the fangs of ferocious beasts. Nowadays such evaluations are apt to be marginalized as if they were fanciful extensions of the 'proper' evaluations that express our attitudes, practical decisions, or desires. But when I was told by a certain philosopher who wanted to explain 'good' in terms of choices, that the good roots of trees were roots of the kind 'we should choose if we were trees', this finally confirmed my suspicion of the kind of moral philosophy that was his."

Libraries: A Case of Practical Incommensurability

In debates in ethics (among new and old natural law theorists for instance) one problem that comes up is the incommensurability of certain types of goods. What this means is that it doesn't seem that in general we can even weigh certain goods against others (for example, say, aesthetic experience and friendship). An apparent problem for utilitarians and consequentialists of various stripes, among others.

I'm not sure how much this has to do with that, but it struck me now as I'm working in my personal library of books how much incommensurability considerations affect me on a concrete, everyday level.

I was sitting here looking over a syllabus, and I realized I will have a week-long break or so in a couple weeks, and I'm going to probably pick a book or two to read over that break. Thinking about that, I realized that at a practical level, I have so many books to choose from that I probably will not read all of them straight through any time soon (if I'm honest with myself, maybe even ever). So it'll be a tough decision. I'm probably going to spend at least half an hour going through my long series of unread books and deciding which category of philosophy to even read from (if I can get myself to the point of deciding to read philosophy instead of something else). Then, supposing I've chosen a category (say, ethics), I'm going to have to decide which of several monographs within that category to choose.

One might think that when I'm choosing which book of ethics to read there must be something which marks one of the books off as more worth reading than the others. But I don't think so; I'm probably just going to have to pick one among equally reasonable options. Assuming the authors I'm considering are all equal on obviously measurable standards such as clarity in writing, intelligence, standards of rigor, etc., what other criteria of goodness would there be here? Quality of subject matter? How can I weigh that? If not that, what else? And even if I were to grant that I do a straightforward weighing along a single or several variables, this suggestion seems much less plausible when I'm making the higher-order choice of which philosophical area to read in (or the choice whether to read philosophy at all!).

You might then think that instead of saying my choice was among incommensurables I had multiple choices with an equal degree of goodness along all the variables of goodness, and my choice was arbitrary. In other words, the choices are commensurable, but they just happen to all be equal on the scale of goodness. But as I mentioned above, the subject matter of the books for instance does seem to play into my decision (at least in my case; I don't know about everyone else), and yet this doesn't seem like something I can literally weigh against the alternatives. Probably other incommensurable considerations play into my decision too.

(Note: Making a choice based on considerations and choosing one among alternatives doesn't by itself imply that the alternatives are commensurable. In fact, even if one choice is more rational than another, that doesn't imply the goods chosen are commensurable with those forgone qua goods. There might be some other considerations, for instance purely about what's constitutive of rationality, that bears on what is to be done.)

Maybe this is a case of incommensurability even within a particular category of basic good (viz. knowledge). I wonder how a utilitarian picture of everyday deliberation of this sort would go. And thinking about my own experience, I wonder whether a utilitarian could give a plausible model of such experience.

Okay, that's enough of that. Back to work.

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.