This paper argues that essentialist discernibility arguments (henceforth EDAs) are unsound. An EDA establishes that something x is distinct from something y by pointing out that x and y differ in their essential properties. Here are three such arguments, informally stated.

Clay. A clay statue is essentially a statue. The lump of clay that constitutes the statue is not essentially a statue. Therefore, a clay statue is not identical to the lump of clay that constitutes it.

Pain. Pain is essentially painful. C-fibers stimulation is not essentially painful. Therefore, pain is not identical to C-fibers stimulation.

Person. I am essentially identical to myself. My body is not essentially identical to myself. Therefore, I am not identical to my body.

As witness by the familiarity of these examples, EDAs are widely used and discussed in contemporary metaphysics and beyond. The problem of material constitution is built around arguments looking very much like Clay (Baker, 1997; Della Rocca, 1996; Fine, 2003; Gibbard, 1975; Koslicki, 2005; Noonan, 1991; Yablo, 1987). Similarly, Pain is a version of the classic modal argument against reductive materialism (Kripke, 1980, pp. 144–155). And instances of Person are found in the literature on personal identity (Lewis, 1971).

Here is a rough outline of my argument for the unsoundness of EDAs. An EDA is invalid unless one of its premises implies that there are properties of a unique and special kind. But such special properties do not exist. Thus, anyone running an EDA faces the following dilemma: either the premise implies the existence of the property and it is false, or it does not and the argument is invalid.

In Section 1, I characterize EDAs and present the claims in the background of these arguments. In Section 2, I state the dilemma just sketched in more details. In Section 3, I argue that the special properties mentioned above do not exist. In Section 4, I review a few strategies one may advance in order to escape the dilemma and show that these strategies all fail. Finally, in Section 5, I expand the results of the other sections to the classical argument for the necessity of identity.

1.

Two points are of central importance for any EDA: the general principle of identity known as Leibniz’s Law and the thesis of essentialism. This section explores each of these points in turn.

EDAs are a species of discernibility arguments. Discernibility arguments establish that something x is distinct from something y by pointing out that x and y differ in their properties (essential or not). Underlying discernibility arguments is the following principle:

Indiscernibility. For everything x and everything y, if x is identical to y, then, for every property, x instantiates it iff y does.

In short: the relata of identity have all the same properties. This is of course the familiar general principle about identity known in the literature under such labels as ‘the Indiscernibility of Identicals’ and ‘Leibniz’s Law’.

Note that here and in what follows, I use ‘property’ as a generic term for relations of any arity, unary or not. Thus, for example, the unary relation being red, the binary relation being loved by and the tertiary relation being located between … and … are all properties. Also, for bookkeeping reasons, I write the English name of properties in italics.

The second claim crucial to an EDA is the thesis of essentialism:

Essentialism. At least some things have at least some essential properties.

Anyone committed to Essentialism is also committed to the truth of some essence statements. An essence statement is a sentence that says of something that it has a particular essential property. Familiar examples of essence statements include ‘Socrates is essentially human’, ‘I am essentially born from my parents’, ‘planets are essentially celestial bodies’ and ‘Venus is essentially identical to Venus’. More generally, an essence statement is often an English sentence of the form ‘a is essentially F’.Footnote 1

Two points of clarification are in order. First, the characterization of essentialism I give here is the most neutral and the most far-reaching I can think of, while still being precise enough for the present discussion. I am well aware that in the literature ‘essentialism’ denotes a wide variety of claims. Thus, someone running an EDA may not have in mind exactly the thesis I just stated. As long as their view implies that some things have at least some essential properties, it falls within the scope of the present paper.

Second, I take the essential properties of an object to be all the necessary properties of the object. That is, I adopt the modal account of essence: for everything, it is essentially F iff necessarily, it is F.Footnote 2 However, my commitment to the modal account is one of convenience, not one of ideology. It is well known that the modal account of essence faces important challenges (Fine, 1994). Thus, some readers may prefer the neo-aristotelian account of essence, according to which some but not all the necessary properties of an object are essential to it. Still, those neo-aristotelian essentialists agree that every essential property of an object is also a necessary property of that object. This is enough for my argument to go through. That only some necessary properties are essential properties in the restricted neo-aristotelian sense is irrelevant to my purpose.

I am now in a position to give a more detailed characterization of EDAs. Let ‘a = b’ be an identity statement with terms ‘a’ and ‘b’.Footnote 3 Let ‘a is essentially F’ be an essence statement saying of a that it has a particular essential property. Then, an EDA is an argument such that its first premise is ‘a is essentially F’, its second premise is the negation of ‘b is essentially F’ and its conclusion is the negation of ‘a = b’.

Although the claim of this paper applies to any EDA, it is convenient to rely on a particular example. For its ubiquity in the literature, I choose the classic case of the clay statue, a version of which was presented at the outset of this paper.

Let ‘Goliath’ be the name of a clay statue and ‘Lumpl’ the name of the lump of clay which constitutes the statue. By hypothesis, Goliath and Lumpl come into existence together and go out of existence together, so that they exist at exactly all the same times. Is Goliath identical to Lumpl? A proponent of the negative answer may advance Statue, an EDA with first two premises.

  1. (1)

    Goliath is essentially a statue.

  2. (2)

    It is not the case that Lumpl is essentially a statue.Footnote 4

and conclusion

  1. (3)

    Goliath ≠ Lumpl.

2.

There is a complication. As is the case for any discernibility argument, the conclusion of Statue is established by pointing out that Goliath and Lumpl do not instantiate all the same properties. However, it is not immediately clear from (1) and (2) that Goliath and Lumpl do not instantiate all the same properties. This is because it is not immediately clear what property the essence statements in (1) and (2) attribute to Goliath and Lumpl.

This difficulty is better seen by focusing on the syntax of essence statements. Alone, (1) and (2) are not enough to reach the desired conclusion, (3). How, then, can one turn Statue into a valid argument? A prima facie plausible suggestion is this: add as a third premise an instance of Indiscernibility such that its antecedent is ‘Goliath = Lumpl’ and its consequent contradicts the conjunction of (1) and (2). (3) follows by modus tollens. But here, trouble arises. Since Indiscernibility quantifies over properties, an instance of this principle will contain the name of a property. However, the essence statements in (1) and (2) do not contain the name of a property. They are English sentences of the form ‘a is essentially F’. They do not have the form ‘a instantiates c’, where ‘c’ is the name of a property. Because of this, it is not immediately clear what the relevant instance of Indiscernibility is in the case at hand.

In itself, this is nothing serious. A similar issue arises even with non-essentialist discernibility arguments. For example, consider:

Country. Switzerland is landlocked. Sweden is not landlocked. Therefore, Switzerland is not identical to Sweden.

What is the relevant instance of Indiscernibility here? Since the first and second premises do not contain the name of a property, this is not immediately clear. Still, there is an easy way out of this difficulty. ‘Switzerland is landlocked’ can be straightforwardly interpreted as saying of Switzerland that it instantiates the property being landlocked. And so, it is obvious what the relevant instance of Indiscernibility is in this case: if Switzerland is identical to Sweden, then Switzerland instantiates being landlocked iff Sweden does.

Given the success of this strategy for non-essentialist discernibility arguments, one may try to apply it to EDAs. However, this raises a further issue. Interpreting the first premise of Statue is not as straightforward as interpreting the first premise of Country. The essence statement (1) says of Goliath that it instantiates a particular essential property. But which property exactly? Contrary to what is the case with ‘Switzerland is landlocked’, there are at least two competing answers here.

On one reading, (1) says of Goliath that it instantiates the property being a statue and also convey that this property is essential to Goliath. On the other reading, the sentence says of Goliath that it instantiates a particular property different from being a statue. For lack of a better name, I label this property ‘being essentially a statue’.

In other words, (1) is ambiguous. It has at least two interpretations.

  • (4) Goliath instantiates being a statue essentially.

  • (5) Goliath instantiates being essentially a statue.

On one interpretation, (1) is equivalent with (4); on the other, it is equivalent with (5). I call the former the A-interpretation and the latter the B-interpretation.

Three points of clarification are in order. First, the distinction between these two interpretations is not the distinction between de re and de dicto necessity. Whether (1) is read as (4) or as (5), it says something of the object Goliath and not of the sentence ‘Goliath is a statue’. The relevant question here is whether (1) says of Goliath that it instantiates the property being a statue and add something about this fact, or if it says of Goliath that it instantiates a different property altogether.

Of course, the de dicto/de re distinction still looms in the background. Notice that (4) is itself ambiguous. On the de dicto reading, (4) says that the sentence ‘Goliath instantiates being a statue’ is necessary. On the de re reading, (4) says of Goliath that it instantiates the property being a statue and add that it does so essentially. Here, I simply assume that (4) is understood according to the de re reading.Footnote 5 Note also that (5) is not ambiguous in the same way as (4): since there is no modal operator in (5), the sentence is no more problematic in that regard than ‘Goliath instantiates being a statue’ is.

Second, the label ‘being essentially a statue’ is something of a convenient placeholder. It should not be taken to indicate anything about the nature of the property it denotes (assuming it denotes a property at all). Presumably, anyone committed to the truth of (5) will take being essentially a statue to be closely related with being a statue, hence the label. But nothing I say below requires that it be so.

Third, nothing I have said about the ambiguity of (1) is specific to this sentence. The same ambiguity is found in any essence statement of the form ‘a is essentially F’. In particular, the essence statement in (2) is ambiguous in the same way. Thus, I can talk about the ambiguity of essence statements in general.

This ambiguity in essence statements leads to trouble for EDAs. Consider Statue. If one wishes to add an instance of Indiscernibility to (1) and (2) in order to infer the desired conclusion (3), they must choose between the A and the B-interpretation of essence statements. But this raises the following dilemma. If essence statements are A-interpreted, the resulting argument is invalid (first horn). If essence statements are B-interpreted, (1) is false (second horn). Thus, whatever interpretation one chooses, the argument is unsound.

The first horn of the dilemma is quickly explored. If essence statements are A-interpreted, Statue will turn into Statue-A, an EDA whose premises are.

  • (4) Goliath instantiates being a statue essentially.

  • (6) It is not the case that Lumpl instantiates being a statue essentially.

  • (7) If Goliath = Lumpl, Goliath instantiates being a statue iff Lumpl does.

and whose conclusion is ‘Goliath \(\ne\) Lumpl’. Note that (4) and (6) are simply the result from reading the essence statements in (1) and (2) under the A-interpretation. (7) is the instance of Indiscernibility relevant in this case.

Statue-A is invalid. The premises are not enough to reach the desired conclusion that Goliath ≠ Lumpl. Even if it follows from (4) that Goliath instantiates being a statue, it does not follow from (6) that Lumpl fails to instantiate being a statue. (6) is only saying that if Lumpl instantiates this property at all, it does not instantiate this property essentially. (Similarly, ‘it is not the case that Socrates instantiates being a philosopher essentially’ does not tell us that Socrates fails to instantiate being a philosopher; only that Socrates fails to instantiate this property essentially.) Thus, from (4) and (6) alone, one cannot reject the consequent of (7). So, from (4), (6) and (7) alone, one cannot reject the antecedent of (7), i.e. one cannot conclude that Goliath ≠ Lumpl.

In short: under the A-interpretation, it does not follow from (1) and (2) that Goliath and Lumpl have different properties, so Indiscernibility does not apply. More generally: under the A-interpretation of essence statements, it does not follow from the first two premises of an EDA, ‘a is essentially F’ and ‘it is not the case that b is essentially F’, that a and b do not share all their properties. Thus, adding an instance of Indiscernibility will not yield the desired conclusion ‘a ≠ b’.

I now turn to the second horn. If one reads essence statements under the B-interpretation, they can run the following EDA, Statue-B. Take as premises.

  • (5) Goliath instantiates being essentially a statue.

  • (8) It is not the case that Lumpl instantiates being essentially a statue.

  • (9) If Goliath = Lumpl, Goliath instantiates being essentially a statue iff Lumpl does.

(5) and (8) result from reading the essence statements in (1) and (2) under the B-interpretation. (9) is the instance of Indiscernibility relevant in this case. Given (5) and (8), the consequent of (9) is false. Therefore, so is the antecedent of (9), i.e. Goliath ≠ Lumpl.

Although it is valid, Statue-B is unsound. This is so because (5) is false. And (5) is false because ‘being essentially a statue’ does not denote a property, so that (5) attributes to Goliath a property that it does not have. I will now spell out this claim in more details and argue in its favor.

3.

First, let me introduce a convenient terminological shorthand. Under the B-interpretation, an essence statement ‘a is essentially F’ says of a that it instantiates a property. I call such a property ‘an essentiality’. (The equivocal ‘essential property’ is already taken.) Thus, by definition, an essentiality is a property attributed to a by the essence statement ‘a is essentially F’, where this statement is B-interpreted. For example, being essentially a statue is an essentiality.

If EDAs are to have any chance of success at all, one must admit that some essence statements are true when they are B-interpreted. And so, one has to accept the existence of at least some essentialities. However, the nature of essentialities is unclear. Assuming that essentialities exist, what are they exactly? If ‘being essentially a statue’ denotes a property at all, what property does it denote exactly? This is a difficult question. Fortunately, a rough answer will be enough for my purposes.

To begin, notice that being essentially a statue is a different property from the familiar being a statue, attributed to Goliath by (4). If it were not so, (5) would itself be equivalent to ‘Goliath instantiates being a statue’. But then, (5) would not be an essence statement and it would turn out that Statue-B is not an EDA after all.

The point generalizes. Statue is not the only discernibility argument for the claim that Goliath ≠ Lumpl. Other properties beside essential ones have been proposed as the source of discernibility. These include being well made, being ugly, being insured, being admired, as well as properties of kind-membership such as being a statue and being a lump of clay.Footnote 6 Now, notice that none of these properties require talk of essence. One can perfectly well agree that Goliath but not Lumpl instantiates being ugly (say), all while denying Essentialism. So, if being essentially a statue is identical to any of these other properties, (5) is not an essence statement. Therefore, Statue-B is not an EDA and the argument falls outside the scope of the present discussion. Thus, if ‘being essentially a statue’ denotes at all, it denotes a different property from such properties as being a statue, being ugly and the like.

None of what I just said is specific to being essentially a statue. The same point applies to any essentiality. Essentialities should not to be identified with sortal, axiological, structural, representational or other standard sortalish (Bennett, 2004) properties. To deny this is to remove what makes the statement an essence statement in the first place. It turns an EDA built from that statement into a standard discernibility argument, similar to Country.

With this in mind, let us now turn to the question of whether essentialities exist. I answer ‘no’ and give two arguments to support this answer.

First, the argument from intuition. To begin, notice that it is common to appeal to intuitions in order to determine the truth value of essence statements. Intuitively, Socrates could not have been an apple pie instead of a human, although he could have been a farmer instead of a philosopher. That is: intuitively, Socrates is essentially human, but not essentially a philosopher. Such intuitions are often used to back up the premises of EDAs. This is common in the literature on material constitution. For example, consider:

it certainly looks as though there are various sorts of changes that only one of [Goliath and Lumpl] would survive. For example, only Lumpl would survive being squashed into a ball, and only Goliath would survive the loss of the bit of clay that forms his nose. (Bennett, 2004, p. 339, my italics)

This passage illustrates the following thought: intuitively, Goliath but not Lumpl has its shape essentially; and intuitively, Lumpl but not Goliath has some particular parts essentially.Footnote 7

Similarly, someone running Statue may observe that (1) and (2) are intuitively true, so that we have a reason to accept these premises. However, this line of argument loses its strength once the ambiguity of essence statements is pointed out. While intuitions may support the premises of Statue, they do not support the premises of Statue-B. Even though it may be intuitive that Goliath is essentially a statue, it is by no means intuitive that Goliath instantiates being essentially a statue. More generally: intuitions do not support the existence of essentialities.

If intuitions allow us to conclude anything at all about the truth value of (5), it is that the sentence is false. Intuitively, there is no such property as being essentially a statue. Rather, intuitions go as follows. There is a property being a statue. This is a familiar property that we attribute to individuals by using the predicate ‘…is a statue’. We attribute it to the Venus de Milo, Rodin’s The Thinker and Eriksen’s The Little Mermaid. We think it is typically had by objects having certain representational (e.g. figuring a mythical creature), aesthetic (e.g. being ugly) and material (e.g. being solid) properties. Moreover, this property is essential to some objects which instantiate it (e.g. Goliath) and not to some other (e.g. Lumpl).

In addition, independent intuitions clash with (5). Intuitively, no two objects are exactly colocated, i.e. they do not occupy exactly the same space throughout their entire existence. In particular, intuitively, if Goliath ≠ Lumpl, Goliath and Lumpl are not exactly colocated. Moreover, as I have already mentioned, it is intuitive that Goliath and Lumpl differ in their essential properties. For example, it is intuitive that Goliath but not Lumpl is essentially a statue. Now, it is often thought that theses two families of intuitions clash. According to the former intuitions, Goliath and Lumpl are identical. But according to the latter intuitions together with Indiscernibility, they are different. This is, very roughly, the problem of material constitution (Rea, 1995).

It should be clear by now that this view is mistaken. The intuitions do not clash. One can perfectly well maintain (1), (2) and Indiscernibility without having to conclude that Goliath and Lumpl are different. Just go with the A-interpretation! Once the ambiguity of essence statements is pointed out, it is easily seen that all our intuitions can be preserved. However, if essentialities exist and (5) is true, we need to reject our intuitions about exactly colocated objects. Thus, ceteris paribus, someone denying (5) is in better position to maintain our intuitions than someone accepting (5) and the existence of essentialities.

Of course, this is not enough to conclude once and for all that (5) is false. There may be independent arguments for the truth of this sentence, arguments before which our intuitions about exactly colocated objects must ultimately bow. My point here is more modest: as long as no independent argument for (5) is provided, we have a good reason to reject this sentence.

This concludes the argument from intuition. Unfortunately, I do not think the argument is a very good one. First, I am not sure that it can be expanded so as to cover every EDA. While intuitions tell us that (5) is false, this may not be the case for every B-interpreted essence statement. Second, the argument only works if intuitions (and in particular intuitions about essence statements) are a reliable guide to truth. But whether they are is a controversial matter. Since an exploration of this point is far beyond the scope of the present discussion, I prefer to take the safer route and not assume that intuitions can be trusted. Moreover, even if intuitions are a reliable guide to truth, some readers might disagree with my statement that (5) and the existence of essentialities are counterintuitive. They might on the contrary find it intuitive that there are essentialities, or have no strong intuitions one way or another. To those readers, I owe another argument.

Still, for all its flaws, the argument from intuition has some virtue. Intuitions are often used in order to determine the truth value of essence statements, especially when these statements are in the premises of an EDA. Now, suppose that someone running an EDA accept that intuitions tell us something about the truth value of (1), (2) and other essence statement of the form ‘a is essentially F’. Such a person has little reason to disagree with the claim that intuitions tell us something about the truth value of (5). On the contrary, if someone refuses to back up essence statements (or the negation thereof) with intuitions, they need to support the premises of their argument in some other way. Thus, the argument from intuition manages to put at least some pressure on the friends of EDAs.

Let us now turn to my second argument against the existence of essentialities, the argument from parsimony. This argument is of a kind familiar to philosophers. Arguments of this kind are commonly used to answer the question ‘does it exist?’ At the heart of the argument from parsimony is the methodological maxim known as Occam’s Razor: do not multiply entities without necessity! Turning this imperative into an affirmative sentence, we can get the following principle.

Parsimony. Nothing is redundant.

The predicate ‘is redundant’ is meant to capture the notion expressed by ‘without necessity’ in Occam’s Razor. Roughly, something is redundant if if it has no explanatory power.

A decent test for redundancy is this. Assume that some particular thing a exists. Take any true sentence which does not say something about a. Ask whether the assumption that a exists makes any difference in supporting the sentence. That is, ask whether someone denying that a exists have a harder time accounting for the truth of the sentence. If the answer is ‘no’, a is redundant relative to this sentence. If a is redundant relative to every sentence which does not say something of a, a is redundant.

Note that most things are redundant relative to a great many sentences. Also, note that if an object is empirically detectable, it is not redundant. For in this case, the existence of the object is a reason to maintain a sentence saying something of a particular instrument, e.g. ‘the scale reads “30 g”’, ‘I feel something smooth’, etc. (Of course, this is not to say that empirical undetectability entails redundancy.)

The argument from parsimony goes as follows. Assume, for reductio, that being essentially a statue exists. This property is redundant. For any sentence which does not say something about being essentially a statue, someone assuming that the property exists is not better off than someone denying that the property exists. This is so because instead of maintaining that something x instantiates being essentially a statue, one can always note that x instantiates being a statue and add that this property is essential to x. In other words, one can always choose the A-interpretation of ‘x is essentially a statue’ without losing any explanatory power. From Parsimony and the claim that being essentially a statue is redundant, contradiction follows.

The following may help see why being essentially a statue is redundant. Assume that being essentially a statue exists. Unsurprisingly, the property is redundant relative to a great many sentences. For example, it is obviously redundant relative to astronomical truths such as ‘Venus is a planet’. How then can we find a true sentence relative to which being essentially a statue is not redundant? A natural starting point is to look for a sentence saying something about a statue’s persistence conditions. Indeed, a well-recognized theoretical role of essence statements is to account for persistence through actual and possible changes.

Let us apply this to the case at hand. According to the passage by Bennett quoted above.

  • (10) Goliath does not survive being squashed into a ball.

is a true sentence. Does the assumption that being essentially a statue exists make any difference in supporting (10)? It does not. Of course, one can use the assumption in support of the sentence. For if being essentially a statue exists, it is instantiated by Goliath. Moreover, one can maintain that being essentially a statue puts some constraints on Goliath. In particular, it ensures that Goliath does not have the shape of a ball. All this is irrelevant however. The important point is that someone denying the existence of being essentially a statue still has the resources needed to support (10). Indeed, all they have to say is that Goliath instantiates being a statue and that this property is essential to it. This suffices to ensure that Goliath is not ball-shaped. Thus, being essentially a statue is redundant relative to (10).

Let me put this point otherwise. (10) is supported by the essence statement ‘Goliath is essentially a statue’.Footnote 8 However, because essence statements are ambiguous, this is not quite the end of the story. We still have to ask whether (10) is supported by the A or the B-interpretation of ‘Goliath is essentially a statue’. And the answer here is: it is supported by both interpretations independently. This in turn makes being essentially a statue redundant relative to (10).

The same can be said of any true sentence about Goliath persistence’s condition. In fact, the point holds of any true sentence relative to which being essentially a statue is not obviously redundant. Any such sentence is one supported by an essence statement ‘a is essentially a statue’, whether the statement is A or B-interpreted. Thus, someone denying the existence of being essentially a statue will have no more difficulty supporting the sentence than one who accepts that the property exists. So, being essentially a statue is redundant relative to this sentence.

I now consider some potential objections against the argument from parsimony and offer rejoinders.

Objection. My claim that every essentiality is redundant is insufficiently supported. Contrary to what I say, the argument from parsimony cannot be generalized so easily. Even if one agrees that being essentially a statue is redundant relative to (10), it does not follow that the property is redundant relative to any sentence. And even if one agrees that being essentially a statue is redundant, this does not suffice to show that every essentiality is redundant.

First reply. One can take the argument from parsimony as a way to reject the burden of proof on the friend of essentialities. Anyone who thinks a given essentiality exists need to come forth with a theoretical role for it. That is, they must provide a true sentence which the assumption that the essentiality exists helps support.

Second reply. There is a meta-philosophical reason to believe essentialities are redundant. The distinction between the A and B-interpretation of essence statements is largely ignored in the literature. If essentialities were not redundant, the failure to take this distinction into account would be quite surprising. However, as things stand, we can easily explain why the distinction does not play a more important role in the literature: since introducing essentialities into one’s ontology provide no theoretical benefits whatsoever, there is simply no reason to bother with them.

Objection. Contrary to what I say, there is at least one theoretical role played by essentialities: they allow to distinguish objects that could not be distinguished otherwise. For example, the assumption that being essentially a statue exists support the true sentence ‘Goliath ≠ Lumpl’.

Reply. This will not do. The truth of ‘Goliath ≠ Lumpl’ is to be established through an argument. This argument is either an EDA or it is not. If it is, the objection begs the question. If it is not, someone denying the existence of being essentially a statue will not have any more difficulty to argue for the sentence than someone assuming the property exists.

Objection. I do not offer sufficient support for Parsimony. This undermines the whole argument.

Reply. I will not try to argue in favor of Parsimony. Whether one accepts Parsimony is a methodological choice point. Still, someone who wishes to do away with Parsimony should be aware that it will be much harder or even impossible to provide an answer to many questions of the form ‘does it exist?’. In particular, it will be much harder to provide an answer to the question ‘does being essentially a statue exist?’. (This becomes even more striking if one thinks the argument from intuition also fails.) Thus, rejecting Parsimony may force us to a stalemate on this question.

Objection. The argument from parsimony only works if one grants that (i) Goliath instantiates being a statue and (ii) does so essentially. But one needs not grant this. A proponent of essentialities may claim that Goliath instantiates the special property being essentially a statue all while denying (i) or (ii). If they are right, ‘Goliath is essentially a statue’ is true under the B-interpretation but not under the A-interpretation. Thus, the A-interpretation of ‘Goliath is essentially a statue’ does not have the same explanatory power as the B-interpretation of the sentence. For example, it cannot be used to support (10). It turns out that being essentially a statue is not redundant after all.

Reply. This objection merely displaces the problem without managing to save essentialities. Indeed, what reasons could a proponent of essentialities have to deny (i) or (ii)? (i) follows from the stipulation that Goliath is a statue and the meaning of ‘is a statue’. And I cannot think of a good reason to deny (ii) that is not also a reason to deny that Goliath instantiates being essentially a statue. Until such a reason is provided, the A-interpretation is available whether the B-interpretation is, so that assuming the existence of essentialities will not make it easier to account for the truth of any sentence.

The argument from intuition and the argument from parsimony allow one to reject the existence of the essentiality being essentially a statue. Thus, the arguments allow one to reject the first premise of Statue when it is read under the B-interpretation. However, a reader may wonder whether the two arguments discussed in this section succeed allow one to reject the existence of any essentiality.Footnote 9 In particular, consider the following sentence:

  • (11) Pain instantiates being essentially painful.

This sentence results from reading the first premise of the EDA Pain under the B-interpretation. Do my arguments against essentialities allow one to reject (11)? I think they do.

Let us first turn to the argument from intuition. The following claim is intuitively true: pain instantiates being painful. In addition, the following stronger claim is also intuitively true: pain instantiates being painful essentially. But it is not intuitive that (11) is true. More generally, intuitions do not support the existence of the essentiality being essentially painful. If intuitions allow us to conclude anything at all about (11), it is that the sentence is false. Intuitively, there is no property being essentially painful – rather, there is the familiar property being painful, the property that is instantiated when one puts their hand too close from the fire or bumps their toe against the table.

However, I grant that the argument from intuition is somewhat more limited when it is applied against (11) than when it is applied against (5), i.e. against ‘Goliath instantiates being essentially a statue’. Indeed, as discussed above, some intuition clashes with (5), namely the intuition that no two different objects are exactly colocated. But there seems to be no corresponding intuition that clashes with (11). In particular, it is not counterintuitive that pain and C-fibers stimulation are different and yet exactly colocated.

Now, let us see how the argument from parsimony fares when applied against (11). Is the essentiality being essentially painful redundant according to the test I proposed? It is. For any sentence which does not say something about being essentially painful, someone assuming that the property exists is not better off than someone denying that the property exists. In particular, instead of maintaining that pain instantiates being essentially painful, one can maintain that pain instantiates being painful, and add that this property is essential to pain. In other words, one can always choose the A-interpretation of ‘pain is essentially painful without losing any explanatory power. This exactly mirrors the above reasoning regarding being essentially a statue. Thus, if the argument from parsimony is successful against (5), it will be successful against (11) as well.

More generally, the argument from parsimony can be expanded so as to cover every essentiality (and so, every B-interpreted EDA). For any essentiality whose existence is assumed, the only sentences relative to which it is not obviously redundant are sentences supported by an essence statement. But any such sentence is supported by both the A and the B-interpretation of the statement. Thus, since the A-interpretation is available, someone assuming that the essentiality exists is not better off than someone denying the essentiality exists. So, it turns out that the essentiality is redundant. From Parsimony, contradiction follows and the initial assumption has to be rejected.

This concludes my rejection of essentialities and my exploration of the dilemma’s second horn. I now review a few strategies that a proponent of EDAs may suggest in order to escape the dilemma. I show that none of these are successful.

4.

Perhaps the most obvious way to escape the dilemma is to claim that Statue and other EDAs should not use Indiscernibility but another general principle about identity.Footnote 10 What principle, though? Prima facie, the following is a plausible candidate.

Substitutivity. For every identity statement ‘a = b’, if it is true, then, if \(\varphi\) is true, \(\varphi [b//a]\) is true; where \(\varphi\) is a sentence and \(\varphi [b//a]\) results from replacing one occurrence of ‘a’ in \(\varphi\) by ‘b’.

In short: coreferring terms can be substituted salva veritate. This is of course the principle known in the literature under such labels as ‘the Substitutivity of Identicals’, ‘the Substitutivity of Coreferring Expressions’ and (somewhat confusingly) also as ‘the Indiscernibility of Identicals’ and ‘Leibniz’s Law’.

Contrary to Indiscernibility, Substitutivity does not quantify over properties. So, an instance of this principle need not contain the name of a property. Thus, one can run Statue (or whatever EDA) without having to worry about the ambiguity of essence statements. However, this strategy will not do. Substitutivity is false and so cannot be used to support its instances. That Substitutivity is false is uncontroversial and is easily shown by way of counterexamples (Quine, 1953a). For example, the following true sentences of English famously generate a counterexample to Substitutivity.

  • (12) Cicero = Tully.

  • (13) ‘Cicero’ is a six-character string.

Still, one may hope Substitutivity can be fixed. Observe that (13) does not attribute a property to the object denoted by the terms of the identity statement (12). This observation can be generalized: for any true identity statement and true sentence, if they generate a counterexample to Substitutivity, the sentence does not attribute exactly one property to the object denoted by the terms of the identity statement. Sometimes it does not attribute a property to the object at all, as is the case for (13). Sometimes it does attribute more than one property to the object, as in ‘Giorgione was so-called because of his size’.

With this in mind, one may try to restrict the substitution of coreferential terms to sentences saying of the object denoted by the terms that it instantiates a particular property. Moreover, because such a principle piggybacks on Indiscernibility, it is plausibly true. Thus, we arrive at the following suggestion.

Substitutivity*. For every identity statement ‘a = b’, if it is true, then, if \(\varphi\) is true, \(\varphi [b//a]\) is true; where \(\varphi\) is an atomic sentence saying of a that it instantiates a particular property and \(\varphi [b//a]\) results from replacing one occurrence of ‘a’ in \(\varphi\) by ‘b’.

Note that Substitutivity* does more than restrict the substitution of coreferential terms to sentences saying of the object denoted by the terms that it instantiates a particular property. It also requires that these sentences are atomic. Without this further condition, the counterexamples to Substitutivity can be easily adapted against Substitutivity*. For example, (12) and.

  • (14) ‘Cicero’ is a six-character string and Cicero is human.

generate a counterexample to Substitutivity*.

However, this added requirement brings back the very problem Substitutivity was supposed to avoid. To qualify as a sentence for which substitution is permitted, an essence statement must not only attribute a property to an object. It must also be an atomic sentence. At this point, the ambiguity in essence statements resurfaces. Indeed, an essence statement is an atomic sentence if it is B-interpreted, but not if it is A-interpreted. So, applying Substitutivity* will yield a valid EDA only if the essence statements in the argument are B-interpreted. But B-interpreted essence statements are false, as I argued in the previous section. Therefore, whether one chooses to go with the A or B-interpretation of essence statements, an EDA formed by applying Substitutivity* is unsound.

I can put the point otherwise. To avoid counterexamples to Substitutivity, one need to restrict the substitution of coreferential terms to sentences that merely attribute a property to the object denoted by the term. But A-interpreted essence statements do not merely attribute a property to the object denoted by the term. They do more than that: they say the object has the property essentially.

Another strategy to escape the dilemma is to protest that EDAs should not be formed by applying Indiscernibility, but rather by applying something like the following principle:

Indiscernibility*. For everything x and everything y, if x is identical to y, then, for every property, x instantiates it essentially iff y does.

This principle allows one to escape the dilemma by the first horn. As outlined above, it does not follow from ‘Goliath instantiates being a statue essentially’, ‘Lumpl does not instantiate being a statue essentially’ and Indiscernibility that Goliath ≠ Lumpl. However, if one instead uses Indiscernibility*, the desired conclusion can be reached from the premises. (Note that appealing to Indiscernibility* is only relevant under the A-interpretation of essence statements. It does nothing to help the friend of essentialities.)

This move will not do either. Applying Indiscernibility* does turn an EDA into a valid argument. But it also unacceptably weakens the argument. For why should we accept Indiscernibility* in the first place? No reason is provided in favor of the principle. And there is at least one reason to think the principle is false. The reasoning here is very similar to the argument from intuition against the existence of essentialities. Indiscernibility* itself is not intuitive. But if the principle is true, we must drop either our intuition that Goliath and Lumpl are not exactly colocated or our intuition that Goliath and Lumpl differ in essential properties. On the contrary, both intuitions can be preserved if we stick to Indiscernibility.

I do not claim that this alone is a reason to reject Indiscernibility*. As I mentioned above, I think one should be wary of appealing to intuitions in order to back up metaphysical claims. Still, it should put pressure on anyone wishing to form their EDA using Indiscernibility*. As far as I know, Indiscernibility is completely uncontroversial, so that one can safely assume it to be true. On the other hand, Indiscernibility* is controversial. Thus, if someone running an EDA uses the expression ‘Leibniz Law’ to denotes Indiscernibility*, it is simply methodologically inappropriate not to provide a reason in support of the principle.

Yet another strategy to escape the dilemma is to deny that the ambiguity of essence statement is as bad as I make it sounds. The thought is that if something has a property x essentially, this thing also has the higher-level property of having x essentially. Thus, if Goliath instantiates being a statue essentially, then Goliath also instantiates being essentially a statue. More generally:

Bridge. For every essence statement, if it is true under the A-interpretation, it is true under the B-interpretation.

If Bridge is true, the dilemma is avoided. Anyone agreeing that ‘Goliath instantiates being a statue essentially’ is true and ‘Lumpl instantiates being a statue essentially’ is false must also conclude that Goliath ≠ Lumpl.

It should be clear at this point that this strategy will not do either. The content of the previous section can be taken as an attack on Bridge. Since there are no essentialities, no essence statement under the B-interpretation is true. Thus, Bridge is either false (if some A-interpreted essence statement is true) or vacuously true (if no A-interpreted essence statement is true). Whatever is the case, it will not help save EDAs.

Although this strategy to escape the dilemma is clearly flawed, I mentioned it in order to dispel a potential source of confusion. Indeed, someone may think Bridge is true because of their familiarity with the resources of λ-abstraction. Roughly, λ-abstraction allows to generate predicates from (possibly complexes) sentences. For example, abstracting from the sentence ‘Socrates is bald and Socrates is human’ — formally: ‘\(Ga \& Ha\)’ —, one gets the monadic predicate ‘…is bald and human’ — formally: ‘\(\lambda x(Gx \& Hx)\)’. The resources of λ-abstraction include the following principle, which states the equivalence between a possibly complex sentence and the atomic sentence formed with the predicate abstracted from the complex sentence.

λ-Conversion. \(\left(\lambda x\varphi \right)a\leftrightarrow \varphi [a/x]\), where \(\varphi\) is a sentence and \(\varphi [a/x]\) is the result of substituting all free occurrences of ‘x’ by ‘a’ in \(\varphi\).Footnote 11

For example, λ-Conversion tells us that Socrates is bald and human iff Socrates is bald and Socrates is human.

Now, notice that an essence statement yields a complex sentence when it is A-interpreted, but yields an atomic sentence when it is B-interpreted. That is,

  • (4) Goliath instantiates being a statue essentially.

  • (5) Goliath instantiates being essentially a statue.

can be formalized respectively as

  • (15) □Ga

  • (16) Ha

where ‘G’ and ‘H’ are monadic predicates and ‘□’ is the essentialist operator. Moreover, by λ-Conversion, it turns out that (15) is equivalent to

  • (17) (λx□Gx)a

So far, so good. However, one needs to be careful here. Noting that (17) has the same form as (16), one may be tempted to see (17) itself as equivalent to (16) and conclude that Bridge is true. But this is clearly a mistake. While (16) says of Goliath that it instantiates being essentially a statue and thus commits one to the existence of essentialities, (17) does not.

The equivalence between (17) and (16) only becomes plausible under the following assumption: if a sentence of the form ‘\(\Pi a\)’ is true (where ‘\(\Pi\)’ is a monadic predicate), ‘\(\Pi\)’ expresses a single monadic property (of a). For if the predicate ‘\(\lambda x\square Gx\)’ in (17) expresses a property at all, what property could this be other than an essentiality? And which essentiality could this be other than the one expressed by ‘H’ in (16)?

However, this will not do. The assumption that every predicate expresses a property is much too strong and controversial. One can agree that Socrates is bald and human without committing oneself to the existence of a property being bald and human expressed by ‘…is bald and human’. Similarly, one can perfectly well accept that Goliath is essentially a statue without committing oneself to the existence of essentialities. (Again, this was the lesson of the previous section.) To put the point otherwise: although the assumption that every predicate expresses a property allows to establish the equivalence between (17) and (16), it undermines λ-Conversion and thus the equivalence between (15) and (17). Indeed, as stated, λ-Conversion is unproblematic. However, the added requirement that ‘\(\lambda x\varphi\)’ ranges only over predicates expressing a single property suffices to turn λ-Conversion into a very controversial principle. Thus, it is methodologically inappropriate to assume the truth of this principle and to leave matters there.

At this point, one might protest that I misunderstood the notion of properties at play in Indiscernibility. In my discussion, I have taken the question ‘what properties are there?’ to be controversial and metaphysically interesting. Moreover, my view allows that some predicates do not express a single property. Even though ‘Socrates is a bald man’ is true, it does not follow that there is a property being a bald man expressed by ‘…is a bald man’. Using a familiar terminology (Lewis, 1986, pp. 56–63): I have taken the notion of property at play in Indiscernibility to be the sparse notion rather than the abundant one. But, the objection goes, this is misguided: in Indiscernibility, ‘property’ should be understood in the abundant sense.Footnote 12 That is, one should accept that for any things, there is a property had by all and only these things.

This move allows one to escape the dilemma by the second horn: if properties are abundant, how could essentialities fail to exist? In fact, the effect of understanding Indiscernibility against the abundant notion of properties is even more dramatic than that. With the sparse notion of property in mind, a friend of the A-interpretation of essence statements may very well agree that ‘Goliath instantiates being a statue essentially’ is true and ‘Lumpl instantiates being a statue essentially’ is false, without also having to agree that Goliath and Lumpl differ in their properties. Thus, this person need not question the respective essence of Goliath and Lumpl or take an anti-essentialist stance. This is precisely where the force of the dilemma’s first horn resides. However, if properties are abundant, there is a property shared by all and only the things that are essentially a statue. So, even a friend of the A-interpretation must agree that Goliath and Lumpl differ in their properties if Goliath is essentially a statue and Lumpl is not. In other words: the abundant notion of property guarantees the equivalence between (16) and (17) without threatening λ-Conversion. Thus, it establishes Bridge and makes the dilemma irrelevant.

I reply that properties are not abundant. This reply is not ad hoc, but is warranted by the two arguments against essentialities I gave above. If properties are abundant, essentialities exist. (If this is false, to maintain that properties are abundant is of no use to escape the dilemma.) Since, as I have argued, essentialities do not exist, properties are not abundant. To put the point otherwise: if one goes with the abundant notion of properties, one must accept the existence of things (namely, essentialities) that do not exist. Therefore, the abundant notion of properties is undesirable and should be dropped. At the very least, if the soundness of EDAs relies on a commitment to abundant properties, this need to be made clear from the start.

5.

Since the main aim of the present paper was to show that EDAs are unsound, I have only discussed the ambiguity of essence statements in relation with EDAs. However, this ambiguity has at least one other important consequence beyond the unsoundness of EDAs: it undermines the classical argument for the necessity of identity.Footnote 13 The issue can be easily seen by focusing on the following version of the argument.

Suppose that α and β are identical. Then [by Indiscernibility], they share all their properties. Since one of β’s properties is that necessarily it is identical with β, this must be one of α’s properties too. So necessarily α is identical with β, and it follows that α and β cannot have been only contingently identical. (Yablo, 1987, p. 263)

Consider: ‘one of β’s property is that necessarilyFootnote 14 it is identical with β.’ As ‘Goliath is essentially a statue’, this sentence is ambiguous. According to the A-interpretation, it says of β that it instantiates being identical with β necessarily. Interpreted in that way, the sentence is presumably true. But it does not follow from it and Indiscernibility that α instantiates being identical with β necessarily, only that \(\alpha\) instantiates being identical with β. Indiscernibility is completely mute on whether a property is instantiated necessarily or not. According to the B-interpretation, the sentence says of β that it instantiates the essentiality being necessarily identical with β. Interpreted in that way, the sentence is false: essentialities do not exist, as I argued above.

In fact, even someone accepting the existence of essentialities or denying that ‘being necessarily identical with β’ expresses an essentiality if it expresses a property at all runs into trouble. The force of the classical argument for the necessity of identity is that its two premises — Indiscernibility and the necessity of self-identity — are both highly plausible. However, once the ambiguity of essence statements is pointed out, it becomes clear that the argument does not yield the desired conclusion unless one adds a controversial premise linking the two interpretations of essence statements. Consider the following version of the argument.

First, the law of the substitutivity of identity says that, for any objects x and y, if x is identical to y, then if x has a certain property F, so does y:

  • (18) \((x)(y)[(x=y)\supset \left(Fx\supset Fy\right)]\)

On the other hand, every object surely is necessarily self-identical:

  • (19) \((x)\square (x=x)\)

But:

  • (20) \((x)(y)(x=y)\supset [\square (x=x)\supset \square (x=y)]\)

is a substitution instance of (18), the substitutivity law. From (19) and (20), we can conclude that, for every x and y, if x equals y, then, it is necessary that x equals y:

  • (21) \((x)(y)((x=y)\supset \square (x=y))\).

(Kripke, 1971, pp. 135–136, numbering adjusted to my text)

The first sentence of the passage makes it clear that ‘Fx’ in (18) is to be translated as ‘x has the property F’ (and similarly for ‘Fy’). Equivalently: if ‘F’ in (18) is a predicate, it is a predicate expressing a single property. Because of this, if (20) is to be a valid substitution instance of (18), an instance ‘\(\square (a=a)\)’ of (19) must only say of a that it has a certain property, i.e. must be read according to the B-interpretation. Thus, if we use λ-abstraction to turn ‘\(\square (a=a)\)’ into an atomic sentence, the resulting predicate ‘\((\lambda x\square (a=x))\)’ must express a property.

The argument itself does not say what this property is. However, we know that it cannot be the familiar relation of identity, lest (21) is the trivial and uninteresting result that for everything x and everything y, x = y if x = y. But if this property is not the identity relation, then (21) does not deliver the expected result, namely: if the identity relation is instantiated, it is instantiated necessarily. In other words: if the argument is to yield the interesting result that identity is necessary, instances of ‘\(\square (x=y)\)’ in (21) have to be read under the A-interpretation; but this reading is not permitted, since instances of ‘\(\square (x=x)\)’ in (19) have to be read under the B-interpretation for the argument to go through.

Unsurprisingly, this difficulty is avoided if one can move freely from the B to the A-interpretation. However, none of the argument’s premises allow us to do so. And any principle linking the two interpretations is bound to be controversial, so that one cannot safely assume it to be true. In particular, since it is required that the predicate ‘\((\lambda x\square (a=x))\)’ expresses a property, λ-Conversion cannot be applied here, as discussed in the previous section.

As promised, the issue does not crucially depend on one’s view about essentialities. Indeed, the core of the issue is the ambiguity of essence statements. The difficulty arises because one cannot straightforwardly go from the A to the B-interpretation of essence statements. Thus, solving the issue requires to add as a premise a further principle — such as a strengthened version of λ-Conversion — which makes this transition possible. But the addition of such principle considerably weakens the argument, as it is much more controversial than the other two premises. Whether the argument can be salvaged from the ambiguity of essence statements remains to be seen.