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On necessary but external relations.

1. Introduction

Consider the following three theses: first, the old philosophical dogma that all necessary relations are internal; second, the core of the Humean doctrine that necessary relations reduce to logical identity; and third, the thesis that there is no necessary connection between distinct entities. Disregarding any other possible relations between these theses, one can safely maintain that if the first is false, the other two cannot be true. In what follows, I challenge the thesis that all necessary relations are internal. Therefore, if my challenge is sound, not only Humeanism, but other metaphysical systems--including different kinds of monism--would lose an essential part of the ground that supports them.

2. Moore on Internal Relations

In his classic paper in the Proceedings of the Aristotelian Society (1919/20) on internal relations, Moore begins his analysis by acknowledging the fact that for any entity, say A, that stands in any relation whatsoever, there is a corresponding relational predicate or property, [PHI], that is true of A. Thus, if A is the father of B, we have the predicate being the father of B that is true of A. Likewise, if A is to the left of B, there is the predicate being to the left of B that is true of A; and so forth. Taking this fact into account, Moore offers the following definition:
   Let [PHI] be a relational property and A a term to which it does in
   fact belong. I propose to define what is meant by saying that [PHI]
   is internal to A as meaning that from the proposition that a thing
   has not got [PHI], it follows [it is entailed] that it is other
   than A. (1)

Through a lengthy part of the paper, Moore explains that the "it follows" in the definition is not material implication, but rather something akin to C.I. Lewis's notion of entailment. So Moore's definition amounts to this; that [PHI] is internal to A means that it is impossible that anything that does not have [PHI] is A. For instance, to say that fatherhood is an internal relation holding between A and B means that it is impossible that anything that is not a father of B is A. As this is a definition of being internal, it follows that if [PHI] is necessary, then the corresponding relation is internal. So, according to Moore, all necessary relations are internal.

Nevertheless, and despite his definition, what Moore in fact argues in his paper is that all internal relations are necessary, but not that all necessary relations are internal. A few pages before his definition (pp. 47-48), Moore writes that internal relational properties are such that if they were not possessed by their terms, the terms would be necessarily different, and so, other. So, the fact that A stands in an internal relation implies that A would not have the same properties if it did not; indeed, it is impossible that anything not standing in that relation is A. Perhaps then, Moore's general formula (on page 53) does not provide a definition of internal relations or internal relational properties after all, but rather a fundamental characteristic of them, namely, that internal relations are necessary--which is what everybody agrees on. To put it differently, (M1) is clearly different from (M2) and (M3):

(M1) If R is an internal relation, then R is necessary. If Rab is internal, then (x) ([logical not] Rxb) entails (x [not equal to] a).

(M2) If R is a necessary relation, then R is internal. If (x) ([logical not] Rxb) entails (x [not equal to] a), then Rab is internal.

(M3) If R is an external relation, then R is not necessary. If Rab is external, then it is not the case that (x) ([logical not] Rxb) entails (x [not equal to] a).

Below each thesis I have added the way that Moore would write it out, using the notion of entailment. It is clear then that Moore accepts--and it seems more than reasonable to agree with him--that if R is necessary, then not standing in R entails not being identical to whatever stands in R. That is, necessity immediately reflects on identity: the loss of a necessary relation/ property is a loss of identity.

As I say, Moore does not justify (M2) or (M3). However, he does justify (M1) by means of a more fuzz y definition, which works by analogy. That fuzzy definition shows, I think, where and why Moore went wrong in concluding (M2) and (M3) when defending (Ml); i.e., in concluding that only internal, not external, relations are necessary.
   The definition is: To say of a given relational property [PHI] that
   it modifies or is internal to a given term A which possesses it, is
   to say that from the proposition that a thing has not got [PHI] it
   follows that that thing is different from [and other than] A. In
   other words, it is to say that the property of not possessing
   [PHI], and the property of being different from [i.e., other than]
   A are related to one another in the peculiar way in which the
   property of being a right-angled triangle is related to that of
   being a triangle, or that of being red to that of being coloured.

The first part of the quote is once again Moore's definition of internal relations as necessary; relations such that not possessing the corresponding property means a loss of identity. However, the second part of the quote explains the relation between not possessing the relational property and being other than whatever does indeed possess it. That is, this second part explains that the relation between not possessing the relational property necessarily implies a loss of identity; i.e., being other.

Moreover, Moore says that the reason why anything that does not have [PHI] would not be A, where A does have [PHI], is that having [PHI] and the identity of A are related in the same way as being a right-angled triangle is related to being a triangle, or in the same way as that in which being red is related to being coloured. In other (my) words: the reason why internal relations are necessary is that internal relations belong to the nature of their terms.

That internal relations belong to the nature of their terms means not only that the terms could not possibly be the same terms without the relation, but also that they are related in the same way as being red is related to being coloured. There is something in the nature of the relata, in the essence or the "what" of them--to use the Aristotelian expression--that includes the other term. If [PHI] is internal to A, then being A includes possessing [PHI]. This is, I believe, why it is so tempting to think that internal relations reduce to the nature of their terms. We could call the corresponding properties (internal properties, or internal relational properties) essential or definitional properties, for internal properties are part of the essence of whatever possesses them. So, in consequence, anything that does not have a property of this type cannot be what has it.

Not possessing an essential or definitional property of some thing (A) could be one reason for not being A--and a very fundamental one indeed! However, although not being what a thing is could be one reason, pace Moore, it need not be the only reason why any thing that does not possess a given property/relation is not identical to the entity that possesses it: perhaps there are other reasons. And, of course, if there are other reasons, essence is not identity.

3. Necessary Relations that Do Not Belong to Essence

Are there necessary relations that do not belong to the essence of their relata? Are there necessary relations that are not like the one that holds between red and being coloured? Are there relations that, even if they do not belong to the essence of their relata, are such that anything that does not possess the corresponding property would not be identical to whatever possesses it?

Yes, there are! In fact, even if they are not explicitly acknowledged as such, some exemplars of these are already common in the literature. For instance, the relation between an entity and its material origin, if it has one, is external but necessary--that is, if one agrees with Kripke that material origin is necessary. If A has origin B, it is necessary that A has origin B. Necessarily, any x such that it does not have material origin B is not A. It would be impossible that, say, the very same table were made from a different chunk of matter: being made from a specific chunk of matter is a necessary relational property of a particular table. (3)

Yet the entity and its material origin are distinct things. The same table could go on existing after all its material parts had been substituted for new ones over the years. As Kripke explicitly acknowledges: "The question whether the table could have changed into ice is irrelevant here. The question whether the table could originally have been made of anything other than wood is relevant." (4) Kripke leaves open the question of whether the table could change into ice. If, however, it is a possibility that entities can go on existing when all their original material has been replaced, then their origin is not one of their essential properties, because no thing can go on existing without its essence, i.e., without being what it is. Thus the relation between an entity and its material origin is external. But it is also necessary, as we have just seen.

Consideration of this example, together with the previous discussion on internal relations, leads to the general thesis that identity differs from essence. Not having its particular material origin would make it a distinct table. Having its particular material origin, however, is not part of the essence of a specific table. So the identity of the table is other than its essence. Similarly, being a colour is part of the essence of red; but the identity of red is also something other than its essence. Without its essence, red would not be what it is; neither would anything that is not a colour be identical to red. Being what it is though, having the nature that it has, is not the same as being this particular thing instead of that other. Difference of essence implies otherness; but the converse does not follow. Moreover, identity is a necessary relation between any entity and itself; but essence is a form: an internal nature and the real definition of things.

In passing, I should say that I disagree with Fine on his conclusion that necessity follows from essence. (5) I think that Moore is right on this: if necessarily A is P, then anything that is not P is not A--where identity and diversity are understood as necessary relations. Now, Moore is wrong in identifying necessity with essence; for it can be true that necessarily A is P, but P is not internal/definitional to A. Fine offers his (already classic) example to support the claim that necessity and essence should be distinguished: it is necessary that Socrates belongs to the singleton Socrates, but Socrates's belonging to the singleton Socrates is not an essential property of Socrates. His example shows, I think, that internality (essence) is not identity. However, Fine reads his example differently. He wants to maintain that essence is identity and that necessity follows from essence. I disagree. For, if it is necessary that Socrates belongs to the singleton Socrates, then anything that does not belong to the singleton Socrates is not Socrates--where identity and otherness are necessary relations. The conclusion should then be that the essence of Socrates and the identity of Socrates are distinct. So, not every necessity follows from essence. Rather, necessity immediately reflects on identity. For the loss of any necessary property/relation is a loss of identity: if A does not belong to the singleton Socrates, then A is not Socrates. But not every necessary property/relation is essential: Socrates's belonging to the singleton Socrates is not an essential property of Socrates, as has already been said. So there are necessary properties that do not belong to the essence of their entities; that is, there are necessary relations that are external.

Gorman also offers a definition of essence where, as I understand him, an essential property is a necessary property that is not external. (6) Gorman's account is more complex than I need recount here, as it includes a distinction between characteristics and features that I do not need, and he introduces a generic external and necessary relation he calls 'explanation'--but I do not see why there should be only one kind of necessary and external relation. His own wording is: "F is essential to x just in case F is (i) a characteristic of x and (ii) not explained by any other characteristic of x." I mention his account here because Gorman also defends the existence of external but necessary relations, even if his aim--unlike mine--is to provide a (negative) definition of essence. In contrast, I think it is important to acknowledge that one could sustain the view that entities stand in necessary relations but deny a world of essences (or accept that the only things that have essences are properties and relations themselves).

Returning to the general point, all relations between a necessary property and its bearer, when the property is not an essential property, are necessary but external to that bearer. For instance, as in the medieval example, it might be necessary that all human beings can laugh. That is, it might be true that any thing that cannot laugh is other than a human being. And yet it can also be maintained that the ability to laugh is not part of the essence of human beings. Again, using another classic: it might be necessary that every entity with a heart also has at least one kidney (because, let us imagine there is some necessary connection between the biological structure of the kidney and the biological structure of the heart). But having a kidney is not internally related to having a heart: having a heart does not belong to the essence of kidneyed entities. So having a kidney could be necessarily but externally related to having a heart.

Equally, it might be necessary that salt dissolves in water, as Bird has maintained. (7) So, anything that does not dissolve in water is not salt. But it is not an essential property of salt that it dissolves in water. It is not part of the what that salt is that it dissolves in water, even if its chemical structure is such that salt necessarily behaves in this way, under certain conditions.

Another example would be the relation between a trope and its bearer--in theories such as those of Moreland, Mertz or Molnar, in which tropes are individuated by their bearers. (8) If the individual white of this page is the individual it is because it is the white of this page, then the white of any entity other than this page is another trope. Thus the relation between the white trope and this page is necessary. But belonging to this page is not an essential property of the trope; it is not part of what the trope is that it belongs to this page.

Singular causality, in realist theories such as that proposed by Ducasse, (9) is also a good candidate for being a necessary but external type of relation. Perhaps, if A causes B, then any thing that does not cause B (or a B-indiscernible) is not A. If having a different effect makes it another cause, causality is a necessary relation. But it could simultaneously be maintained that causing B (or a B-indiscernible) is not an essential property of A: it is not part of the real definition of A that it causes B-indiscernibles. So the relation between A and B is not internal. So, again, causal relations do not belong to the nature of the relata. (Thus, assuming that properties are the proper ontology for causation, causal theories of properties such as Shoemaker's and others result in error: (10) neither do causal features of properties

--i.e., the contributions of the properties to a causal fact--belong to the essence of properties, nor are properties individuated by their causal features.)

My final example is just a daring suggestion. There is an long-standing difficulty concerning the relation between matter and form that Aristotelian scholars deal with when trying to achieve a consistent reading of Metaphysics Z, which perhaps could be solved if the matter-form relation were understood as being external but necessary. (11) The problem is that Aristotle seems to make these two apparently incompatible claims:

(A1) Forms are necessarily en-mattered.

(A2) Forms are, by definition, matter-independent. (12)

Now, if one allows that necessary relations can be external, then one could accept (A1) and say that form and matter are necessarily related even if, given (A2), they are not internally related. Form is essentially distinct from matter. And then, even if form must occur in matter, matter would not be part of the nature of sensible substances, under the supposition that essence is form; but at the same time, any sensible substance with matter other than its own would be another substance. (13)

I cannot finish without a word about a generalized worry concerning necessary but external relations. The worry is that externalism precludes any explanation of necessity. For, being external to its terms, the necessity in the relation cannot derive from the nature of them. As Fales wonders in the case of causation, if causal relations are not simply a consequence of the natures of their terms, then what determines that there are these causal connections and not others? (14)

There is, however, a clear-cut answer available to this worry, which is that these are causal connections, and the "others" are not. (15) Causality, the relation itself, explains the necessity in causal connections; and one is allowed to maintain this precisely because, being external, causality is a real relation. To complain that there are no necessary and external relations because their necessity should be grounded in the natures of their terms is only to beg the question.

4. Conclusions

I have argued for the existence of necessary properties/relations which do not belong to the essence of their entities; that is, I have argued for the existence of necessary relations which also are external relations. The thesis and deeply rooted fundamental metaphysical dogma that all necessary relations are internal is, consequently, false. (16)


Aristotle (1984), Metaphysics, in The Complete Works of Aristotle: The Revised Oxford Translation, Vol. 2. J. Barnes (ed.). Princeton, NJ: Princeton University Press.

Bird, A. (2001), "Necessarily, Salt Dissolves in Water," Analysis 61(4): 267-274.

Bird, A. (2005), "The Dispositionalist Conception of Laws," Foundations of Science 10: 353-370.

Cartwright, N. (1989), Nature's Capacities and Their Measurement. Oxford: Oxford University Press.

Ducasse, J. C. (1924), Causation and Types of Necessity. Seattle, WA: University of Washington Publications in the Social Sciences.

Ellis, B. (2001), Scientific Essentialism. Cambridge: Cambridge University Press.

Fales, E. (1993), "Are Causal Laws Contingent?," in J. Bacon, K. Campbell, and L. Reinhardt (eds.), Ontology, Causality and Mind: Essays in Honour of D. M. Armstrong. Cambridge: Cambridge University Press, 121-51.

Fine, K. (1994), "Essence and Modality," Philosophical Perspectives 8: 1-16.

Galluzo, G., and M. Mariani (2006), Aristotle's Metaphysics Book Z: The Contemporary Debate. Pisa: Edizioni della Normale.

Garcia-Encinas, M.J. (2003) "A posteriori Necessity in Singular Causation and the Humean Argument," Dialectica 57(1): 41-55.

Gorman, M. (2005), "The Essential and the Accidental," Ratio 18(3): 276-289.

Kripke, S. (1977), "Identity and Necessity," in S. P. Schwartz (ed.), Naming, Necessity, and Natural Kinds. Ithaca, NY: Cornell University Press, 66-101.

Kripke, S. (1980), Naming and Necessity. Cambridge MA: Harvard University Press.

Mertz, D. W. (1996), Moderate Realism and its Logic. New Haven, CT: Yale University Press.

Molnar, G. (2003), Powers: a Study in Metaphysics. Oxford: Oxford University Press.

Moore, G. E. (1919/20), "External and Internal Relations," Proceedings of the Aristotelian Society 20: 40-62.

Moreland, J. P. (1985), Universals, Qualities, and Quality-Instances. Boston, MA: University Press of America.

Shoemaker, S. (1998), "Causal and Metaphysical Necessity," Pacific Philosophical Quarterly 79: 59-77


(1.) Moore 1919/20, 53. My italics.

(2.) Moore 1919/20, 48. My italics.

(3.) Kripke 1971; Kripke 1980.

(4.) Kripke 1980, 114-5, n. 57.

(5.) Fine 1994.

(6.) Gorman 2005, 284.

(7.) Bird 2001.

(8.) Moreland 1985; Mertz 1996; Molnar 2003.

(9.) Ducasse 1924. See also Garcia-Encinas 2003.

(10.) Shoemaker 1998; Cartwright 1989; Ellis 2001; Bird 2005.

(11.) I thought of this in 2008 while attending to a talk in Geneva given by Peramatzis on the same problem, although his proposal was markedly different. I also thank Allan Silverman for helping me with these complex matters.

(12.) Everybody agrees that Aristotle maintained (A1). The following is a text that sustains (A2): "As regards that which is compounded out of something so that the whole is one not like a heap, however, but like a syllable--the syllable is not its elements, ba is not the same as b and a, nor is flesh fire and earth; for when they are dissolved the wholes, i.e., the flesh and the syllable, no longer exist, but the elements of the syllable exist, and so do fire and earth. The syllable, then, is something--not only its elements (the vowel and the consonant) but also something else; and the flesh is not only fire and earth or the hot and the cold, but also something else. (...) But it would seem that this is something, and not an element, and that it is the cause which makes this thing flesh and that a syllable." (Metaphysics Z 17, 1041b11-28.) Now, this alternative text could, perhaps, be brandished as an argument against (A2): "people look for a unifying formula, and a difference, between potentiality and actuality. But, as has been said, the proximate matter and the form are one and the same thing, the one potentially, the other actually." (Metaphysics H 6, 1045b18.) Yet I say 'perhaps' because, perhaps, the "one and the same thing" in this last quote is the whole composite of matter and form, so matter and form are both together this composite whole; the former potentially, the latter, actually. See, for instance, Galluzo & Mariani 2006, 119.

(13.) I think that this would give a straightforward reading of the famous passage: "And when we have the whole, such and such a form in this flesh and in these bones, this is Callias or Socrates; and they are different in virtue of their matter (for that is different), but the same in form; for their form is indivisible." (Metaphysics Z 8, 1034a6-8.)

(14.) Fales 1993, 140. However, despite this amazement, Fales also suggests that nomological relations between universals should be understood as necessary but external relations.

(15.) In personal communication, Thaddeus Robinson pointed out a similar worry to me. He claims that in all worlds in which c and e exist, it will appear as if c produces e. For, if in fact there is nothing about c that makes it true that c causes e, all we have is the appearance of ontological production. However, my claim is that if c causes e, then there is a real relation--a causal relation--that holds between them. Necessity is an aspect of the relation itself, and this is shown by the fact that c would lose its identity if it did not stand in its causal relation to an external e, even if no definitional aspects of c and e are involved.

(16.) This paper was written thanks, in part, to financial help from the project FFI2011-29834-C03-02, sponsored by the Spanish Ministry of Education. I am very grateful for comments on previous drafts and parts of this paper to Allan Silverman and Thaddeus Robinson (as already mentioned) as well as to those who attended different meetings held in Granada and Chapel Hill.


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Author:Garcia-Encinas, M.J.
Publication:Review of Contemporary Philosophy
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Date:Jan 1, 2013
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