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An Argument Against Naturalism from Abstract Objects

Some naturalists, like Quine, feel compelled to admit abstract objects, like numbers, sets, and propositions, into their ontology. But I’ve always had the sense that abstract objects are incompatible with naturalism. Here I will lay out some premises about abstract objects and naturalism that appear fairly intuitive to me. I will then represent those premises in logical notation, and demonstrate that they do, indeed, serve as a defeater for naturalism.

1. If a thing1 is natural then it is possibly not the case that there exists a thing2 where thing2 is natural and identical to thing1.

In other words, if a think is natural, then it possibly doesn’t exist.

2. There exists some proposition such that necessarily that proposition is true.

For example, the mathematical proposition ‘2 + 3 = 5’ is necessarily true and cannot be false.

3. For all propositions, necessarily, if a proposition is true, then there exists some thing1 such that thing1 is abstract and thing1 is identical to that proposition.

This is just to say that it is necessary that if a proposition is true, then it is an existing abstract object. After all, it would seem odd to predicate a truth-value to a proposition, but deny that said proposition doesn’t exist.

4. If there exists a proposition that is necessarily true, and everything is natural, then for all proposition, necessarily, if there exists a thing1 that is an abstract object identical to that proposition, then there exists a natural thing2 identical to that proposition.

I defend this premise on the grounds that, if a proposition is natural, then in every world where it obtains, it obtains as a natural proposition. That is, if ‘natural’ is predicated of a proposition, it is essentially predicated of it, which is to say that in every world where that proposition exists, it exists as a natural object.

From these four premise, we can conclude that naturalism, which I take to be the claim that everything is natural, is false.

The deduction is as follows:

Nx – x is natural
Tp – p is true
Ax – x is an abstract object

1. (∀x){Nx ⊃ ♢~(∃y)[Ny & (y=x)]} (premise)
2. (∃p)☐Tp (premise)
3. (∀p)☐[(Tp ⊃ (∃x)(Ax & (x=p))] (premise)
4. (∃p)(☐Tp & (∀x)Nx) ⊃ (∀p)☐{(∃x)[Ax & (x=p)] ⊃ (∃y)[Ny & (y=p)]} (premise)
5. (∀p)☐{(∃x)[Ax & (x=p)] ⊃ (∃y)[Ny & (y=p)]} (IP)
6. ☐Tu (2 EI)
7. ☐[(Tu ⊃ (∃x)(Ax & (x=u))] (3 UI)
8. ☐(∃x)(Ax & (x=u) (6,7 MMP)
9. ☐{(∃x)[Ax & (x=u)] ⊃ (∃y)[Ny & (y=u)]} (5, UI)
10. ☐(∃y)[Ny & (y=u)] (8,9 MMP)
11. (∃y)[Ny & (y=u)] (10 NE)
12. Nv & (v=u) (11 EI)
13. Nv ⊃ ♢~(∃y)[Ny & (y=v)] (1 UI)
14. Nv (12 Simp)
15. ♢~(∃y)[Ny & (y=v)] (13,14 MP)
16. (v=u) (12 Simp)
17. ♢~(∃y)[Ny & (y=u)] (15,16 ID)
18. ~☐(∃y)[Ny & (y=u)] (17 MN)
19. ☐(∃y)[Ny & (y=u)] & ~☐(∃y)[Ny & (y=u)] (10,18 Conj)
20. ~(∀p)☐{(∃x)[Ax & (x=p)] ⊃ (∃y)[Ny & (y=p)]} (IP 5-19)
21. ~(∃p)(☐Tp & (∀x)Nx) (4,20 MT)
22. (∀p)~(☐Tp & (∀x)Nx) (21, QN)
23. ☐Tu (2 EI)
24. ~(☐Tu & (∀x)Nx) (22 UI)
25. ~☐Tu ∨ ~(∀x)Nx (24 DeM)
26. ~~☐Tu (23 DN)
27. ~(∀x)Nx (25,26 DS)
28. (∃x)~Nx (27 QN)

Line 28 is our conclusion, namely that something exists that is not natural. I take this to be incompatible with naturalism. Therefore, I take the existence of abstract objects, like propositions, to be a defeater for naturalism. I suspect that the naturalist will take issue with one or more of the premises, but at a cost. Likely (4) will require the most defending. Again, (4) says that, given naturalism and the existence of necessary truths, it is necessarily the case that if a proposition exists as an abstract object, it will exist as a natural object. If one denies this, then it would seem possible that an abstract object be natural and possibly not natural. But then in what way is it the same sort of thing? It seems odd to me that a proposition is a natural thing in this world, but a non-natural thing in other possible worlds. For it seems to me that the property of being natural is an essential property. If something is natural, it is necessary that it is natural. Thus, if naturalism is true, then all abstract objects are natural and essentially natural. But our argument shows, by indirect proof, that it is possible for there to exist an abstract object that is not natural. Giving up on (4) entails that ‘natural’ is non-essential to some things, and I find that to be implausible.

(My thanks to Skepticism First on Twitter, who dialogued with me on this argument and pushed me in some new directions)

Thoughts on David Charles and Aristotle regarding Scientific Definitions

According to David Charles, one difference between Aristotle and modern essentialists is what he calls the Existence Assumption (EA), which states:

If one understands the term ‘water’, one must therein know that the kind has instances (Charles 2000, 16).

It is the modern essentialist who believes that the determining factor of any necessary part of the sense of a term is the referent, and so knowledge of the terms instances is required even in the earliest stages of scientific investigation.

For, one cannot understand ‘water’ without knowing that water is in fact instantiated by certain specific examples (Charles 2000, 17).

Charles contrasts this with Aristotle, whom he claims rejects EA.  That is, one need not have knowledge either of the instances of a kind or whether there are instances of a kind to know the sense of a term.  In fact, Charles goes so far as to say that Aristotle faces a challenge, “…of showing that one can determinately grasp natural-kind terms without knowing (even derivatively) of some cases that they are instances of that kind” (Charles 2000, 18).  But why think Aristotle should be burdened with such a challenge at all?  Consider, for instance, the goat-stag, of which he says  “…you may know what the account or the name signifies when I say goat-stag, but it is impossible to know what a goat-stag is” (Aristotle 1985, 152; Post. An. 92b4).  I’ve always taken this to mean that one cannot give an essential definition to a goat-stag since it lacks a referent.   One can provide a sense to the term “goat-stag”, but the resulting definition will not be anything more than a nominal definition.  And Aristotle is quite clear that definition alone does a demonstration make.  In other words, a definition does not necessarily provide the what-it-is of a thing, nor proves whether there is such a thing.  Thus:

For definitions do not in addition make clear either that what is said is possible, or that it is that of which they say they are definitions, but it is always possible to say ‘Why?’ If, therefore, the definer proves either what a thing is or what its name signifies, then if a definition has nothing at all to do with what a thing is, it will be an account signifying the same as a name. But that is absurd. For, first, there would be definitions even of non-substances, and of things that are not–or one can signify even things that are not (Aristotle 1985, 153; Post. An. 92b23-29).

Charles picks up on this and notes that this seems to suggest that, for Aristotle, definitions are not restricted objects and kinds, but can extend to non-existents as well (Charles 2000, 28).   Key to understanding Charles will be his three stages of enquiry (Charles 2000, 24):

Stage I: This stage is achieved when one knows an account of what a name or another name-like expression signifies (section [A]: 93b30-2).

Stage II: This stage is achieved when one knows that what is signified by a name or name-like expression exists (section [B]:93b32).

Stage III: This stage is achieved when one knows the essence of the object/kind signified by a name or name-like expression (section [B]: 93b32-3).

Charles offers three interpretations of Aristotle based on four questions (Charles 2000, 30):

(A) Does Aristotle accept the three-stage view?  Does he think that every case of scientific enquiry involves a first stage where one need not know of the existence of the kind, but must know an account of what the name signifies?

(B) Are any Stage I accounts identical in content (in Aristotle’s view) with any definition of a kind?

(C) Does Aristotle think that if one grasps an account which is in fact definitional one knows that it is so?  Is the definitional nature of that account transparent to the person who grasps it?

(D) Are all accounts of what names signify regarded by Aristotle as definitional in some way?

Defending what he calls the “Liberal” view, Charles answers in the affirmative to (A), (C), and (D), but denies (B).  He also develops to “Restrictive Views”: Restrictive View 1 answers affirmatively to (A) and (B) , but in the negative to (C) and (D), and Restrictive View 2  answers in the affirmative to (B) and (C), but negatively to (A) and (D) (Charles 200, 30).

I am inclined to accept (A) and (D) as true.  I find a progressive development of signification to be plausibly found in Aristotle, and certainly the above quotes suggest that even the goat-stag account is definitional in some way, i.e. what some may call a nominal definition.

Charles argues in favor of (C) by saying:

…[D]efinitions are accounts which reveal what something is (B.3, 91aI) and, thus, make its nature known to us (B.3, 90B16).  Thus, if we grasp an account which makes known to us the nature of something, we grasp its definition.  There is no more to grasping a definition than grasping a knowledge-giving/revealing account of this type.  If so, there can be no case in which we grasp an account which makes a kind’s nature known to us but do not grasp its definition (Charles 2000, 30; fn14).

He entertains two ways to avoid this argument.  The first is to say that we might not label a particular account a definition, but Charles thinks this is a mere semantic objection and discounts it.  I think this is unduly rejected, since the issue at stake is epistemic transparency for all who grasp accounts.  If semantics are sufficient to render the matter opaque, we must deny (C).  The second objection is even more troubling, that we might know the nature of a thing, but not know that we know.  Charles points out that Aristotle defines definitions as making things known to us (Post. An. 90b16).  Definitions need not make us know that we know!  However, there does seem to be a third objection that Charles appears to side-step, namely that his own argument only appears to apply to Stage II-III accounts, i.e. to accounts of extents that have natures to be revealed.  But if Stage I is an account, there is no reason to think (C) would hold.

I also have some concerns about (B), and Charles notes that his entire book is fundamentally a defense of affirming (A) and denying (B)–(C) and (D) being related but not central issues.  The denial of (B) seems too strong.  For it suggests that no Stage I account could even happen to be identical in content with any definition of a kind.  But suppose I were to come upon a magic lamp, and relate to a magical genie my Stage I account of a goat-stag saying, “Genie, I wish the content of my definition to be identical to a definition of a kind that has instances.”  Surely my nearly omnipotent genie could satisfy this simple request with a quick nod and blink… that is, unless Charles is somehow equivocating on “content”, i.e. that in a Stage I account the content is somehow merely verbal, whereas the content of, say, an essential definition, has another mysterious ontological status.  If that is so, then I am not really sure I am clear on what he means by “definition” and whether he is using it univocally throughout his stages.

He never labeled the position of taking (A), (B), and (D) as true, and (C) as false.  And this seems to be my position, as I am struggling to accept (C).  Now it seems possible that (D) is true, but that not everyone accepts (D) (as is evidenced by the two restrictive views above) and that such a person might see the definitional nature of a particular account as opaque.  But it seems to me that such a person would only be so insistent if she were only to admit Stage III accounts as definitions.  And one might imagine that such a person might hear a Stage III account, say from a relevant, reputable, and authoritative scientist without recognizing it as an actual Stage III account.  Thus, such a person might not accept that it is definitional, even when it fits her restricted notion of what may count as definitional in fact.  Take, for instance, the Young Earth Creationist (YEC) who also holds to a restricted view of Aristotle’s Prior Analytics B.8-10–a coincidence of views one might imagine is not uncommon among the home-schooled.   She grasps the definition of some missing link between humans and the great apes, but may deny that there were any such instances.  The YEC might be able to recite the “definition” with the same precision as the Darwinian scientist, though it is the scientist’s “definition”, not hers.  The YEC thinks homo heidelbergensis is none more real than the goat-stag.  And so Charles’ argument in defense of (C) would fail on the YEC, since the YEC is unaware of any revealing nature of the homo heidelbergensis by which she might be made aware that she grasps a definition and knows it to be definitional.  And while I may affirm (D), that both the YEC and the scientist use the name “homo heidelbergensis” definitionally, our YEC denies (D) on the grounds, say, that an account requires instances in order to be definition.  This, of course, isn’t a defeater for (D) as much as it helps explain why she might not know that she has a definitional account even when she clearly is aware that the name has a sense.

Given all of this, I am still not clear on why Aristotle is said to reject EA.  For it seems that EA applies to any definitional account that rises above Stage I.  It is, after all, the essentialists assumption.  And it’s evident from the above quotes from Aristotle that goat-stags and other non-substances are not defined essentially–as they lack essences.  EA is an assumption that only underlies Stage II-III, to put it in Charles’ terms.  Others would say that, on Aristotle, chimeras can only hope for nominal definitions and those definitions carry no existential import and cannot be used in demonstration.  It seems, then, that Charles may be pushing a distinction without a difference.  His point seems to be that Aristotle does not require instances in definitions, so long as the definitions are non-essential. Is there a modern essentialist who would disagree with that?  Would Kripke really hold that “goat-stag” rigidly designates across possible worlds?


Aristotle. 1985. “Posterior Analytics“. In The Complete Works of Aristotle, vol. I. Ed. J. Barnes. New York: Oxford University Press

Charles, D. 2000. Aristotle On Meaning and Essence.  New York: Oxford University Press