The Incompleteness of the World and Its Consequences
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In the recent literature we find various arguments against the possibility of absolutely general quantification. Far from being merely a technical question in the philosophy of logic, the impossibility of absolutely general quantification (if established) would have severe consequence for ontology, for it would imply the non-existence of the world as traditionally conceived. This paper will investigate these implications for ontology and consider some possible ways of addressing them.
1 Realism and Anti-Realism, Local and Global
It is often argued that anti-realist positions can only be made sense of as local anti-realist positions. You can be an anti-realist about numbers or moral norms if you hold that there really are no such things, but that we mistakenly identify or hypostasize other things to fit their role. We might argue that when speaking about numbers we really only mean facts about marks on paper, and when we speak about the moral and immoral we in fact, like Hemingway, only mean what we feel good or bad about afterwards. But, it is claimed, you cannot hold such a position tout court, for there has to be something all these mistakenly identified things are correctly identified as. Some things must exist absolutely for other things to exist in a merely derived manner. The only feasible ontological position therefore is a general realism, possibly supplemented by local anti-realisms regarding specific areas (such as mathematics, ethics, and so forth).
On the other hand, one might also argue that inconsistency does not just result from taking anti-realism to be a global theory, but also from understanding realism in this way. The only consistent philosophical positions, it is argued, is to be a local realist. This position is based on worries about the consistency of the notion of the world, or the universe, understood as the collection on absolutely all there is. If these worries are justified, one would then be allowed to make local existence claims, saying that all numbers or all moral norms existed, since these claims are always restricted by the relevant sortals. But one would not be allowed to speak in an unrestricted manner, using statements that required speaking about absolutely everything (such as saying that everything exists is material, or that it is all mental).
Yet this argument from incompleteness against global realism is not anything that would be of great help to the defender of global anti-realism, since any way of spelling out such a position in a consistent manner (such as claiming that everything is mind dependent, a social construction, and so forth) has to make the very kind of universal claims that have just been criticized as problematic. If we cannot talk about everything, we cannot say that we should be anti-realists about everything either.
In this essay, I want to look in more detail at this argument from incompleteness, investigate how it might be supported, and focus especially on its ontological consequences which turn out to be more far-reaching than is traditionally assumed.
2 The Argument from Incompleteness
Philosophers (and ontologists in particular) frequently make reference to everything when elaborating their theories of the world. Eliminative materialists argue that all that exists is material, dualists claim that there are precisely two kinds of substances, the material and the mental, and that everything belongs to either kind, absolute idealists say that only mental things exists, solipsists claim that only one mental thing, one’s own self exists.
Yet such references are more problematic than they appear at first sight. When we talk about all men, we interpret this in terms of a domain, which is some sort of collection of all men, but when we speak about the collection of all things (the world, the universe) we cannot similarly understand this as acquiring its meaning by talking about a domain of all things, since this collection constitutes an additional item not included in the original collection. If we want to make our talk about everything exhaustive we have to enlarge the domain to include this collection, but then we need a larger domain to explicate this more comprehensive meaning of “all things”, and this process will go on indefinitely. We never seem to be able to talk about absolutely everything.
Three sources are commonly identified as the cause for difficulties with absolute generality. The first two of these derive from results in mathematical logic. First of all appeals to collections of absolutely everything are likely to lead contradictions along the lines of Cantor’s or Russell’s paradox. Secondly, absolute generality is afflicted by a specific kind of indeterminacy fundamentally due to the Löwenheim–Skolem theorem. Finally, conceptual relativity is identified as a source of problems for quantifying over absolutely everything. If all our cognition is relative to some conceptual scheme, whenever we think we have collected together all objects we have not included the scheme that lets us distinguish the objects in the first place. Absolute generality requires a perspective outside of any scheme which is precisely something the conceptual relativist does not allow.
Philosophical opinion on whether all (or any) of these three considerations provide sufficient evidence for rejecting quantification about absolutely everything is divided. In a recent landmark volume dedicated exclusively to this question (Rayo and Uzquiano 2006), about half of the contributors argued in favour and half against it.1
My aim in the following pages is not to present new possible fixes for the problems resulting from quantification over absolutely everything or to argue for fresh considerations of why such fixes do not work. I rather want to explore the consequences resulting from the view that absolute generality leads to contradictions, that cures for these are generally worse than the disease, and that we should therefore forego engaging in such quantification.
3 An Argument from Cantor’s Theorem
Some of the most obvious ways of deriving a contradiction from the idea that we can generalize over absolutely everything are based on Cantor’s theorem. This allows us to produce a fairly simple argument for the incompleteness of the world, at least as long as we assume that the kind of “collection” the world constitutes is sufficiently set-like. As soon as you have put together all the things in the world in order to form the universal collection you realize that there is one more thing: the collection of all the subcollections one can form by grouping together the different things in the universal collection in different ways. This “power-collection” will be strictly larger than the original one and therefore cannot already be a part of the universal collection.2 We can, of course, enlarge the universal collection by adding the “power-collection” to it, but then there will be another “power-collection” for this extended one, and so on. You are never able to bring everything together in one big collection.
4 Avoiding the Cantorian Problem
There are various obvious ways of avoiding this conclusion, such as denying that the universal collection is sufficiently set like for the Cantorian result to hold.3 Yet there appears to be the remaining problem that if we want to speak about absolutely everything there is we want to do so from the outside. This is immediately obvious if we want to supply our talk with the familiar kind of semantics: we have to introduce a domain our quantifiers range over, and this domain constitutes an additional object.
At the very least, we want to replicate on the inside how somebody from outside would speak about it. If we count the number of people in a room we are in, we want to count it in the way somebody not inside the room would count it (i.e., by including ourselves). We would therefore look at a possible world which contained our world (or rather a virtually indistinguishable counterpart) as a proper part, together with a being that was not part of that smaller world. The conception that this being would have of a proper part of its actual world would then be the conception we would want to have when thinking about the totality of all that exists in our world. Yet this approach is unlikely to be successful for we are not talking about the actual world at all, but just about a possible world. Now in this world, the universal generalizations about “all things” are not problematic simply because they are not absolutely general, but only speak about a subset of that world. But there is no reason to assume that these generalizations cease to be unproblematic once we re-import them into the actual world which differs from the possible world in important ways.
Even on this understanding, however, we appeal to something sometimes referred to as the “All-in-One Principle”, the idea that if we generalize over some things there must be some unity, some domain, to which everything we quantify over belongs. And this principle is controversial.4
Let us now look at another argument, a variation of Russell’s paradox developed from the assumption that absolutely general quantification is possible. The advantage of this argument is that it does not make the assumption (essential to the Cantorian case) that the world forms any kind of set-like collection.
5 Semantic Argument for Incompleteness
This short argument for incompleteness is based on the notion of an interpretation. An interpretation of a language is a set of meta-level statements that assigns individuals to the names in the language, and sets to its predicates. Interpretations can of course have properties themselves. They have the property of being an abstract object, of being a set-theoretic entity, and of being essential for assigning truths to sentences in a language. This allows us to introduce the property of modesty for interpretations. An interpretation i (we are here restricting ourselves to interpretations of predicates) is modest if it does not interpret a predicate P in such a way that P applies to i itself. Thus, if i is modest it is not the case that it interprets some P as being an abstract object, being a set-theoretic entity, and so forth. If we want to talk about the property of modesty in a language we can introduce a predicate M such that Mi if and only if there is no predicate letter P such that P-interpreted-by-i applies to i. Doing this is of course just providing an interpretation of M (let us call this j). Now the question is: Is j modest? Assume Mj. This means that there is no predicate letter P such P-interpreted-by-j applies to j. But there is, namely M, which j interprets as being the property of modesty. So -Mj. But if it is not modest then there must be some predicate letter P such that it, interpreted by j, applies to j. But as M is the only predicate letter interpreted by j, M must be this letter, so that Mj.
We are thus faced with a contradiction. It is obvious that the property of modesty (applied to models) is a close cousin of that of heterologicality (applied to words). A word is heterological just in case it does not apply to itself (such as “monosyllabic”), autological otherwise (such as “polysyllabic”). If “heterological” was heterological, it would not apply to itself and therefore would not be heterological. Yet if it is not heterological it is autological, and thereby expressing a property that also applies to itself, in which case it is heterological after all. There appears to be something fundamentally wrong with the notion of heterologicality, as there appears to be with the notion of a modest interpretation. This difficulty results from the attempt to quantify over all interpretations of a language. Doing so appears to be indispensable for making certain fundamental philosophical points (for example that the inference “P→P implies P” is a logical truth because on all interpretations of the sentence-letter the resulting sentence comes out as true). Yet as soon as we can do that, there is nothing that keeps us from defining predicates like modesty for interpretations, leading directly to a paradox. What gave rise to the paradox is the fact that we did not take into account that as we talk about objects, then about interpretations of terms, then about properties of interpretations, the scope of what we mean by “all” widens. By moving from an object-level statement about tables and chair to a meta-level statement about the interpretation of the terms “table” and “chairs”, we talk about more things: first, we talked about a universe with tables and chairs in it, now we talk about a bigger one that also features interpretations. As we can always repeat the move to a meta-language, there is always more to talk about than we initially thought we were talking about. Full generality is therefore forever beyond our grasp.
6 Arguments from Conceptual Relativity
The argument from conceptual relativity boils down to the point that in order to talk about all things in an intelligible manner, we need to have some clear conception of what a “thing” is in the first place. Unless you already know how to tell the things from the non-things, you cannot even start collecting them together into one big class. Unfortunately, the conception of a thing is not anything given to us by the universe, but something arising from our conceptual engagement with the world. Yet this implies that conceptions of thinghood can differ as our conceptualizations differ, giving rise to differing accounts of what the “collection of all things” amounts to. But if this is true we have lost the notion of absolute generality, of quantifying over absolutely everything, since we can now only speak about everything relative to such-and-such understanding of what a thing is. (To use the Carnapian terminology, we can only understand the question concerning the collection of everything as an internal question, though we would want to conceive of it as an external question.)
The three arguments given above at least raise our suspicion that there is something problematic with quantification over absolutely everything. Even though the corroboration for that suspicion given here is far from comprehensive, let us assume we agree that quantification over absolutely everything leads to sufficiently significant conceptual problems and should therefore be given up. Assume that we cannot refer to a complete list or theory of all there is, or to a complete universe comprising all things or all facts. What would the consequences of such a view be?
7 Incompleteness as Self-Refuting
First of all, we might worry that the claim that the world is incomplete appears to come dangerously close to self-refutation. In fact, some would argue that it is self-refuting. One thing the defender of incompleteness wants to assert is that there is no set or class or collection containing everything. But this is just the same as saying of all things that they cannot be put in a set, class, or collection. But if we agree with the idea that the notion of the complete universe, of “all things” is incoherent,5 then the previous sentence contained a meaningless term, and is therefore itself meaningless. So embracing incompleteness at the same time robs us of the ability to express it. The proponent of incompleteness is literally unable to express his position.6
8 An Immediate Reply
But perhaps this is a bit too quick. After all, can we not run the same argument against someone who says that there are no square circles, pointing out that by his own lights “square circle” is a contradictory concept, so that his statement “There are no square circles” is meaningless for that very reason? The opponent of square circles could reply that his statement is just to be understood as the claim that the properties of squareness and circularity cannot be co-instantiated, so no reference to contradictory entities is required. Could we equally say that the properties “being complete” and “being the world” can never be co-instantiated? An obvious way of spelling out what that means is that for all objects it is not the case that they collectively have the first property as well as the second, and we are thus back to square one.
9 A Second Reply
The opponent of square circles could also try a different approach, claiming that what his position really amounts to is that whenever somebody else postulates the existence of a square circle he has a way of reducing this to a contradiction. By analogy, the claim about the incompleteness of the universe would not have to be understood as a claim about the universe but as an open-ended claim about one’s own abilities of developing philosophical demonstrations: whenever someone comes up with a theory that postulates the existence of a complete universe (in terms of a universal collection, an omniscient knower, an assembly of all facts) we are able to derive a contradiction from this. Thus the statement “the world is incomplete” is not to be interpreted according to the familiar model-theoretic semantics, but as an existential statement asserting the existence of a reductio argument against any purported claim of completeness.7
10 Methodological Justification
What is the reason for this peculiar philosophical methodology? Why can’t the defenders of completeness and incompleteness pitch their positions against each other in honest philosophical combat, instead of the latter acting as an argumentative sniper who aims at completeness claims whenever they rear their heads? The explanation lies in the fact that both disagree about a fundamental philosophical assumption.
The defender of completeness assumes that there is a final account of the world. “Final” is here to be understood in two senses. Firstly, this account leaves nothing out; it does not have to be supplemented by other additional theories. This does not mean that every true statement must be a part of it, but that such statements could always be derived from the statements which are part of it. Secondly, the statements included in the final account are all primary statements. Primary statements do not depend for their truth on other statements; they encompass “what is true anyway”, what is “true no matter what”. (Secondary statements, on the other hand, are true only because certain other statements are true.)
It is evident that the thesis of incompleteness creates difficulties for both senses of finality. The deductive closure of the final account is supposed to encompass all truths, yet the defender of incompleteness denies that there is such a thing as “all truths”. Secondly, there is a conflict between incompleteness and the notion of primary statements. Suppose we have some primary statement X that describes part of how the world is fundamentally (say “electrons exist”). It is only primary if there are no other statements Y, Z such that X depends for its truth on them. So what it means for X to be a primary truth is that it is true that there are no truths distinct from X such that X is only true because they are. That statement is a generalization over all truths. This reveals an interesting connection between primary statements and absolute generalizations. It appears that we can only speak of primary truths if we accept some absolutely universal statements as truths as well.8
11 Incompleteness Does Not Mean Something Is Missing
It is important to understand the claim of incompleteness not in a way in which we might, e.g., say that an encyclopedia is incomplete, as if there were certain important objects missing in the world, but as saying that the very notion of the complete world is incoherent.
Saying that the world is complete is entailed by saying that there is an ultimately true theory. The incompleteness thesis is not about something missing, but says that we cannot give an account of “the world how it is anyway”, for such an account can only be made sense of if we understand what “being true independent of everything” means, and to do this we have to understand absolutely general statements. The distinction we are after here cuts much deeper than the familiar realist/anti-realist distinction. You could be an anti-realist of an extreme kind (e.g., one who believes that everything is a human construct) and still hold and that saying so fundamentally accounts for all there is, so that there is an ultimately true theory (namely this particular brand of anti-realism). Virtually all metaphysical theories agree with this meta-metaphysical assumption, claiming that there is an ultimately true theory accounting for the whole of reality. This is hardly surprising given the way we usually understand theorizing with ontological categories. If there were sections of reality we could not account for by appeal to theories of these, we would regard the theory based on them at best as a partial ontological account of the world.
To use a different example, the disagreement between the two opponents on the matter of completeness is not like the disagreement between two art historians, one claiming that a certain collection of paintings is incomplete, because there is written documentation that the painter painted more, the other claiming that it is complete, since the evidence of “additional paintings” is spurious. In this case, the defender of incompleteness could adopt the “sniper methodology” in trying to find fault with his opponent’s assertions whenever the opponent was trying to claim that the paintings in the collection are all the artist ever painted by introducing evidence that contradicts this claim. But he could also simply defend the positive claim that the present collection is just a part of the painter’s output, that some works we know something about are lost, and that there might well be others about which we know nothing whatsoever.
Yet the defender of ontological incompleteness would not be able to defend such a positive claim, since in this example there is a presupposition that both art historians share, whereas in the case at issue the corresponding presupposition is not shared by both ontologists. In the example, this is the presupposition that there is some fact to the matter regarding what this painter painted, some final theory (though it may always be unknown to us) that contains information about his artistic output. The disagreement only concerns whether the paintings his theory mentions and the paintings we have in our collection form the same class.
But the defender of ontological incompleteness disagrees with the presupposition that there is some big fact to the matter regarding what there is in the world, some final theory that encompasses all truths. It is now evident that the appearance of self-refutation only arises for the defender of incompleteness because we still rely on the opponent’s presupposition. We might reply to someone who believes that there is no final theory by saying: “Well, is this your final theory? And if it isn’t, why should I be convinced by it, as it is eventually going to be replaced by another one (since that is what it means to say that is not final)?” But this reply only carries any force if we assume that not being final speaks against a theory, because other theories are final. Yet, this is exactly the point at issue.
It is thus becoming clear that the claim of ontological incompleteness is not the claim that there is “something missing” in the world in the same way in which something is missing in an incomplete art gallery. Its key claim is rather that there cannot be a final theory of the world, since such a theory would encompass the totality of what there is, and since the primary status of its claims can only be spelt out in terms of a collection of all truths.
But since there is no such totality, there cannot be such a theory. The distinction we are looking at here is significantly more fundamental that the one involved in the debate about realism. For even the most extreme global anti-realist, one believing that everything (including Mt Everest, electrons, and the moon) are social constructs would agree that there is a fact to the matter regarding what there is in the world (social constructs all the way), that there is some final theory that gives a record of all there is (i.e., his particular brand of social constructivism).
12 Hyper-Antirealism and Its Consequences
The incompleteness claim supports a view which we shall call hyper-antirealism, a position relative to which familiar kinds of realism and anti-realism have to be considered as equally “realistic” since they both affirm that there is a final account of the world correctly captured by these respective theories.
The most obvious consequences of hyper-antirealism concern ontology. If ontology is understood in the traditional Quinean sense as providing an account of “what there is”, and if this does not just mean accounting for some of what there is, but for all of it, it appears as if ontology is impossible. For if no sense can be made of the notion of a collection of all things on any plausible understanding of “collection”, if there is no final account of what things there are in the world, how is it possible to come up with an ontological theory that provides precisely this? There are some who claim (rightly, it seems to me) that the Quinean definition of ontology sacrifices accuracy for conciseness. For ontology is not concerned with either a partial or a comprehensive laundry list of all there is,9 but with an account of very general structural features of what exists. But even this more sophisticated understanding of ontology is difficult to reconcile with the idea of the incompleteness of the world.
A prominent part of ontology concerned with ontological categories deals with a set of structural features that are supposed to tell us about the dependence relations between general kinds of being in the world. Ontological theories construct accounts of dependence relations by taking certain categories as primitive or primary, showing how other categories can be understood as derived or secondary.10
13 List of Categories as Closed
For such a constructive view of derived categories to be informative, it has to be supplemented by the assumption that the list of basic categories is closed, that no other categories in the world are basic. For the interest in these axiomatic constructions (similar in this respect to deriving theorems from axioms) lies in developing them from a restricted basis. If it was the case that the world contained all kinds of entities unfamiliar to us, including some additional basic categories, any particular construction of a derived category would be less interesting (for perhaps there are shorter constructions of these categories based on these additional, basic ones, or perhaps the supposed derived categories do in fact belong to the additional basic categories) and any claim that a certain category could be constructed would not be likely to remain stable under this enlargement.
14 List of Categories as Complete
In addition, the list of basic categories should be derivationally complete: all kinds of things in the world that are not basic should be derivable or constructable from the basic kinds. But of course such a claim cannot be made plausible on the assumption of an incomplete universe, for if we could not even have a final account of all the kinds of things that there are, how could we make a claim that they are all derivable from a fixed list of basic categories?11
Does this mean that the assumption of the incomplete universe precludes us from making any ontological claims? This might be too rash a generalization; it is clear, however, that the difficulties apply to ontological claims that are also universal statements, such as the claim that apart from these categories all are non-basic, or that all kinds of things in the world are derivable from a specific set of categories.
However, as the majority of ontological claims either takes the form of universal statements, or relies on such statements (as in the case of claims of ontological reducibility, which rely on universal statements about the set of basic categories), it appears as if at least a great part of ontology is thrown into doubt given the hypothesis of the incompleteness of the world.12
We could still save much of what we usually regard as ontological truths by rephrasing them as hypothetical truths, as truths that would hold if the universe was in fact complete, if there was a final theory and a collection of all that is the case. (Though there would be the difficulty that this would be a necessarily false propositions, and we would therefore have to adopt a non-classical logic in order to prevent everything being true under the hypothesis.)
15 Universal Explanations
Another consequence of hyper-antirealism is that encounter difficulties when developing a theory of all truths as a whole, and of their relation to the world as a whole. Since the assumption of incompleteness denies the existence of either of these, such an enterprise will be doomed from the start.
Problems arise, for example, if we tried to develop some answer to the question why the world exists at all,13 or why it is the way it is. This is not asking for the explanation of individual truths, such as why tomatoes are red, or why bananas are curved, but for an explanation of all truths taken as a whole. To answer this question, it is not sufficient that we can provide an explanation for each individual truth, but we want an explanation of why everything is as it is, that is an explanation of all truths taken together. And nothing but the set of all basic sentences or fundamental truths is able to provide such an explanation. But if there is no such thing, there will also not be a collective explanation for everything.
This also entails that what is probably the best argument for ontological foundation becomes questionable. Some philosophers argue that the reason why there cannot be an infinitely descending sequence of dependence relations is that we could never arrive at a comprehensive explanation of why the world is the way it is.14 We can always find an explanation for each particular fact, just by referring to a more fundamental fact it depends on, but it would be impossible to come up with an explanation of the whole. Yet if the world is in fact incomplete there could never be such an explanation, so that arguing that an infinite descent of dependence relations forbids such an explanation loses its force.
16 Incompleteness as Open-Endedness
An obvious but unsatisfactory suggestion would be to regard the claim of ontological incompleteness as something analogous to Zermelo’s picture of the realm of set theory, consisting of “an open-ended but well-ordered sequence of universes, where each universe is strictly more inclusive than its predecessor”.15 Yet on this theory, there is precisely one final and complete account of what the world is like, and that is that it has this open-ended structure.
17 Incompleteness Restricted to Representation
Nor would it make sense to regard the incompleteness as only applicable to the representation, and not to the representatum. There is an important difference between the incompleteness of representational items (lists, models, languages, formal systems, knowledge, and truth) and non-representational items (facts and states of affairs). For representational items, their incompleteness amounts to the representata outrunning the representanda (not every part of what is out there can be modeled, described in language, thought of) or the true outrunning the provable. But for non-representational items, their incompleteness cannot be such a “mismatch”. Facts or states of affairs do not stand for anything, so their collection cannot be incomplete in the sense that there are not enough to go round. Grim (1991: 122) notes that “What the arguments above ultimately indicate is not merely that all truths, somehow unproblematically referred to, fail to form a set. What they show, on the contrary, is that the very notion of all truths—or of all propositions or of all that an omniscient being would have to know—is itself incoherent.”16
Incompleteness understood as concerning representations has consequences which are just the opposite of incompleteness as understood concerning the world. The former supports realism by claiming the existence of something that goes beyond all our representational resources.17 The latter does not, saying that what an incomplete word amounts to is a world in which conceptual schemes give rise to objects, and schemes form a plurality, so that there is no scheme of schemes.
18 Semantic Ascent
A final reply consists in understanding the incompleteness claim not as a statement about the world, but as a statement at the meta-level speaking about our concept of the world. This statement would then be saying that the world being the way it is, and our conception of collections etc. being the way it is, the world turns out to be incomplete. In this case, our concept of the world would not include absolutely everything there is, but only everything at the object level. Should we now produce a concept “world+” really incorporating all there is (which would thus function as the domain of the language in which our meta-level assertions are phrased) we would of course just replicate the problem we encountered when thinking that the world incorporated all there is. We therefore need a statement at the meta-meta-level to tell us what the meta-level statement means. But since this process continues, we have the case of an infinite ascent through higher and higher meta-languages, none of which is ever furnished with a domain. To which extent this is problematic is not immediately clear. We certainly have, for each level, a statement that tells us what the statements at that level mean. (This is comparable to the case of an infinite descent of levels of explanations where, even though there is no fundamental explanation, there is an explanation for every level.)
With such an infinitely ascending chain of languages, we would never acquire a universal semantics in the sense of a theory that supplied the meaning for all the languages in the chain. And indeed there are some who claim that such a thing is impossible in the first place.18 The basic thought behind this claim is that a formal semantic theory will face the difficulty of having to postulate some kind of universal domain in terms of which all of the language is interpreted, and such a domain cannot exist if the world really is incomplete. On the other hand, if we choose an informal semantic theory as a final reference point to determine meaning,19 there would be no reason why there could not be a language powerful enough to quantify over whatever resources the informal theory employs (whether these are plural noun phrases or some other devices) in which case we could easily generate a version of Russell’s paradox, as this language would have to be interpreted in the same language in which the informal semantic theory is formulated.20 Yet if the idea of a comprehensive semantic theory applicable to all languages is an inconsistent fiction, we could hardly find fault with the infinite hierarchy of languages just described for failing to provide us with such a semantics.
The method of semantic ascent appears to be the most promising of the three ways out presented, yet it is doubtful whether it will be of much use in defending the ontological enterprise. For it presupposes conceding that when we are seemingly talking about the world, we are de facto talking about something else, namely about our concept of the world. Of course if concepts were all there is, talk about concepts would qualify as genuinely ontological talk, but if that is the consequence we have to accept in order to ward of hyper-antirealism many might regard the cure as worse than the disease.
We are thus left with a dilemma: either the arguments for incompleteness are insufficient (though it is hard to come up with a refutation that is successful against all three kinds) or ontology as traditionally conceived is a fundamentally deficient enterprise (which is a conclusion we might be reluctant to accept). We thus have a choice between either coming up with a robust argument demonstrating that despite contrary appearances, absolutely general quantification is unproblematic, or with developing a revised conception of ontology compatible with an incomplete world. Which of these is the more promising strategy, only further investigation will be able to tell.
We should note that there is a third possible position, that of the dialetheist, which argues that we should neither rule out absolute generality nor strive to find a way to make it paradox-free, but to accept that the resulting inconsistencies represent true contradictions. I am not going to explore this possibility further in this essay.
This distinguishes the case of the universal collection from that of the universal list. If you try to construct a long laundry list of everything there is in the universe you will at the end discover that there is one more thing to be included, namely the list itself. But including this will increase the universe, so that we now need a larger list, to cover the new universe + list. But unlike the Cantorian case, this does not have to go on forever. There is no inconsistency in assuming that one of the things that exist in the universe is the following list: “the earth, Mt Everest, New York, ...., this list”, where “…” has to be filled in by all the remaining objects in the universe. There is no inconsistency because there is no reason for assuming that the list is strictly larger than the universe without the list. A more difficult situation arises if we want to include a smaller map or model of all there is within the world. If this is a maximally accurate model, that is if everything in the world has an equivalent in the model the model would have to be infinitely complex. For at the place in the model corresponding to where the model is located in our world, it would have to contain a model of itself (and thus of the entire world) which would have to contain another copy of itself and so on ad infinitum.
Alternatively, we could argue that the members of the “power-collection” are mere entia rationis, thought constructions that are not to be regarded as fully real (perhaps because they are reducible to other, nonmental objects).
See also Williamson 2003: 433–434.
This is the reason why “the world is incomplete” cannot be a primary truth. For if it is, then (a) there is no collection of all truths and (b) “the world is incomplete” does not depend on other truths for its truth. But (b) means that for all truths it is the case that no subset of them is such that “the world is incomplete” is only true because that subset is. But this contradicts (a).
“Metaphysics is about what there is and what it is like. But it is not concerned with any old shopping list of what there is and what it is like. Metaphysicians seek a comprehensive account of some subject-matter—the mind, the semantic, or, most ambitiously, everything—in terms of a limited number of more or less basic notions.” (Jackson 1998:4); “Because the ingredients are limited, some putative features of the world are not going to appear explicitly in some more basic account. The question then is whether they nevertheless figure implicitly in the more basic account, or whether we should say that to accept that the account is complete, or is complete with respect to some subject-matter or other, commits us to holding that the putative features are merely putative. In sum, serious metaphysics is discriminatory at the same time as claiming to be complete, or complete with respect to some subject matter, and the combination of these two features of serious metaphysics means that there are inevitably a host of putative features of our world which we must either eliminate or locate.” (Jackson 1998:5). “Es geht in der Ontologie um die Grundstrukturen des Seienden. […] [This includes the possible and the impossible.] Ansonsten würde man dem Anspruch der Ontologie nicht gerecht, eine allgemeinste Wissenschaft von allem überhaupt zu sein.” (Meixner 2004).
As such, ontological theory-building can be understood along the lines of constructing axiomatizations of theories, the axioms corresponding to the fundamental categories and the theorems to the derived categories.
Of course this presupposes that the incompleteness of the world is an incompleteness in kinds, a qualitative, rather than quantitative incompleteness. The assumption that the world could be qualitatively complete, yet quantitatively incomplete, so that there would be a complete collection of kinds of things, while there would no collection of kinds subsumed amongst them does not seem to be very promising. If our argument for incompleteness relies on the view that our knowledge of the world is based on some conceptual scheme, and that any attempt at forming a collection of all the elements of the scheme would take us outside of that scheme this would still apply in the case of kinds. The collection of all kinds is a kind not included amongst the original kinds, which means that these kinds could not have exhausted all there is.
Compare Williamson (2003: 435). I do not share his belief that this is not much of a problem, as “most of our life, even most of our intellectual life, does not appear to depend on the prospects for traditional metaphysics”. This appears to underestimate the foundational role generally ascribed to ontology within the architecture of philosophical disciplines.
“If you believe in metaphysical explanation, you should believe it bottoms out somewhere. […] it is better to give the same explanation of each phenomenon than to give separate explanations of each phenomenon. […] if there is an infinitely descending chain of ontological dependence, then while everything that needs a metaphysical explanation (a grounding for its existence) has one, there is no explanation of everything that needs explaining. […][A common metaphysical explanation for every dependent entity can be given] only if every dependent entity has its ultimate ontological basis in some collection of independent entities.” (Cameron 2008: 12).
Note that “there need be no suggestion, by the way, that truths be thought of as linguistic entities in even the most attenuated or metaphorical sense. Nothing in the arguments demands, for example, that the truths at issue be in any way linguistically expressible. The result would thus be the same for ‘true propositions’, for ‘actual states of affairs’, or for ‘facts’ in place of ‘truths’: there can be no set of all true propositions, no set of all facts, and no set of all actual states of affairs.” (Grim 1991: 93–94).
Though we seem to have a choice here between realism about the representata and realism about the representanda. Compare Yourgrau (1999: xiv): “I formulate a question that no one else has yet addressed, to wit: Why did Gödel conclude, in the case of T, intuitive arithmetic truth, that the limitation lies with formalized mathematics, whereas in the case of t (intuitive time) he concluded, not that relativity theory has intrinsic limitations, but rather that intuitive time itself is an appearance or illusion?” On page 107, he notes that “I don’t think we really know enough, at present, to resolve this question satisfactorily.”
“[T]here may in the end be no universally adequate formal (or for that matter informal) semantics. […] Thus is may be that there is and can be no X such that an X-theoretic semantics would prove adequate in all cases.” (Grim 1991: 153).
Along the lines of Boolos (1984).
Grim (1991: 153).
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