Abstract
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.
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Notes
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).
Cartwright 1994.
Grim 1991:122.
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).
References
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Ernst Zermelo: "Über Grenzzahlen und Mengenbereiche", Fundamenta Mathematicae 16, 1930, 29–47.
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Westerhoff, J. The Incompleteness of the World and Its Consequences. Int Ontology Metaphysics 14, 79–92 (2013). https://doi.org/10.1007/s12133-012-0113-y
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DOI: https://doi.org/10.1007/s12133-012-0113-y