Musings on Saint Anselm’s Dilemma
Reflexiones sobre el dilema de San Anselmo
J.-Martín Castro-Manzano
josemartin.castro@upaep.mx
Abstract: In this contribution we suggest two simple contentions: i) that Saint Anselm’s ontological argument, as presented in the Proslogion, only makes sense if embedded in an ordered and bounded ontology; and ii) that an interpretation of the main premise of the argument, within such ontology, produces a dilemma that demands a new revision of the argument.
Keywords: Ordered ontology; bounded ontology; greatness.
Resumen: En este trabajo proponemos dos afirmaciones: i) que el argumento ontológico de San Anselmo, como se presenta en el Proslogion, sólo tiene sentido al interior de una ontología ordenada y acotada; y ii) que una interpretación del argumento, dentro de tal ontología, produce un dilema que demanda una nueva revisión del argumento.
Palabras clave: Ontología ordenada; ontología acotada; grandeza.
1. Introduction
We share the opinion that the Proslogion should prove as much a philosophical as it does a mathematical work of art[1], for it presupposes an ontology that requires assumptions close to order theory, namely, assumptions of order and bounds. Following these assumptions, in this short contribution we suggest two simple contentions: i) that Saint Anselm’s ontological argument, as presented in the Proslogion, only makes sense if embedded in an ordered and bounded ontology (§2); and ii) that an interpretation of the argument, within such ontology, produces a dilemma that demands a new revision of the argument (§3).
2. Remarks on the Anselmian ontology
For the purposes of this contribution we define an ontology by a pair 〈D, R〉 where D is a non-empty domain and R is a relation defined on D. In order to describe the Anselmian ontology, OA for short, we consider Pros. II, and Pros. III−which contain subtle versions of the ontological argument (cf. Malcolm, 1961) −, along with Pros. V, Pros. XV, and Mon. I, and Mon. III.
From these passages we can infer, given the uses of maius/melius (cf. Brecher, 1974), that the Anselmian ontology is defined by the pair OA=〈B, ≥〉 where B is a non-empty bounded set of beings and the maius/melius relation, namely ≥, is a partial order relation on B. We can summarize these features with the following facts:
Fact 1. OA is a partially ordered ontology.
Fact 2. OA is a bounded ontology.
These facts provide evidence to the claim that St. Anselm’s ontological argument only makes sense if embedded in an ordered and bounded ontology[2]. Indeed, we could not account for the maius/melius relation without an order relation (Fact 1); and we could not account for the notion of “that than which nothing greater can be conceived” (or “the greatest being” for short) without (upper) bounds (Fact 2). However, the very notion of “the greatest being” is not naive and requires further attention because, for it to make sense, OA must comply with two further remarks:
Remark 1. OA has at least one maximal being.
Remark 2. OA has at most one greatest being.
Consider, for Remark 1, that a maximal being in a bounded and partially ordered ontology, such as the Anselmian, would be a being bx in the domain for which there is no being by such that by≥bx. So, take an arbitrary being bi in OA. Suppose, for reductio, that bi is not maximal. Then we can find another being, say bi+1 (i.e. a next being), s.t. bi+1>bi (i.e. a next being that is greater than the previous one). Now, if bi+1 is not maximal itself, we will find yet another being bi+2 s.t. bi+2>bi+1>bi. If we continue looking for greater beings in this way, we will have to stop eventually at a bound, call it g, corresponding to a being bg, since OA is bounded above, by Fact 2. Therefore, it is impossible to have bi>bg for any i in OA. Hence, there is at least one being bg that is maximal in OA.
Additionally, for Remark 2, since OA is a partially ordered ontology, by Fact 1, we also have at most one greatest being. A greatest being in a bounded and partially ordered ontology would be a being bx in the domain such that for all by, bx≥by. So Remark 2 is clear, for suppose there are two greatest beings bg and bk in OA, then bg≥bk and bk≥bg. But then, by the antisymmetry of the order relation, bg=bk, that is to say, there is at most one greatest being in OA, namely, bg.
Together, these Facts and Remarks make sense of the following proposition:
Proposition 1. If bg is both the maximal and greatest being in OA, then bg is the supremum in OA.
Proposition 1 clarifies the meaning of the expression “the greatest being” and specifies that the domain of beings B should behave like the closed-to-the-right interval [b0,bg][3], for any beings i in B, where bg denotes the supremum; otherwise OA would comply with Fact 1 (order) but not with Fact 2 (boundedness), which would imply the failure of Remarks 1 and 2.
At this point, some numerical examples will help clarify this last proposition. Consider the open interval (−10,10). This interval is bounded above, for instance, by the numbers 11, 12, …, 100. This is because upper bounds need not belong to the given set. But in the interval (−10,10], for example, 10 is the least upper bound and, since it belongs to the set, it is also the supremum. Thus, the interval (−10,10) has a least upper bound (number 10) but no supremum; while (−10,10] has both a least upper bound and a supremum, namely, number 10. The interval (−10,+∞), on the other hand, is not bounded above and it has no supremum.
To further illustrate Proposition 1 let us consider a more concrete example. Imagine we have a domain of books, say a library, of different height, say b1, b2 and b3, ordered with respect to their height in such a way that b3≥b2≥b1. Clearly, we can identify “the highest book” in the library, that is to say, that book “than which no higher book can be conceived” in that library, namely, b3. Now, what must be stressed in this situation is that the only way in which it makes sense to talk about “the highest book” is if the domain is both ordered and bounded (i.e. if Facts 1 and 2 obtain). But notice that if the domain of books is bounded, then b3 is bounded itself, since the bound of the domain is determined by b3’s actual height, for otherwise the domain would be unbounded. So, for instance, the intervals (−10,10] and (−10,10) are clearly bounded above because 10 is bounded itself; had the number 10 no bounds of its own, both intervals would behave like the unbounded interval (−10, +∞), which would require the number 10 to behave like +∞, which is absurd, for +∞ is not even a number. Likewise, in the case of the books, had the book b3 no bounds of its own, the domain of books would look like the unbounded interval (b1,+∞), which would require the book b3 to behave like +∞, which is also absurd, for +∞ denotes the interval’s behavior, not an actual book or entity. Hence, the book b3 is a bound of the domain because it has a bound of its own, say its actual height denoted by the definite number 3. This is an important step for what comes next because, suppose, for reductio, that bg is the supremum of OA but is not bounded itself. If bg is the supremum of OA, the domain B must behave like a closed-to-the-right interval, say (b0,bg]. But if bg is not bounded itself, bg would behave like +∞, in which case B would behave like the open interval (b0,+∞), but that would contradict Fact 2. We call this the bounded-bound condition. The bounded-bound condition can be best “viewed” with the aid of the following diagrams:
From left to right, the first diagram shows a representation of a bounded space as a ball of finite radius so that for all bx, bg≥bx, i.e., the distance d(bx,by)≤bg for any beings by and bx. This represents a situation that considers that we may admit a possible infinite set of beings in B, but B is still bounded by bg. The second diagram represents OA as a bounded chain. In both diagrams the domain is bounded because it is true that for all bx, bg≥bx.
3. A loop and a dilemma
At this point, nothing seems to be too controversial, especially regarding the previous examples because, for one, the notion of “the highest book” is scarcely problematic. However, what happens if we think, not of “the highest book”, but of “the greatest being”? Let us entertain this question for a moment. Suppose bg is the greatest being, according to OA. Then, bg is precisely the supremum, by Proposition 1. Now, if bg is the supremum, bg is bounded itself, by the bounded-bound condition. However, if bg is bounded in itself, it has some sort of limit, namely the limit defined by its own bound (as in the example of the book, it’s very height was its own bound); but if bg, “that than which nothing greater can be conceived”, has bounds of any sort, could not we think of something greater than bg, namely, something not constrained by the bound defined by the limit g? After all, why would the greatest being have bounds of any sort?
In other words, if we say bg is the supremum, then we would have to admit that bg has some sort of own bound or limit. If this is the case, however, we could think of a being beyond that bound, in which case we could extend B indefinitely (just as we can think of a number greater than 10 when considering the interval (−10,10]). On the other hand, if we say there is some being greater than any other being, we would have to admit that there is no supremum whatsoever. If this is the case, nevertheless, we could extend B indefinitely and then B would be unbounded; but this would be odd, because then there would be no supremum at all; but if there is a supremum, then such being would have to be bounded, but then we could think of a being beyond that bound again, namely, a greater being, and then we could extend B again, and so on and so forth. However, this turns out to be kind of paradoxical because it leads to evident inconsistencies regarding the notion of “the greatest being”. To visually explain this situation, consider the next flow chart:
The flow chart above starts by asking whether greatness has bounds of any sort. If greatness has no bounds, it does not make sense to talk about “the greatest being”. But, on the other hand, if greatness has bounds, the notion of “the greatest being” makes sense, but then we can ask if we can think of something above those bounds; and in case we can, then it does not make sense to talk about “the greatest being” again; and in case we cannot, we get into a loop and we return to the original question that starts the flow chart.
With this loop schema in mind, we can now expound what we call “Saint Anselm’s dilemma”:
1. To avoid the loop, either we maintain the assumption that “bg is the greatest” while dropping the notions of order and bounds, namely OA; or we maintain such notions, namely OA, without dropping the assumption that “bg is the greatest”.
2. The problem with the first horn of the dilemma, of course, is that we cannot abandon OA without dropping the thesis that “bg is the greatest”, since such abandonment would render the very proposition “bg is the greatest” meaningless (due to Proposition 1), and then the Anselmian ontological argument would be unsound (because of an ill-defined premise).
3. But then we should consider the other horn of the dilemma and drop the claim that “bg is the greatest” while keeping OA, but this strikes us as odd because the thesis that “bg is the greatest” seems to be both the main premise of the Anselmian ontological argument and a natural consequence of OA (due to Remarks 1, 2, and Facts 1, 2), but then the Anselmian argument would be unsound (because it would be self-defeating).
Since, in both cases, the Anselmian ontological argument turns out to be unsound, but we know the argument is logically valid (cf. Oppenheimer and Zalta, 1991), the problem must come from the layman use of the notion of “the greatest being”, and so this notion has to be recast with more attention in order to avoid any inconsistencies or loops. For instance, an obvious objection that undercuts the previous remarks is the use of the “bounded-bound condition”, since it seems to be an artificial and controversial add-on to Saint Anselm’s original proposal. Granted. This so-called bounded-bound condition is not to be found within the Anselmian corpus. However, since the notion of “the greatest being” produces these artificial, yet troublesome considerations, the bounded-bound condition must not be discarded altogether as an alien one. It may certainly be artificial, but it is also not strange to OA, due to Remarks 1, 2, and Facts 1, 2.
4. Concluding remarks
In this contribution we have suggested two simple contentions. The first one, rather typical, states that Saint Anselm’s ontological argument, as presented in the Proslogion, only makes sense if embedded in an ordered and bounded ontology; the second one, which we think is rather controversial, focuses on an interpretation of the argument within such ontology that produces a dilemma that demands a new revision of the argument due to the ambiguity of the notion of “the greatest being”.
The dilemma shows, we think, that unrevised assumptions of order and bounds can render some propositions unsound if we do not handle them with care. And this is an important and delicate issue, especially if we consider, for instance, any conception of God as a being defined in terms of greatness[4], for such greatness would have to be embedded in an ordered and bounded ontology in order to make any sense; but then we would need to revise our order assumptions more carefully, because the notion of greatness, if not managed carefully and precisely, might lead to strange loops where none should be.
References
Barth, K. (1960) Anselm: fides quaerens intellectum, trans. by Ian W. Robertson. Richmond: John Knox Press.
Brecher, R. (1974) ‘Greatness’ in Anselm’s Ontological Argument, The Philosophical Quarterly 24, 95, pp. 97-105.
Gilson, É. (1934) Sens et nature de l’argument de saint Anselme, Archives d’Histoire Doctrinale et Littéraire du Moyen-âge 9, pp. 5-51.
Harsthorne, Ch. (1961) The Logic of the Ontological Argument, The Journal of Philosophy 58, 17, pp. 471-473.
Malcolm, N. (1960) Anselm’s Ontological Arguments, Philosophical Review LXIX, pp. 41-62.
McMahon, R. (2006) Understanding the Medieval Meditative Ascent. Washington: The Catholic University Press.
Niemeyer Findlay, J. (1948) Can God’s Existence Be Disproved? Mind 57.
Oppenheimer, P. and Zalta, E. (1991) On the Logic of the Ontological Argument, ed. J. Tomberlin, Philosophical Perspectives 5: The Philosophy of Religion. Atascadero: Ridgeview.
Plantinga, A. (1974) The Nature of Necessity. Oxford: Oxford University Press.
Stolz, A. (1967) Anselm’s Theology in the Proslogion, ed. John Hick and Arthur C. McGill, The Many-Faced Argument. New York: McMillan.
Recepción: 05.01.2019
Aceptación: 25.02.2019