Conceptual Closure in Anselm’s Proof: Reply to Professor Roark
Let me begin my reply to Professor Roark’s objections in good old scholastic fashion, by a distinction. Philosophical objections can be good in two senses. In the first, trivial sense, a good objection is one that convincingly shows the presence of a genuine error in a position or reasoning. Such objections are useful, but uninspiring. In the second, non-trivial sense, a good philosophical objection broadens and deepens our understanding of the problems at issue, whether or not they manage to refute the opponent’s position. In this reply I am going to argue that even if Roark’s objections may not be necessarily good in the first, trivial sense, they are very good in the second, non-trivial sense.
The first objection argues that the formal reconstruction of the argument, although it is valid, contains an informal ambiguity not represented by the formalization. Once this informal ambiguity is brought to light in a more detailed reconstruction, however, it can be seen that the argument cannot support the intended conclusion, namely, that God exists in reality. The ambiguity in question resides in the relational expression: ‘x can be thought to be greater than y’, which in the formal reconstruction was left unanalyzed, represented by a simple relational parameter. To bring out the ambiguity in question, Roark further analyzes this relational expression, distinguishing its three different readings. Rejecting the first two, he accepts the third reading, but claims that it renders the third premise of my reconstruction implausible, whereas providing an intuitively plausible interpretation of the third premise in accordance with this reading of the phrase requires a corresponding reinterpretation of the second premise, which then supports at best a weaker conclusion, namely, that God cannot be thought to exist only in the intellect, but not the intended conclusion, namely, that God exists in reality.
It is certainly true that in the formal reconstruction I only exploited what later medieval logicians would call the ampliative force of the “modal-pistic” component of Anselm’s phrase, in order to allow the variables of the reconstruction to range over objects of thought, whether they are mere objects of thought or also objects simpliciter. Accordingly, further considerations concerning the intrinsic structure of the relational phrase involved in Anselm’s description were left on an intuitive level. Therefore, further stringent analysis of the logic of this phrase definitely may bring out further, even unexpected results.
The unexpected result according to Roark’s argument in this case would be that, if we accept the only plausible reading of the originally unanalyzed phrase and we correspondingly modify the third premise in order to keep it intuitively valid, then my reconstruction can at best prove that God cannot be thought to exist only in the intellect, but not that God exists in reality.
However, I am not convinced that this is indeed the inevitable conclusion of Roark’s analysis. He supports his conclusion with the following: “In order simultaneously to render the sufficiency claim in Premise Three plausible and to accommodate (γ), the predicate ‘I( )’ must be understood to include a modal-pistic component: ‘. . . can be thought to exist only in the intellect’. So it appears that the proper conclusion of the argument is not that God exists, but rather that God cannot be thought to exist only in the intellect.”
The original interpretation of the predicate ‘I( )’ in the reconstruction was ‘( ) is only in the intellect’, which I expounded further by saying that an x is only in the intellect in this sense if and only if x is thought of, but does not exist in reality. So, this predicate does contain a certain “pistic component”, namely, the component that x is thought of, which of course entails the “modal-pistic component” that x can be thought of. Now, if g is only in the intellect in this sense, then it seems clear that something greater than g can be thought of in the sense of Roark’s interpretation (γ) by a thinking subject S who assumes premise (2). For S, by virtue of assuming premise (2), is thinking that g is in the intellect and does not exist in reality. Therefore, S can obviously think of something with “a greater cardinality”, whether g itself or anything else, by simply thinking, or counterfactually assuming, that that thing does exist in reality. (Say, by thinking: ‘OK, now I think that g is only in the intellect, so I am assigning to g the cardinality of 0, but I could also assume that g existed in reality, in which case I would assign to it the cardinality of 1, which is greater’ – in accordance with Roark’s interpretation (γ).) But then, if S also assumes the description of g in the first premise, then he has to conclude that by being able to think of something greater than g he would have to be able to think of something greater than that than which nothing greater can be thought, so he has to abandon the assumption from which this absurdity follows.
Accordingly, the argument does have to prove its conclusion for any thinking subject S, provided S assumes all the premises in the required senses, interpreting the phrase ‘x can be thought to be greater than y’ as expounded by Roark. The important point here is that what S has to conclude on the basis of the premises thus interpreted is not that he simply cannot think that g exists only in the intellect, but that it is not true that g exists only in the intellect, from which he further has to conclude that, since g is in the intellect and not only in the intellect, g also has to exist in reality.
To be sure, an external observer E, listening to the reasoning of S, can describe what she observes by saying that S had to conclude that g exists because S cannot consistently think that g does not exist. And E may further claim that she is not thus committed to accepting S’s conclusion, for S can plausibly argue only for himself, since he is the one who makes the comparisons of his own thought objects regarding their assumed cardinalities within his own “modal-pistic” context.
But then, this result seems to make perfect sense in the larger context of the paper. After all, my main argument in the paper is that Anselm’s argument can genuinely work only for those who are willing to make constitutive reference to God. But for them it is indeed an inevitable conclusion that they cannot consistently think of God and think that he does not exist. So they have to conclude without any pistic-modal component in their conclusion that God exists. Viewed from this angle, it seems to me that Roark’s first objection just reinforces this broader point, bringing out very clearly why anyone who herself is not making constitutive reference to g as described in the argument, but rather studies the argument as something taking place in the mind of a sophisticated believer, would never be persuaded by the argument, even if she is able to appreciate its compelling force in the mentality of the believer.
The question then, as I argued in the paper, really is whether this constitutes an unbridgeable gap between the mentality of those who are willing to make such constitutive reference and that of those who are not. But of course this presupposes that it is possible to make constitutive reference to God as required by Anselm’s description. It is this presupposition that is the target of Roark’s second objection.
When I first read Professor Roark’s paper, I was inclined to think that the second objection was based on a rather weak analogy. But then in a quick correspondence he kindly introduced to me Modest. After this introduction it became clear that the analogy is actually stronger than at first it seemed to be, insofar as we do have a genuine paradox to deal with here. Indeed, so much so that I would base my “primary defense strategy” against the objection on the strength of the analogy.
For as the medieval treatments of the Liar paradox and in particular John Buridan’s ingenious solution to it indicate, Tarski’s solution may be an unnecessary overkill, which works only for artificial languages anyway. But then, by analogy, if the “semantic closure” discovered by Tarski need not require a global ban on self-reference in a language, but the paradoxes emerging from certain self-referential propositions can be treated “locally”, then the “conceptual closure” ingenuously discovered by Professor Roark may not demand a similar global ban on constitutive reference to thought objects applying conceptual sortals, but may likewise be treated “locally”.
To be sure, I do not want to criticize here either Tarski’s or Roark’s “global solutions”, because it would be quite impossible to do so on this occasion. Therefore, in closing, I would just briefly indicate how I would treat Modest without a total ban on constitutive reference with the application of the conceptual sortal of real existence, and add a reason why I think such a total ban would have some undesirable consequences.
Modest in my view is unthinkable (cannot be thought to exist/cannot be thought of – I’m using now these phrases interchangeably), despite possible appearances to the contrary. The unthinkability of Modest is shown precisely by Roark’s Paradox. Therefore, anyone who first thinks Modest is thinkable may first think that she is able to think that Modest exists. But after understanding Roark’s paradox she cannot consistently think that Modest exists. To be sure this still leaves open for her, or for anyone else, the possibility to refer to Modest parasitically, using her previous, mistaken belief about the possibility of Modest’s existence. In fact, this sort of consideration applies not only to Modest’s case. After all, “the greatest prime number”, on the face of it, seems like a pretty well-behaved possible object of thought, yet, after understanding Euclid’s proof no one can consistently refer to “it”, except parasitically.
So, rather than banning all constitutive reference with conceptual sortals, I would simply say that the paradoxes that some of these generate just show us that such putative possible objects of thought are in fact not possible objects of thought, and so, after this realization we can only make parasitic reference to them. But from this “local treatment” of the emerging paradoxes nothing seems to follow concerning Anselm’s description, unless one shows that this description is inconsistent. But that is a tough call.
On the other hand, to justify a total ban on constitutive reference with conceptual sortals, which would affect Anselm’s argument, one would have to show that any such reference has to lead to a paradox. But that, again, is a tough call.
In addition, it seems that a total ban on constitutive reference to any object of thought that involves the notion of existence in its conceptual content would lead to the unwelcome conclusion that we could not make constitutive reference to anything to which we would normally attribute real existence. For we must not forget that the relevant notion of existence we are using in this context is the notion of a first-order predicate, an Aristotelian transcendental covering all real beings in the Aristotelian categories. Therefore, this notion is part of the conceptual content of anything that can be referred to within these categories, at least by implication. So a global ban on constitutive reference with this notion of existence would kill such reference in all real sciences. And that certainly seems to be unnecessary overkill.
Read at the Pacific APA meeting, Seattle, March 27-30, 2002.