Tarski and Klima: Conceptual Closure in Anselm’s Proof


Gyula Klima has recently offered a novel and sophisticated interpretation of Anselm’s ontological argument, which he believes to be valid proof of God’s existence.[1] The interpretation exploits a medieval conception of intentional reference (and consequent notion of intentional quantification) that is very different from the twentieth-century notion of reference advanced by Russell and others. I have no qualms with Klima’s interpretative claim: for all I know, he’s got Anselm’s argument just right. What I do dispute is the claim that the argument as interpreted by Klima succeeds in establishing the existence of God. I do not intend here to attempt to demonstrate that the argument is unsound, for I do not claim to know that Klima’s version of Anselm’s proof is unsound, and doing so would require more space that I have available here, anyway. Rather, I wish to highlight two different difficulties the argument faces. The first has to do with a certain ambiguity in Klima’s formulation of the argument. The second draws on the notion of semantic closure introduced by Tarski in his “Semantic Conception of Truth.”[2] Given the complicated issues involved, I can do little more than sketch each line of objection, but I hope that my remarks are at least provocative.


I.     Intentional Reference and the Proof


       The novelty of Klima’s interpretation of Anselm’s proof lies primarily in its employment of a distinctively medieval notion of reference, according to which the reference of a linguistic expression is determined by the intention of its user in the context of use. This conception of reference differs from the Russellian notion at least to the extent that, on the intentional conception, one can successfully refer to mere objects of thought, which do not exist in extramental reality and consequently are not objects tout court. On the Russellian conception, reference is inextricably linked with existence, such that an individual who attempts to refer by using a singular term that fails to connect up with anything in the world simply fails to refer. On this view, “refer” is a success verb.

But such is not the case on the intentional conception of reference. Indeed, on this view one always refers to objects of thought (entia rationis), only some of which are objects simpliciter (entia).

       Adopting this intentional conception of reference, one can devise a quantificational scheme in which variables range over thought objects, any one of which may or may not be an object simpliciter. Klima formulates his interpretation of Anselm’s proof in just such quantificational scheme. Adopting otherwise conventional notation, where “I( )” translates “... is only in the intellect”, “R( )” translates “... can be thought to exist in reality” , “M( )( )” translates “... can be thought to be greater than ...”, and “g” translates “God”, Klima gives us the following formulation and assessment of Anselm’s proof:

(1)             g =df ix.~($y)(Myx)

(2)             Ig

(3)             ("x)("y)((Ix & Ry) ® Myx)

(4)             Rg

(a)      Mgg                                            [2,3,4, UI, &I, MP]

(b)     ($y)(Myg)                                   [a, EG]

(5)             ($y)(My(ix.~($y)(Myx))             [1,b, SI]


... Abbreviating ‘($y)(My( ))’ as ‘P( )’, (5) will look like ‘P(ix.~Px)’, i.e., ‘($x)(~Px & ("y)(~Py ® x=y) & Px)’, which implies ‘($x)(~Px & Px)’, an explicit contradiction. But then, since (1), (3) and (4) have to be accepted as true, (2) has to be rejected as false. So it is not true that God exists only in the intellect. But since to exist only in the intellect means to exist in the intellect but not in reality, not to exist only in the intellect means either not to exist in the intellect, or to exist in the intellect and also in reality. Therefore, since God, being thought of, does exist in the intellect, he has to exist also in reality.[3]


The well formulated argument is clearly valid. Short of simply denying one of the premises, one can object by claiming that it is in some way an inadequate formulation of the original argument. My first objection proceeds in this way.


II.    Ambiguity in the Argument


       I want to begin by drawing out an ambiguity in one of the three predicates that appear in Klima’s argument, and a certain incongruity among the trio.

The relation “... can be thought greater than ...” is at least triply ambiguous. On the face of it, the relation is analyzable into two distinct logical components: a modal-pistic operator (“it can be thought that ...”), and the greater than relation. I take the former operator to mean simply that the sentence it modifies can be regarded by some subject as being true. It’s not clear whether the latter operator takes as its arguments ordinal, or cardinal values. I will suppose without argument that it takes cardinal values.

The relation “M( )( )” is obviously not irreflexive. Were it so, Klima’s having derived the statement on line (4a) would have been sufficient to reject the assumption on line (2) that funded the derivation. But he clearly does not think it sufficient, since the argument proceeds by employing the definition of God on line (1) in order to derive an explicit contradiction. So let “b” denote some thought object such that “Mbb” is true, and let “Cb” denote b’s cardinality with respect to whatever factor is regarded as relevant by all interested parties. Now it can’t possibly be the case that Klima is willing to maintain that “Mbb” means:


       (a)  It can be thought that (Cb > Cb)


for that is obviously absurd: no one can think that n > n, for any number n. Or, at any rate, someone who appeared capable of forming the thought would not be one whose cognitive capacities would figure favorably in the context of Anselm’s argument. Only those who recognize the obvious and necessary falsity of the claim that n > n (for any n) are candidate beneficiaries of Anselm’s argument, and it cannot seriously be maintained that these individuals might wittingly think a necessary falsehood to be true.

In properly interpreting the predicate, then, the scope of the modal-pistic operator must be reduced to one side or the other of the principal function. I think that there are only two plausible alternatives here. The first is a conjunction that employs a single modal-pistic operator modifying the expression that recurs on the left-hand side of the greater than function in the right-hand conjunct:

       (b)  It can be thought that (C1b), and C1b > C2b


where “C2b” denotes b’s actual cardinality. Here, an agent who forms the thought thinks that b has a certain cardinality, where that cardinality happens to be greater than the cardinality that b actually enjoys. It is not required, of course, that the agent have cognitive access to b’s actual cardinality: it might simply be the case that he has a sincere, though inaccurate, estimation of b’s greatness.

       Although this reading of the predicate does not fall prey to the objection of absurdity to which reading (a) is subject, it is nonetheless clear that it will not serve Klima’s purposes. For on (b), the description of God deployed in the argument would be the following: God is that thought object whose actual cardinality is no less than the cardinality one might think any other thought object to have. But this is quite obviously question begging: given the connection between greatness and existence advanced in Premise Three, Klima would just be stipulating God’s existence. Consequently, reading (b) must be rejected.

       The final reading employs a pair of modal-pistic operators, each modifying an expression that recurs as one of the arguments in the greater than function in the right-hand conjunct:


       (g)   It can be thought that (C1b), and it can be thought that (C2b), and C1b > C2b


where neither “C1b” nor “C2b” need denote b’s actual cardinality.

       I take this to be the correct reading of the predicate “M( )( )”. It accounts for the non-irreflexivity of the relation, and it brings to the surface the incongruity I mentioned earlier. In order that the predicate “... can be thought greater than ...” be true of some subject S with respect to some pair of thought objects x and y, it must be the case that S can think of x as having a certain cardinality (with respect to the relevant criterion) C1, and S can think of y as having a certain cardinality C2, and C1 > C2. Now reconsider Premise Three in the light of our proper understanding of the relation that figures in it:


("x)("y)((Ix & Ry) ® Myx)


It is here maintained that x’s simply being a mere thought object along with y’s being such that it can be thought to exist in reality, constitutes a sufficient condition for its being the case that x can be thought greater than y. But we can now see that this claim is plausible only if one presupposes reading (b), which employs the actual cardinality of one of the two relata. However, we also know that this an untenable interpretation of the predicate “M( )( )”, because it renders the description of God on line (1) question-begging. More generally, although “R( )” has the right kind of character to fund half of the relation denoted by “M( )( )”—namely, a modal-pistic component—“I( )” utterly lacks this character. So on Klima’s interpretation of the predicates, the claim of sufficiency in Premise Three is specious.

It’s easy to miss the incongruity on account of the fact that Premise Two (“Ig”) is asserted in the context of a reductio assumption. The required modal-pistic component is smuggled into the argument by way of the context in which the claim is made: “Suppose that God exists only in the imagination.” The ear is anesthetized just enough to let Premise Three go by unchallenged.

In order simultaneously to render the sufficiency claim in Premise Three plausible and to accommodate (g), 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 consequences of this conclusion are far from clear. But I won’t pursue this issue any further, because I think that Klima’s argument may face a more telling objection.


III.   Constitutive Reference and Conceptual Closure


In order to frame this objection, we need to consider the escape route that Klima offers to the atheist (77-79). He says that the consistent atheist can verbally accept Anselm’s description of God without accepting all of the logical consequences of that description so long as he uses Anselm’s description to refer parasitically rather than constitutively. This distinction is rather like the distinction between semantic reference and speaker reference, which, as Kripke pointed out, often come apart from one another. For example, Smith remarks to Jones about George W.’s foreign policy strategy, referring to him as “the most intelligent president the U.S. has ever had.” Jones, given to sarcasm, might co-opt Smith’s description and reply by saying “‘The most intelligent president the U.S. has ever had’ is a blasted idiot.” Jones does not contradict himself; he employs Smith’s preferred description to refer to Smith’s intended referent, even though Jones does not believe that the description is actually true of the intended referent. Smith uses the description to refer constitutively to George W.: the description constitutes part of Smith’s conception of Bush, at least on this occasion. Jones uses the description to refer parasitically: he borrows part of the conceptual content of Smith’s conception of Bush in order to refer to the man, but he does so without adopting that conceptual content as part of his own conception of Bush.

Thus, the atheist can refer parasitically to God under the theist’s preferred description (viz., as that than which nothing can be thought greater) without thinking that the description actually applies to the referent, and he therefore is not obliged to accept any of the logical consequences of that description. In Klima’s words:

[T]he atheist, when speaking about God, is constantly making parasitic reference to the theists [sic] object of thought, using the theist’s beliefs to refer to this thought object, but without ever sharing them. Accordingly, he will be willing to admit that whoever thinks of something as that than which nothing greater can be thought of also has to think that this thing exists in reality, and that it cannot even be thought not to exist in reality. Being a consistent atheist, however, he himself will think of nothing as that than which nothing greater can be thought of (whence that than which nothing greater can be thought as such will not be in his mind). But he still will be able to think of what theists think of as that than which nothing greater can be thought. (79)


But while Klima sees this move as a way for the atheist to temporarily evade the point of the argument, a condition to be remedied through a gradual process that will result in his acquisition of the correct conception of God, the situation may well be more worrisome for the theist than he acknowledges. For when the definite description is used to make constitutive reference to God, I maintain that the conceptual system of the referring agent becomes conceptually closed, and therefore inconsistent. By conceptual closure, I have in mind a certain condition of a conceptual system that is an analog of what Tarski called semantic closure. A semantically closed language is one that contains antinomies like the Liar’s Paradox (“This sentence is false.”). Tarski’s characterization of semantic closure is that of a conflation of object- and meta-languages, where the object-language contains terms (such as “true”) that include in their extension expressions in the object-language.[4] While his discussion of viable semantic theories is of course vastly more complicated than this, the normative upshot is not to be missed: semantically closed languages are inconsistent, and are therefore to be avoided when possible.      

For my present purposes, I would like to offer a characterization of semantic closure that employs two basic notions: semantic content and semantic sortal. The semantic content of a linguistic expression is just the meaning of that expression as it is commonly used by the speakers of the language. For example, the semantic content of the statement “It will rain this afternoon” is a certain prediction—namely, that it will rain this afternoon. A semantic sortal is a semantic property that partitions the statements of the given language (perhaps exhaustively). Truth, on this account, is a semantic sortal. With these two notions in place, I characterize semantic closure as follows:


(SC)       A language is semantically closed if it contains a statement whose    semantic content includes a semantic sortal for that language.


This characterization implies that any language capable of expressing the Liar’s Paradox is semantically closed, because part of the semantic content of the statement of the paradox is a semantic sortal (viz., falsity).

       The considerations behind semantic closure apply in an analogous way to domains of thought objects, or conceptual systems. Whereas a language consists (in part) of statements, a conceptual system consists (in part) of thought objects. Whereas statements have semantic content, thought objects have conceptual content. And whereas statements are subject to partitioning by semantic sortals (for example, not all statements are true), thought objects are subject to partitioning by conceptual sortals (for example, not all thought objects are objects simpliciter, enjoying extramental existence). Now I contend that conceptual systems are subject to a variety of closure that is for them what semantic closure is for languages. To be more specific:


(CC)       A domain of thought objects is conceptually closed if it contains a thought object whose conceptual content includes a conceptual sortal for that domain.


My suggestion is that the conceptual system required for Klima’s argument satisfies this condition for conceptual closure, and can therefore be expected to produce paradoxes analogous to the Liar. Let me indicate the sort of thing I have in mind.

Let Modest be the thought object x such that x is a non-mere thought object (Klima’s “~Ix”) just in case x cannot be thought to exist in reality (Klima’s “~Rx”). Modest is a curious fellow: an omniscient, but terribly shy genie. Indeed, Modest is so shy that if someone other than himself can so much as conceive of him as actually existing, he “extinguishes” himself. Such an individual is certainly peculiar, but unlike the round square, it doesn’t appear to be an impossible object. At any rate, Modest seems like a perfectly acceptable thought object.

But does Modest actually exist? Suppose he does. In that case, Modest cannot be thought to exist in reality (by virtue of our description of him). But that contradicts our supposition that he does in fact exist. So it looks like Modest cannot even be thought (employing constitutive reference) to actually exist without contradiction. So Modest cannot be thought to exist in reality. But in that case, Modest is a non-mere thought object; that is, Modest actually exists. But by asserting Modest’s existence, it’s quite clear that we can think Modest to exist in reality. Hence, Modest doesn’t actually exist.

This paradox arises from the conditions that are required for Klima’s interpretation of Anselm’s proof to go through. When an agent refers constitutively to God under the description deployed in the argument, he has in mind a thought object that contains as part of its conceptual content the notion of greatness. That’s how constitutive reference works. But if one conceives of greatness in terms of existence (as Premise Three requires), by referring constitutively to God as that than which nothing can be thought greater, one conceives a thought object that has existence as part of its conceptual content, and one’s conceptual system therefore becomes closed.

And if conceptual closure is to be avoided as steadfastly as is semantic closure, the atheist has available a toothy tu quoque response to Klima’s offered escape route. Klima says that the atheist isn’t moved by the argument, because he uses the requisite description to make parasitic reference to God. The atheist’s response ought to be that his doing so isn’t a matter of stubbornness; it’s an attempt to avoid paradox and inconsistency. Klima maintains that one may in this case employ parasitic reference; the atheist maintains that one must.

       This is but the merest sketch of how such an objection might run. There are numerous details that must be worked out in order to render the objection as potent as is might be. But I believe that this line is worth pursuing further, and considered alongside the first objection I’ve laid out, I want to claim that it constitutes grounds for doubting that Anselm’s argument succeeds in establishing God’s existence, as Klima says it does.

[1] G. Klima. “Saint Anselm’s Proof: A Problem of Reference, Intentional Identity and Mutual Understanding.” In Medieval Philosophy and Modern Times, Holmström-Hintikka, ed. Kluwer (Dordrecht: 2000), 69-87.

[2] A. Tarski. “The Semantic Conception of Truth.” Philosophy and Phenomenological Research, 4 (1944): 342-375.

[3] Klima, 73-74. I’ve removed parentheses that surround each predicate’s argument(s) because I consider them superfluous.

[4] See Tarski (348-349) for a fuller characterization of semantic closure.