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)