Anselm’s Argument and Berry’s Paradox

We argue that Anselm’s ontological argument (or at least one reconstruction of it) is based on an empirical version of Berry’s paradox. It is invalid, but it takes some understanding of trivalence to see why this is so. Under our analysis, Anselm’s use of the notion of existence is not the heart of the matter; rather, trivalence is. Anselm’s ‘proof’ of the existence of God goes like this: I have the concept of that than which nothing greater can be thought. If this concept were not instantiated, something greater could be thought namely the same concept with the property of existence. This is a contradiction, therefore that than which nothing greater can be thought is instantiated God exists. We suggest that this argument is based on an empirical version of Berry’s paradox. It is invalid, but it takes some understanding of trivalence to see why this is so. Under our analysis, Anselm’s use of the notion of existence is not the heart of the matter; rather, trivalence is (as far as we know, this distinguishes the present reconstruction from its predecessors, notably Lewis 1970; see also Oppy 2007 for references). For the sake of clarity, we start by explaining with some simple examples why someone who always reasoned within a bivalent framework would make systematic errors when faced with non-trivial empirical paradoxes. In particular, we develop an empirical version of Berry’s paradox which is similar in structure to Anselm’s argument. With these tools in place, we offer a simple reconstruction of Anselm’s argument, one in which the notion of existence is entirely innocuous. 1 Paradoxes and Trivalence Four important lessons can be drawn from the recent study of semantic paradoxes (Kripke 1975): (i) A semantics for a paradoxical object language should be (at least) trivalent, so that paradoxical statements can be given an indeterminate truth value (neither true nor false). (ii) Paradoxes may arise in devious ways. The Liar (This very sentence is not true) is particularly simple because it involves direct self-reference and a truth predicate. But paradoxes may be obtained with a denotation or a satisfaction predicate instead (McGee 1992), and direct self-reference may be replaced with circularity (= indirect self-reference), or with self-referential quantification (paradoxes without circularity can also be obtained when infinite series of sentences are considered, as in Yablo 1993). (iii) Some statements may display a pathological behavior even though they may sometimes obtain a classical truth value. This is in particular the case of empirical Liars, which only display a pathological behavior when certain empirical conditions hold. A particularly simple example is provided by a sentence named ε, which says: It is raining or ε is not true. When it is raining, the first disjunct is true, and hence the sentence is too (on the assumption, which we will make throughout, that sentences are evaluated within Kleene’s strong trivalent logic); * For discussion and comments, I wish to thank Paul Egré, Béatrice Lenoir, and an audience at École Normale Supérieure (Lyon). The author gratefully acknowledges the financial support of the American Council of Learned Societies (‘Ryskamp Fellowship’) and of UCLA. 1 In fact, it follows from results reported in Schlenker 2007 that under broad conditions every semantic phenomenon and hence every paradox obtained with self-reference can be emulated without it.