St. Anselm of Canterbury, Proslogion (1077-78). In chapter II of Anselm’s argument for God’s existence, he argues that God is “something than which nothing greater can be thought.” He first argues regarding the fool (one who “in his heart says that there is no God”; Psalm 14:1), that “something than which nothing greater can be thought exists at least in the understanding.” He reasons through this assumption – that such a being only exists in the understanding alone – to the conclusion of such a being existing not only in reality and in the understanding, but is actually unable not to exist (ch. III).
Rene Descartes, The Meditations (1637). Found in The Objections; replies to his two friends, Bannius and Bloemart. Descartes rejected Anselm’s ontological argument and furthered his own, suggesting that “the existence of God is demonstrated a posteriori from the mere fact that the idea of God is in us” (Proposition II). He demonstrates this proposition by arguing that ideas require causes. Ideas, as he defines them, are “the very thing thought, insofar as it exists objectively in the intellect.” In Axiom V., he notes that “the objective reality of our ideas requires a cause which contains this very same reality, and not merely objectively, but either formally or eminently.” However, our idea of God is an objective reality of which exists neither formally or eminently, nor can it exist outside of God, therefore, our idea of God requires God to be its cause – therefore, God exists.
Immanuel Kant, The Critique of Pure Reason (1787). In Book II (“Of the Dialectal Conclusions of Pure Reasons”), chapter III (“The Ideal of Pure Reason”), section IV, entitled, “Of the Impossibility of the Ontological Proof,” Immanuel Kant offers what is known as the classical refutation to the ontological argument for the existence of God. In this section, Kant furthers his famous critique of “existence is not a predicate.” In other words, by merely stipulating that such a being as God exists, it does not add to the reality of the actual existence of that being. In other words, predicates do not add actual existence to their subjects. He also argues that in denying the predicate to “God exists,” no contradiction arises in that denial. Therefore, God is not a necessary being.
Gottlob Frege, The Foundations of Arithmetic (1884). In his section “Solution and its Difficulty” and the “Distinction Between Component Characteristic of a Concept and its Properties,” (s. 53) Gottlob Frege offers his critique of the ontological argument by suggesting that since “existence is a property of concepts the ontological argument for the existence of God breaks down.” According to Frege, there is first-order existence and second-order existence. The latter involves existence as a predicate, while the former makes existence claims that are essentially meaningless. Furthermore, the ontological argument is what Frege considers a first-order existence claim.
Mortimer J. Adler, How To Think About God (1980). Throughout a number of chapters trying to understand the proper route as to how we should think about God, chapter 8 (“How to Formulate a Definite Description of God”) begins Adler’s discussion on Anselm’s argument and the value it has for finding a coherent definite description of God (although he finds the argument itself invalid). However, chapter 11 is where Adler argues that no “existential proposition can be self-evidently true or necessarily true.” For instance, when “we have direct perceptual acquaintance with a particular individual, the existence of that individual is evident to us, but the proposition which asserts that the individual exists is not self-evident, nor is its truth necessary” (104). It is in the same way to the proposition “God exists” cannot be necessarily true if understood existentially.
Gordon Clark, Three Types of Religious Philosophy (1973, 1989). In his chapter on Rationalism (ch. II), Clark first presents Anselm’s argument (only by quoting a passage from Anselm at length) and then immediately goes on to address Kant’s classic criticism to the argument. He argues that not only was Kant’s argument irrelevant – in that Kant supposedly wasn’t even addressing Anselm’s argument – but that it was also an incomplete criticism (i.e., Kant was too vague on his terms and wasn’t clear in linking his argument together).
Alvin Plantinga, God and Other Minds (1967). Alvin Plantinga dedicates two chapters to addressing/presenting and critiquing certain objections to Anselm’s argument. In the first of the two chapters (ch. II), Plantinga writes that “[w]e have no reason to believe, therefore, either that existence in reality cannot be predicated of a being presupposed to exist in the understanding, or that Anselm’s argument necessarily involves predicating real existence of a such a being.” In critiquing certain opponents (Kant, Ayer, Alston, etc.), Plantinga finishes by saying, “I think the conclusion to be drawn is that we do not have yet a general refutation of Anselm’s ontological argument.” However, in the second chapter, Plantinga goes on to critique Norman Malcolm’s modal ontological argument.
A.J. Ayer, Language, Truth, and Logic (1936). Though other parts in the book consider different objections to existence of God, chapter I of Ayer’s book (“The Elimination of Metaphysics”) contains his critique of the ontological proof (one that Plantinga (1967) considered in his book as well). Ayer agrees with Kant that “existence is not an attribute.” However, as Ayer states a bit differently than Kant, “when we ascribe an attribute to a thing, we covertly assert that it exists: so that if existence were itself an attribute, it would follow that all positive existential propositions were tautologies, and all negative existential propositions self-contradictory; and this is not the case.” You can find the Dover Books edition (1952) of Ayer’s critique starting on page 42.
Walter Kaufmann, Critique of Religion and Philosophy (1958). In chapter V of Kaufmann’s book (“The God of the Philosophers”), section 46 (“Perfection and the Ontological Argument”), we see his objection that “perfection” is a very general or vague term when it is applied to being. However, he comes to agree that God can be perfect in the sense of “the absence of flaws which generally characterize human beings.” However, he thence considers Kant’s objection immediately after this, but says “let us attempt a slightly different approach.” Through this “different approach” he comes to the conclusion that “[f]rom the definition of God we can only learn how he is to be thought of, not whether he exists.”
Anthony Kenny, The Unknown God (2004). Kenny’s addressing of Anselm’s argument should be understood as more so a discussion rather than a presentation or a critique (for instance, in the opening paragraphs of his essay Kenny writes: “I shall not in this essay be concerned with the validity of the ontological argument”). In prior essays discussing the nature religious language, Kenny turns to Anselm for help in finding coherency for our language regarding the infinite. He first considers his description of God (“something than which nothing greater can be thought”), and tries to find out if Anselm is justified in that description. Although this isn’t necessarily a presentation or critiquing piece of literature, but moreso a religious language oriented subject in Kenny’s book, I would still consider this book meritous for discussion.
George H. Smith, Why Atheism? (2000) In chapter nine of his book (“Metaphysical Muddles”), George Smith goes on to address the Ontological Argument(s) for the existence of God (Anselm, Descartes,) and its critiques (Kant, Hobbes, Gassendi, etc). His critique is more so focused on Anselm’s attack on the justification of the “fool.” In this chapter, he sides with Thomas Aquinas and objects that the ontological argument is “factual, not theoretical.” In other words, Anselm’s argument hinges off an empirical observation: that all men have a conception of God. Where Anselm suggests that “even the fool understands what he hears”, Smith says: “This is clearly an empirical assertion, since it depends on the truth of a psychological observation about human consciousness” (156).
Alvin Plantinga, The Nature of Necessity (1974). This is perhaps the book (or one of the books) that set the stage for our modern understanding of modal logic and its application to the ontological argument. In section X (see 7) we find Plantinga’s “Victorious Modal Version”, where his argument runs as the following: (1) It is possible that there is a maximally great being, that is, a being that is omniscient, omnipotent, and perfectly good in every possible world. (2) Therefore, there is an omniscient, omnipotent, and perfectly good being. Plantinga himself concedes that “not everyone who understands and reflects on its premise [ … ] will accept it.”
Robert Maydole, The Ontological Argument (2009). Robert Maydole’s essay in The Blackwell Companion to Natural Theology (2009) is an almost 40 page essay covering objections and different versions of the ontological argument. He not only presents Anselm’s argument, but also Norman Malcolm’s, Leibniz’, Descartes’, Godel’s, Hartshorne’s and Plantinga’s (although the last three are relatively shorter than the others). Good presentations of this argument can be found in Maydole’s essay.