Saint Anselm of Canterbury was an Intalian Benedictine abbot, monk, philosopher and theologian of the Catholic Church who held the office of Archbishop of Canterbury from 1093 to 1109. Before he held the office of Archbishop of Canterbury, Saint Anselm published the Proslogion in which he proposed the first Ontological Argument in the Western Christian tradition.
An Ontological Argument is an argument for the conclusion that God exists. Saint Anselm’s conclusion that God exists in reality (Theorem 3) is formulated based on the following definition and theorems:
Definition 1: God is something than which nothing greater can be thought.
Theorem 1: If something is that than which nothing greater can be thought, then it exists in reality.
Theorem 2: There is something than which nothing greater can be thought.
In this post, I describe Saint Anselm’s formulation of the Ontological Argument. In particular, I elaborate on the definition and theorems mentioned above.
Definition 1. Let \(G\) be a predicate for something that is God, \(P\) be a predicate for something than which nothing greater can be thought and \(x\) be a variable. God is something than which nothing greater can be thought. \[ \forall x \quad G(x) \leftrightarrow P(x) \] In the Proslogion, Saint Anselm defines God as “something than which nothing greater can be thought” (Anselm, 1708, p. 99).
Theorem 1. Let \(R\) be a predicate for something which exists in reality. If something is that than which nothing greater can be thought, then it exists in reality. \[ \forall x \quad P(x) \rightarrow R(x) \] Proof. In the Proslogion, Saint Anselm argues that if something is that than which nothing greater can be thought, then it exists in reality:
“And surely that than which a greater cannot be thought cannot exist only in the understanding. For if it exists only in the understanding, it can be thought to exist in reality as well, which is greater. So if that than which a greater cannot be thought exists only in the understanding, then the very thing than which a greater cannot be thought is something than which a greater can be thought. But that is clearly impossible.” (Anselm, 1708, p. 100)
To paraphrase, suppose that there is something than which nothing greater can be thought. If it does not exist in reality, then it is not something than which nothing greater can be thought. This is because it is now something than which something greater can be thought. Therefore, if there is something than which nothing greater can be thought, then it exists in reality: \[ \forall x \quad P(x) \rightarrow R(x) \]
which was to be demonstrated.
Theorem 2. There is something than which nothing greater can be thought. \[ \exists x \quad P(x) \]
Proof. In the Proslogion, Saint Anselm argues that there is something than which nothing greater can be thought that exists in the understanding.
“The fool has said in his heart, ‘There is no God’” (Psalm 14:1; 53:1)? But when this same fool hears me say “something than which nothing greater can be thought,” he surely understands what he hears; and what he understands exists in his understanding, even if he does not understand that it exists [in reality.] (Anselm, 1708, pp. 99-100)
To paraphrase, because the “fool hears […] something than which nothing greater can be thought” and “understands what he hears”, he understands “something than which nothing greater can be thought”. And because “what he understands exists in his understanding”, “something than which nothing greater can be thought […] exists in his understanding”. Therefore, there is something than which nothing greater can be thought that exists in the understanding: \[ \exists x \quad P(x) \land U(x) \] where \(U\) is a predicate for something which exists in the understanding.
If there is something than which nothing greater can be thought that exists in the understanding, then there is something than which nothing greater can be thought: \[ \exists x \quad P(x) \] which was to be demonstrated.
Theorem 3. God exists in reality. \[ \exists x \quad G(x) \land R(x) \]
Proof. If there is something than which nothing greater can be thought, then it exists in reality (Theorem 1). There is something than which nothing greater can be thought (Theorem 2). Therefore, there is something than which nothing greater can be thought that exists in reality: \[ \exists x \quad P(x) \land R(x) \]
Because God is something than which nothing greater can be thought (Definition 1), there is something that is God that exists in reality: \[ \exists x \quad G(x) \land R(x) \]
In other words, God exists in reality. Which was to be demonstrated.
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