How does St. Anselm of Canterbury’s ontological argument reason from the idea of God to God’s existence?

St. Anselm of Canterbury’s ontological argument, famously presented in his Proslogion, reasons from the concept of God to the necessity of God’s existence through the following logical steps:

• Definition: Anselm defines God as "that than which nothing greater can be conceived" (aliquid quo nihil majus cogitari possit).12
• Understanding: Even those who deny God's existence, such as the "fool" mentioned in the Psalms, understand this definition. Therefore, the concept of God exists within the human intellect.34
• Comparison of Existence: Anselm argues that it is greater to exist in both reality and the understanding than to exist in the understanding alone.34
• The Contradiction: If God existed only in the intellect, one could conceive of a greater being—one that exists in both reality and the intellect. This would contradict the initial definition of God as the greatest conceivable being.12
• Conclusion: To avoid this contradiction, God must necessarily exist in reality.12

Philosophers have debated this argument for centuries, particularly regarding the assumption that existence is a predicate or attribute that can be added to a concept to make it "greater".2