Existence Is Not a Predicate

Ordinary grammar suggests that "exists" is a predicate like any other: just as "Socrates is bald" attributes baldness to Socrates, "Socrates exists" seems to attribute existence to him. But Russell argues this is an illusion. If existence were a genuine property of individuals, then "Pegasus does not exist" would attribute the property of non-existence to Pegasus — which seems to require Pegasus to be there to receive the property. Conversely, if existence is just a property every actual thing has, it is trivially true of everything and carries no information. Neither option is satisfactory.

Russell's resolution: "exists" is a predicate of propositional functions (or, in modern terminology, predicates), not of individuals. "Socrates exists" means "the propositional function 'x = Socrates' is sometimes true" — i.e., there is at least one thing identical to Socrates. "Pegasus does not exist" means "the propositional function 'x is Pegasus' is never true" — i.e., nothing satisfies the description associated with the name. On this account, existence claims are really claims about how often a predicate is satisfied — they are existential quantifications, not attributions of a property to an individual.

The most celebrated consequence is the refutation of the ontological argument for God's existence. Anselm's argument — that God, being the greatest conceivable being, must exist (since a non-existent God would lack a perfection) — presupposes that existence is a perfection that can be included in a concept. If existence is not a genuine property of individuals but a second-order claim about predicate-satisfaction, then we cannot include it in God's concept any more than we can include it in the concept of a unicorn. The argument attempts to move from conceptual analysis to existential conclusion, and Russell's theory shows why this move is logically invalid.

The claim that existence is not a genuine first-order predicate was made by Kant (Critique of Pure Reason, A598/B626) and forms part of Kant's refutation of the ontological argument. Russell's theory of descriptions provides the rigorous logical framework within which the insight can be fully articulated. Frege makes the same point in "Foundations of Arithmetic" §53 (1884).