John Owen's Argument for Limited Atonement (Syllogism)

Either Christ died for:
1. All of the sins of all people
2. Some of the sins of all people
3. All of the sins of some people

However,

1. If Christ underwent punishment for all of the sins of all people, then no one will be in Hell.
2. Some people will be in Hell.
Therefore 3. Christ did not undergo punishment for all of the sins of all people.

1. If Christ underwent punishment for only some of the sins of all people, then all people will be in Hell.
2. Not all people will be in Hell.
Therefore 3. Christ did not undergo punishment for some of the sins of all people.

1. If some people will not be in Hell, then Christ underwent punishment for all of the sins of some people.
2. Some people will not be in Hell.
Therefore 3. Christ underwent punishment for all of the sins of some people.

Owen anticipates a response:
1. Christ underwent punishment for all sins
2. Unbelief is a sin
Therefore 3. Christ underwent punishment for the sin of unbelief