The following is the Doctrine of Particular Redemption as I understand it. (Yes. I know there is more than one variant of this doctrine. I am going with the non-Equivalentism version).
Particular Redemption:
(1) Christ died for the non-elect and the elect.
(2) Christ died for the non-elect and the elect as regards the matter of common grace;here we are NOT talking about His dying for their sins.
(3) Christ died for the elect as regards the matter of special grace;here we are talking about His dying for their sins.
(4) There is no sense in which Christ died for the non-elect in which He did not die for the elect.
(5) Additionally, this atonement was sufficient for all, but efficient only for the elect.
Common Grace : involves such things as rain or food. It also involves such things as the free offer.
* Add Venn diagram
The impression I get from Romans is the all of Creation fell when Adam sinned. Hence it is subject to futility.
"For the creation was subjected to frustration, not by its own choice, but by the will of the one who subjected it, in hope that the creation itself will be liberated from its bondage to decay and brought into the glorious freedom of the children of God." ~Romans 8:20-21
What this means is that even animals are in some respects fallen.
So here is the thought. We are saying that common grace involves non-salvific (or non-sin canceling) benefits to the righteous and unrighteous (Psalm 145). So when we consider common grace, we consider a sense in which Christ died for all – elect and non-elect.
So Christ’s atonement makes possible common grace.
Now - This may sound strange. We can even go so far as to say that Christ died in some sense for animals. Why? Psalm 104 teaches us that common grace involves benefits even for animals.
"He makes grass grow for the cattle, and plants for man to cultivate — bringing forth food from the earth..."(v.14). See also Psalm 145.v9 in particular.
Additionally, we can say that Christ also died for all of the creation in the sense that it is His death that is going to free a Creation that is fallen, i.e. subject to bondage and futility and decay.
