Yes. The book of life is the inscription of those ordained to eternal life, to which one is directed from two sources: namely, from predestination (which direction never fails) and from grace (for whoever has grace, by this very fact becomes fitted for eternal life). This direction fails sometimes because some are directed by possessing grace, to obtain eternal life, yet they fail to obtain it through mortal sin.
Therefore those who are ordained to possess eternal life through divine predestination are written down in the book of life simply, because they are written therein to have eternal life in reality; such are never blotted out from the book of life. Those, however, who are ordained to eternal life, not through divine predestination, but through grace, are said to be written in the book of life not simply, but relatively, for they are written therein not to have eternal life in itself, but in its cause only.
Yet though these latter can be said to be blotted out of the book of life, this blotting out must not be referred to God, as if God foreknew a thing, and afterwards knew it not; but to the thing known, namely, because God knows one is first ordained to eternal life, and afterwards not ordained when he falls from grace.