What mortification is not …

The Question that John Owen is seeking to answer is:

Suppose a man to be a true believer, and yet finds in himself a powerful indwelling sin, leading him captive to the law of it, consuming his heart with trouble, perplexing his thoughts, weakening his soul as to duties of communion with God, disquieting him as to peace, and perhaps defiling his conscience, and exposing him to hardening through the deceitfulness of sin, — what shall he do? what course shall he take and insist on for the mortification of this sin, lust, distemper, or corruption, to such a degree as that, though it be not utterly destroyed, yet, in his contest with it, he may be enabled to keep up power, strength, and peace in communion with God?

Before he answers he gives a number of descriptions as to what mortification is NOT, including the two below.

To mortify a sin is not utterly to kill, root it out, and destroy it, that it should have no more hold at all nor residence in our hearts. It is true this is that which is aimed at; but this is not in this life to be accomplished. There is no man that truly sets himself to mortify any sin, but he aims at, intends, desires its utter destruction, that it should leave neither root nor fruit in the heart or life. He would so kill it that it should never move nor stir any more, cry or call, seduce or tempt, to eternity. Its not-being is the thing aimed at. Now, though doubtless there may, by the Spirit and grace of Christ, a wonderful success and eminency of victory against any sin be attained, so that a man may have almost constant triumph over it, yet an utter killing and destruction of it, that it should not be, is not in this life to be expected.

Occasional conquests of sin do not amount to a mortifying of it. There are two occasions or seasons wherein a man who is contending with any sin may seem to himself to have mortified it:–

[1.] When it hath had some sad eruption, to the disturbance of his peace, terror of his conscience, dread of scandal, and evident provocation of God. This awakens and stirs up all that is in the man, and amazes him, fills him with abhorrency of sin, and himself for it; sends him to God, makes him cry out as for life, to abhor his lust as hell, and to set himself against it. The whole man, spiritual and natural, being now awaked, sin shrinks in its head, appears not, but lies as dead before him: as when one that hath drawn nigh to an army in the night, and hath killed a principal person, — instantly the guards awake, men are roused up, and strict inquiry is made after the enemy, who, in the meantime, until the noise and tumult be over, hides himself, or lies like one that is dead, yet with firm resolution to do the like mischief again upon the like opportunity. Upon the sin among the Corinthians, see how they muster up themselves for the surprisal and destruction of it, 2 Cor. 7:11. So it is in a person when a breach hath been made upon his conscience, quiet, perhaps credit, by his lust, in some eruption of actual sin; — carefulness, indignation, desire, fear, revenge, are all set on work about it and against it, and lust is quiet for a season, being run down before them; but when the hurry is over and the inquest past, the thief appears again alive, and is as busy as ever at his work.

[2.] In a time of some judgement, calamity, or pressing affliction; the heart is then taken up with thoughts and contrivances of flying from the present troubles, fears, and dangers. This, as a convinced person concludes, is to be done only by relinquishment of sin, which gains peace with God. It is the anger of God in every affliction that galls a convinced person. To be quit of this, men resolve at such times against their sins. Sin shall never more have any place in them; they will never again give up themselves to the service of it. Accordingly, sin is quiet, stirs not, seems to be mortified; not, indeed, that it hath received any one wound, but merely because the soul has possessed its faculties, whereby it should exert itself, with thoughts inconsistent with the motions thereof; which, when they are laid aside, sin returns again to its former life and vigour.

Owen is saying that mortification of sin is neither a complete killing of it so that it has no power to work in a person anymore nor is it the immediate rejection of a sin in the shock caused by its eruption or judgment.

source: John Owen, The Mortification of Sin in Believers, chapter 5.