Concise Bible Dictionary:

There are three Hebrew words translated to forgive.
1. kaphar, “to cover” (Deut. 21:8; Psa. 78:38; Jer. 18:23). It is also translated “atonement.”
2. nasa, “to bear,” take away [guilt]: used by Joseph's brethren when they asked him to forgive them (Gen. 50:17); and used of God as "forgiving iniquity and transgression and sin" (Ex. 34:7; Num. 14:18); and in describing the blessedness of the man "whose transgression is forgiven, whose sin is covered" (Psa. 32:1).
3. salach, “to pardon,” used only of the forgiveness that God gives. It is employed for the forgiveness attached to the sacrifices: "it shall be forgiven him" (Lev. 4:20, 26, 31, 35; Lev. 5:10, 13, 16, 18; &c). It occurs in the prayer of Solomon at the dedication of the temple (1 Kings 8:30, 34, 36, 39, 50. Also in Psa. 103:3; Jer. 31:34; Jer. 36:3; Dan. 9:19).
In the N. T. two words are used: ὔφεσις., from ἀφίημι, “to send from, release, remit,” several times translated REMISSION; and χαριξομαι, “to be gracious, bestow freely, forgive.” Both words are applied to the forgiveness granted by God, as well as that between man and his fellow.
There are two aspects in which forgiveness is brought before us in scripture.
1. The mind and thought of God Himself towards the sinner whom He forgives. On the ground of the sacrifice of Christ, God not only ceases to hold those who have faith in Christ's blood as guilty before Him, but His favor is towards them. "Their sins and iniquities will I remember no more" (Heb. 10:17). Thus all sense of imputation of guilt is gone from the mind of God. "God for Christ's sake hath forgiven you" ἐχαρίσυτο, graciously forgiven, (Eph. 4:32). So in the O.T., "I will heal their backsliding, I will love them freely" (Hos. 14:4).
2. The guilty one is released, forgiven. "That they may receive forgiveness of sins" (Acts 26:18). "As far as the east is from the west, so far hath he removed our transgressions from us" (Psa. 103:12). "Your sins are forgiven you for his name's sake" (1 John 2:12). Hence it is true of all Christians, that their sins are forgiven. Another thought is included in the forgiveness of sins, namely, that having redemption by Christ, which brings into a new state, the whole guilty past is forgiven, removed from us, so that there is no hindrance to the enjoyment of that into which redemption brings.
The general principle as to forgiveness is stated in 1 John 1:9: "If we confess our sins, he is faithful and just to forgive us our sins;" and to this is added, "and to cleanse us from all unrighteousness." This involves honesty of heart, whether in a sinner first coming to God, or in a child who has grieved the heart of the Father by sinning. The two aspects above referred to are here also. The faithfulness and righteousness of God in forgiving, and the cleansing us from all unrighteousness. God is faithful to His own blessed character of grace revealed in His Son, and righteous through the propitiation which He has made.
3. If a Christian is “put away” from the assembly and is repentant, he is forgiven and restored (2 Cor. 2:7, 10). This of course is different from the act of God in forgiving sins, and may be called administrative forgiveness in the church; and if the act of discipline is led of the Spirit, it is ratified in heaven: (Compare John 20:22-23). This is entirely different from any pretended absolution that may be pronounced over poor deluded unconverted persons.
4. There is also a governmental forgiveness in connection with the government of God here below in time, both on God's part, and toward one another (Isa. 40:1-2; Luke 17:3; James 5:15-16; 1 John 5:16). We are called upon to forgive one another; and if we indulge in a harsh unforgiving spirit, we must not expect our Father to forgive us in His governmental dealings (Matt. 6:14-15).

From Anstey’s Doctrinal Definitions:

The underlying meaning conveyed in “forgiveness” is “to release or to remit someone from a debt.” It is sometimes translated “remission” to convey that idea. W. Kelly said, “Forgiveness is the remission of sins of those who believe in Jesus through faith in His blood” (The Bible Herald, vol. 1, p. 234).
W. Potter indicated that there are five aspects of forgiveness in Scripture (The Christian, 2006 January):
1) Judicial Or Eternal Forgiveness
This has to do with a pardon that a person receives from God through faith that delivers him from the eternal judgment of his sins. Thus, he has “the remission” of his sins (Matt. 26:28; Luke 24:47; Acts 2:38; 10:43; Heb. 9:22) or “the forgiveness of sins” (Acts 5:31; 13:38; 26:18; Eph. 1:7; 4:32; Col. 1:14; 1 John 2:12). In receiving this pardon, the believer’s conscience is purged as to his guilt (Heb. 9:14; 10:2, 22), and thus he has a conscious knowledge that his sins have been divinely forgiven. This blessing is a present possession of every believer on the Lord Jesus Christ (Eph. 1:7).
This eternal aspect of forgiveness of sins was first announced when the Lord rose from the dead (Luke 24:47). Prior to that, in Old Testament times and during the Lord’s ministry on earth, forgiveness was granted to persons purely in a governmental sense (Ex. 32:32; Lev. 4:20, 26, 31, 35; 1 Kings 8:34-39; Psa. 86:5; Jer. 36:3; Matt. 9:2-6; 6:14; Luke 7:47-48; 23:34). Thus, Old Testament saints did not know this eternal aspect of forgiveness of sins. Consequently, they lived with uncertainty as to whether their sins would be visited in judgment by God (Psa. 25:7, etc.). H. E. Hayhoe stated, “Previous to the first advent of Christ, the truth of eternal forgiveness of sins was not made known. Generally speaking, forgiveness as referred to in the Old Testament was governmental—that is, it had to do with this life but not eternity” (Present Truth For Christians, p. 10).
This doesn’t mean that the Old Testament saints aren’t in heaven, but that they didn’t possess a conscious knowledge of their sins being forgiven, as Christians do, because they didn’t know of the finished work of Christ. Their sins were held in abeyance through “the forbearance of God,” waiting for the time when God would place them on the Lord Jesus on the cross, whereby they would be judicially dealt with according to the claims of divine justice (Rom. 3:25). But the Old Testament saints didn’t know that. Thus, their sins were “covered” (Psa. 32:1), but today with the work of Christ having been accomplished and the coming of the Holy Spirit, we have a fuller revelation through the gospel as to what God has done with our sins. We know that our sins are “forgiven” (Acts 13:38) and that they have been “put away” (Heb. 9:26) and “taken away” (1 John 3:5).
2) Governmental Forgiveness
This refers to a pardon that God grants to a person whereupon a governmental judgment that He has laid on him on account of his sinful ways (Matt. 18:23-25; Gal. 6:7; 1 Cor. 11:29-32; 1 Peter 1:16-17; 3:12; 4:17; 1 John 5:16-17) is lifted and forgiven (Psa. 103:10-11; Matt. 18:26-35; John 5:14; James 5:15).
This aspect of forgiveness is conditioned upon two things on the part of the forgiven. Firstly, there must be genuine repentance (Lev. 26:40-41; 2 Chron. 12:7, 12; 33:11-13, 19; Jonah 3:5-10; 1 John 1:9). Secondly, there must be the holding of a forgiving spirit toward others (Matt. 6:12, 14-15; 18:23-35; Mark 11:25-26; Luke 6:37). J. N. Darby pointed out that these two requisites are illustrated in the life of Job. He repented in dust and ashes (Job 42:6) and he prayed for his three friends who had accused him unjustly, asking God that he would forgive them (Job 42:10). When God saw these two things with Job, He lifted his discipline and “turned” his “captivity.” A person does not even need to be a real believer to experience this kind of governmental forgiveness from God. This was the case with King Ahab (1 Kings 21:27-29). Hence, it is possible for a person to be governmentally forgiven, but not eternally forgiven! (Collected Writings of J. N. Darby, vol. 31, p. 362).
When we see the children of God sinning, under normal circumstances, we ought to pray that it would “not be laid to their charge,” and that they would be forgiven governmentally (2 Tim. 4:16; 1 John 5:16a). There may be occasions when discernment would dictate that we should not pray for a person in that way, thus acquiescing in the wisdom of God in His governmental dealings with His people (1 John 5:16b).
It is important to understand that both governmental judgment and governmental forgiveness have to do with God’s dealing with men while they live on earth; these dealings do not affect their eternal destiny. God’s governmental dealings with believers concerns their fellowship with Him, not their relationship with Him.
3) Restorative Forgiveness
J. N. Darby treats this as a subset of governmental forgiveness (Synopsis of the Books of the Bible on 1 John 1:9 – footnote). It has to do with God’s lifting of His governmental discipline which has been felt by one of His erring children in the way of having his or her communion severed through their own carelessness and sin, whereby communion is once again enjoyed. It is lifted because there has been self-judgment and the confession of sins (1 John 1:9). Mr. Darby explains this form of governmental action by God as follows: “If we speak rashly to our brother, or walk abroad carelessly through the streets and see some vanity, we shall find the effect of it in our own souls at the end of the day with God. If any angry word escapes me, I feel the effect at the end of the day with God; but grace will restore us” (Nine Lectures on the First Epistle of John, p. 15). Thus, even something as simple as having wrong thoughts or uttering angry words will cause the interruption of communion. God allows us to feel this loss as a governmental dealing with us. When we judge it and it is confessed, communion is restored.
While this could be taken to be part of governmental forgiveness, it is slightly different in the sense that communion may be restored to a person, but he still may have to carry the consequences of his actions governmentally. This was the case with David. God told him that because of his sin with Bathsheba and the murder of her husband, the sword would not depart out of his house forever (2 Sam. 12:9-10). This judgment went with David all his days. But the Lord said that He had put away David’s sin because of his repentance, and thus David was restored to fellowship and communion with the Lord (2 Sam. 12:13). This is evident by the fact that he wrote many psalms of praise and worship after his restoration to fellowship with God.
4) Brotherly Forgiveness
This has to do with forgiving our brother if he has wronged us (Matt. 18:21-22; Eph. 4:32). There are two things here that mustn’t be confused. Firstly, we are to forgive the person who has done us wrong from our hearts (Matt. 18:35). That is, we are to hold a forgiving spirit toward him, even if there is no sign of repentance in him. This is important because if we don’t do this, bad feelings could arise in our hearts toward that person. And then, secondly, when the person who has wronged us repents and apologizes for what he has done, then we are to forgive him by expressing it to him audibly or formally (Luke 17:3-4).
Luke 17:3-4 has been used to justify the holding of a unforgiving spirit toward a person, because the person has not apologized. The offended party will say, “I’m not going to forgive him until he repents, because that’s what Scripture says I am to do.” However, Matthew 18:35 shows that regardless of whether the person has apologized or not, we are to hold a forgiving spirit toward him. Scripture warns that those who will not forgive their brother from their hearts will have the Lord’s governmental forgiveness withheld from them in the wrongs that they have done (Matt. 6:14-15; 18:23-35; Mark 11:25-26).
5) Administrative Forgiveness
This refers to the assembly acting administratively in matters of discipline toward erring individuals. The apostles were given this authority from the Lord to act administratively for Him in matters of retaining a person’s sins and the remitting of them (John 20:23). Assemblies gathered to the Lord’s name have this power too (Matt. 18:18-20; 1 Cor. 5:4). If they act to “put away” a person from the fellowship of the saints (1 Cor. 5), and if that person is repentant, they are to lift the censure placed on him and “forgive” him (2 Cor. 2:6-11). (See Judgment, Administrative.)

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