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Forgiveness is the act of pardoning an offender or an offense. When used in the scriptures it can mean a vertical form of forgiveness, God forgiving man, or a horizontal form of forgiveness, man forgiving man. When man receives forgiveness from God, it is made possible only through the atonement of Jesus Christ.
The Doctrine of Forgiveness From God
The need for forgiveness from God comes from offending God through sin, or acting contrary to the will of God. All are in need of forgiveness, since all have sinned (Ecclesiastes 7:20, Romans 3:23). Forgiveness comes as a result of repentance (Doctrine and Covenants 58:42) and is only possible because of the Atonement of Jesus Christ (Mosiah 4:2, Doctrine and Covenants 60: 7).
Receiving Forgiveness From God
Perhaps the first necessary element in receiving forgiveness from God is repentance  (Doctrine and Covenants 138:19). In addition to or in conjunction with repentance, other things seem to be important in order to receive forgiveness for our sins. These include acknowledgment of our sin (Psalms 32: 5), a willingness to worship God over anything or anyone else (Joshua 24: 19), prayer (1 Nephi 7:21, Enos 1: 5), and forgiving those who have offended us (Luke 6:37, Doctrine and Covenants 82: 1).
It is interesting to note that while the acknowledgment of sin and prayer are closely intertwined with the doctrine of repentance, the principle of forgiving others in order to obtain forgiveness is unique to forgiveness. Forgiving others of their sins or trespasses against us is considered below.
Many examples of forgiveness can be found throughout the scriptures, including God's forgiveness of Adam, Moses, Paul, Alma, Joseph Smith and many others. Another example that may be useful to consider is Enos, whose brief account can be found in the Book of Mormon.
Enos, the son of the prophet Jacob and the nephew of Nephi, had gone into the mountains to hunt, and while hunting, he began to ponder the teachings of his father regarding happiness and eternal life. As he pondered, he began to feel a need to pray to God, which he did. Enos eventually received confirmation from God that his sins were forgiven, but not before he underwent what he calls a "wrestle" with God "after praying all the day long". For Enos forgiveness was something that he had to truly desire and work for in addition to whatever steps he had surely taken earlier to repent.
The Effects of Forgiveness From God
When one obtains forgiveness from God, the forgiveness is complete and total. It is so total that it is likened unto God "forgetting" our sins  (Hebrews 8:12, D&C 58:42). It is also compared to taking something that is stained scarlet red, and cleansing the stain so completely as to be pure white, like snow (Isaiah 1:18).
Forgiveness From Man
As mentioned above, one requirement for forgiveness that is separate from the requirements of repentance is to forgive others. In regard to forgiving others, the scriptures teach men to forgive those who truly repent of the things that they have done (Mosiah 26:29), regardless of who they are (Doctrine and Covenants 64:10) or how many times they have offended (Matthew 18:21, Doctrine and Covenants 98: 40Moroni 6:8). 
A failure to forgive others impacts an individual's capacity to receive forgiveness. The Savior taught very clearly that if an individual is unwilling to forgive other people, that individual will not be forgiven (Matthew 6:15 3 Nephi 13:14).
The Savior taught a valuable parable regarding the principle behind forgiving others. He told of a king who found that one of his servants owed him a vast sum of money, and he prepared to have all of the servant's belongings sold or impounded to help pay the debt. The servant begged the king for more time, and the king, having compassion, forgave the servant of the entire debt.
This same servant left the king and eventually found someone who owed him money, although it was a minor sum. In much the same manner as the original servant begged for mercy, this lessor debtor also begged for mercy. Unfortunately the plea fell on deaf ears, and the original servant cast the lessor debtor, who owed him the small sum, into prison. This information made its way to the king, who became very angry with the servant. The king felt that if he was willing to forgive a vast sum of money, the servant should be willing to forgive a lessor sum of money. The king then revoked his compassion and forgiveness and gave the servant over to the "tormentors" until everything was paid.
The Savior finishes the parable by clearly telling his disciples, "So likewise shall my heavenly Father do also unto you, if ye from your hearts forgive not every one his brother their trespasses" (Matthew 18:35).
Not Eligible for Forgiveness
There are some unfortunate cases wherein individuals have committed sins that are ineligible for forgiveness. These "unforgivable sins" are understood to be denying the Holy Ghost (Mark 3:29) and murder.
Denial of the Holy Ghost
Denying the Holy Ghost requires more than a typical hardening of the heart to Holy Ghost and the testimony that comes through faith alone. The Prophet Joseph Smith taught that in order to sin against the Holy Ghost, a person "must receive the Holy Ghost, have the heavens opened unto him, and know God, and then sin against Him. After a man has sinned against the Holy Ghost, there is no repentance for him. He has got to deny Jesus Christ when the heavens have been opened to him, and to deny the Plan of Salvation with his eyes open to the truth of it."  When this statement is compared to Paul's description of the unpardonable sin in Hebrews 6:4-6 and Doctrine and Covenants 76:35, one can get a sense of the extensive knowledge required to commit the sin of denying the Holy Ghost. It requires a special knowledge, beyond faith, that the Savior is real, that he is the Son of God, etc. By rejecting him after having this sure knowledge, it is as if we would actually crucify the Savior again, which would be to shed the most innocent of blood.  after having the most explicit knowledge. This is where those who sin against the Holy Ghost differ from the Roman soldiers, who crucified the Savior. Their belief in his divinity, if in fact they did believe, would have been based on faith; the knowledge required to deny the Holy Ghost appears to be an actual knowledge of the reality and divinity of the Savior. The nature of the extensive knowledge required to deny the testimony of the Holy Ghost is such that few obtain such a testimony in their probationary state.
Murder, or the unlawful taking of an innocent life, is second only to the denial of the Holy Ghost in terms of sins that are unforgivable. It appears to be second to the aforementioned sin because of a slight difference in scriptures regarding the consequence of murder.
It is clear that an individual who murders after having entered into the New and Everlasting Covenant will fail to have his name written in the Lamb's Book of Life (Doctrine and Covenants 132:13). However, Alma 39: 6 relates that for those who murder, it is "... not easy for him to obtain forgiveness." (emphasis added) Alma's characterization that it is "not easy" for someone to obtain forgiveness is slightly different from many of the other scriptures regarding the consequence of murder, suggesting that perhaps there are some rare instances where murder may be forgiven. It is clear, however, that those who have entered into sacred covenants, and violate them by unlawfully taking an innocent life, will be unable to receive a forgiveness of the murder from God.
Other Resources on Forgiveness
Scriptures for Forgiveness
A Mormon Guide to Forgiveness
- Repentance is a topic that is comprehensive enough that it should be discussed on its own.
- while this brief synopsis may be helpful in understanding the concept of forgiveness, even more benefit can be gained by studying the story of Enos directly.
- Although the concept of God forgetting our sins is helpful in understanding the depth of His forgiveness, to suggest that God would actually forget something is contrary to the nature of God, esp. His clairvoyance or all-knowing power. It is perhaps better to suggest He no longer remembers those actions as sins, but merely actions with no negative consequence attached.
- While a strict reading of the scripture suggests that mankind is only obligated to forgive someone 490 times, the context suggests that the Savior was trying to teach that forgiveness is not something that is attached with a number, as the religious scholars of Christ's day tried to do. To be "counting" the number of times a person has been forgiven runs contrary to the principle of forgiveness where the offense is "forgotten."
- See Matthew 12:32 for the difference between speaking against Christ and speaking against a sure testimony of Christ.
- Teachings of the Prophet Joseph Smith, p. 358
- see Doctrine and Covenants 132:27 for a further explanation of the consequence of shedding innocent blood.
- Modern scriptures mention only one mortal being who has qualified to be considered a "son of perdition," and that is Cain.