We believe that men will be punished for their own sins, and not for Adam's transgression.
We believe that through the Atonement of Christ, all mankind may be saved, by obedience to the laws and ordinances of the Gospel.
Both the second and the third articles of faith go together in explanation. They are both related to the fall of Adam and Eve, and the atonement of Jesus Christ. And so, to begin, I will explain the fall and the atonement.
In the beginning God created Adam and Eve in the garden called Eden. When Eve, and later Adam, ate the fruit of the Tree of Knowledge of Good and Evil against the commands of God they were then sent out from the garden and the presence of God. Their eating the forbidden fruit brought about mortality to the world. Mankind was no longer fit to dwell in the kingdom of God, and should there not be some atonement for this fallen state of man, no one would again be able to return to Celestial Kingdom.
To help explain the implications of the fall, and the importance of an atonement, I'll quote the prophet Alma with my own commentary.
"And now, ye see by this that our first parents were cut off both temporally and spiritually from the presence of the Lord; and thus we see that they became subjects to follow after their own will. Now behold, it was not expedient that man should be reclaimed from this temporal death, for that would destroy the great plan of happiness."
The fall of our first parents brought about two death: temporal and spiritual. The first is to physicall die, which became possible when the mortal body was obtained The second is to be cast out of the presence of God, which happened when they were driven from the garden of Eden.
"And now remember, my son, if it were not for the plan of redemption, (laying it aside) as soon as they were dead their souls were miserable, being cut off from the presence of the Lord. And now, there was no means to reclaim men from this fallen state, which man had brought upon himself because of his own disobedience."
Because mankind is no longer worthy to be in the presence of God, having sinned, if they were to die without a way for forgiveness, they'd spend eternity in the misery of knowing their sins.
"Therefore, according to justice, the plan of redemption could not be brought about, only on conditions of repentance of men in this probationary state, yea, this preparatory state; for except it were for these conditions, mercy could not take effect except it should destroy the work of justice. Now the work of justice could not be destroyed; if so, God would cease to be God. And thus we see that all mankind were fallen, and they were in the grasp of justice; the justice of God, which consigned them forever to be cut off from his presence."
What Alma is teaching here is that because man hadsinned, mankind would have to be punished for their sins or God, being just, would no longer be God. But because God is merciful he provided the plan of redemption, a way to be forgiven. The problem is that even with the plan of redemption someone has to pay for the sins committed, because justice cannot go unanswered.
"An now, the plan of mercy could not be brought about except an atonement should be made; therefore God himself atoneth for the sins of the world, to bring about the plan of mercy, to appease the demands of jstice, that God might be a perfect, just God, and a merciful God also."
To answer the requirements of justice, God being just, a Savior was chosen to pay the price for sin. This is Jesus Christ, God's son. But the question could be asked, "Why did God's son need to be sent to atone for man's sins? Could not a man atone for his sins?" Alma continues,
"Now, repentance could not come unto men except there were a punishment, which also was eternal as the life of the soul should be, affixed opposite to the plan of happiness, which was as eternal also as the life of the soul." (Alma 42:10-16.)
The son of God needed to atone for sins because the atonement had to be as eternal as the fall. The fall, if no atonement, would be eternal. Mankind would be in a fallen state forever. Therefore the atonement had to be eternal. The atonement had to cover all the sins of all mankind. To further explain why Christ needed to atone for mankind Alma teaches, on another occasion:
"It must be an infinite and eternal sacrifice. Now there is not any man that can sacrifice his own blood which will atone for the sins of another. Now, if a man murdereth, behold will our law, which is just, take the life of his brother? I say unto you, Nay. But the law requireth the life of him who hath murdered; therefore there can be nothing which is short of an infinite atonemetn which will suffice for the sins of the world." (Alma 34: 10-12.)
The third article of faith says that mankind may be saved through the atonement, explained above, by obedience to the laws and ordinances of the gospel. Alma now explains the importance of having laws, and how following them brings our salvation.
"Now, how could a man repent except he should sin? How could he sin if there was no law? How could there be a law save there was a punishment?"
This is straightforward: A law has two parts - if you obey the law you are rewarded, and if you break the law you are punished. But you cannot be punished for breaking a law if there is no law, and you cannot be rewarded for following a law if there is no law.
"Now, there was a punishment affixed, and a just law given, which brought remorse of conscience unto man. Now, if there was no law given - if a man murdered he should die - would he be afraid he would die if he should murder? And also, if there was no law given against sin men would not be afraid to sin. And if there was no law given, if men sinned what could justice do, or mercy either, for they would have no claim upon the creature?"
As explained before, if there was no law then man cannot be punished, nor rewarded. Also, however, if there was no law then man would not be afraid to sin.
"But there is a law given, and a punishment affixed, and a repentance granted; which repentance, mercy claimeth; otherwise, justice claimeth the creature and executeth the law, and the law inflicteth the punishment; if not so, the works of justice would be destroyed, and God would cease to be God." (Alma 42:17-22.)
The law was set in the garden of Eden, which Adam and Eve broke, bringing about the punishment of forever being in an eternal state. But because God is merciful he allowed a way for repentance by giving his Son to atone for the sins of the world, if they would obey his commandments. So if people obey the commandments of God they can be forgiven of their sins through the atonement that Christ paid. This repentance allows us to return to the presence of God with a clear conscience. However, because God is just as well as merciful, if a person does not follow God's commandments then it is as if an atonement for their sins was not made for them and they have to pay the price of justice.
And now, having that lengthy explanation given, we can go back to the two articles of faith and explain them one at a time.
The second article of faith says, "We believe that men will be punished for their own sins, and not for Adam's transgression." As shown previously Adam's transgression (the fall) has been taken care of by way of the atonement. The fall brought spiritual death to all people, but the atonement repairs the bridge back across that spiritual death. It is only the sins that a person commits himself that he is judged for.
The third, continuing on from where the second leaves off, is that not only will men be judged only for their own sins, but man can even be saved from these sins if they are obedient to God's will. The demands by justice for our sins have been paid for through the atonement if we repent of them. This is the blessing of mercy and justice working together.