Jimmy Akin has presented a logical argument for purgatory. His argument is, as follows:
Premise 1: There will be neither sin nor attachment to sin in heaven.
Premise 2: We (at least most of us) are still sinning and are consequently attached to sin at the end of this life.
Conclusion: Therefore, there must be a period between death and heavenly glory in which the saved are cleansed of sin and their attachment to sin.
In other words, after death those who have achieved eternal salvation but still have some residue or attachment to sin need to undergo purification before entering the joy of heaven. The purification is necessary to achieve the holiness necessary to stand in the presence of God.
To rebut this deductive argument and its conclusion one or both of the premises must be shown to be wrong. That is, to argue against premise 1 one must take the position that sin or attachment of sin is present in heaven. And, to argue against premise 2 one must take the position that sin or attachment of sin does not exist in this life.