Purgatory is defined in the Catechism of the Catholic Church as -
III. THE FINAL PURIFICATION, OR PURGATORY

 1030 All who die in God's grace and friendship, but still imperfectly purified, are indeed assured of their eternal salvation; but after death they undergo purification, so as to achieve the holiness necessary to enter the joy of heaven.

 1031 The Church gives the name Purgatory to this final purification of the elect, which is entirely different from the punishment of the damned.606 The Church formulated her doctrine of faith on Purgatory especially at the Councils of Florence and Trent. The tradition of the Church, by reference to certain texts of Scripture, speaks of a cleansing fire -
The purification process necessary to attain holiness for heaven is a process that occurs between death and the beatitude of heaven.

 Argument for the Existence of Purgatory.

 The Lords prayer states "Thy will [of the Father] be done in earth, as it is in heaven (Matt 6:5-15; Luke 11:1-13).
The Father's will is that of God.
God's will is impeccable (Isaiah 6:3).
Therefore the Father's will is impeccable.

 What is impeccable is without fault.
What is without fault is without sin.
The Father's will is impeccable and thereby without sin.

 Jesus gave Christians the Lord's prayer to show Christians the will of the Father.
The will of the Father is that Christians have an impeccable will as that of the Father (Matt 5:45, 1 Pet 1:15) and as indicated in the Lord's prayer (Matt 6:5-15; Luke 11:1-13).
What Jesus gives, is to be followed to attain heaven, for Jesus as God is the perfect teacher.

 Heaven is to see God face to face as the supernatural last end of man.
To see God face to face, infers man is in God's presence.
But to be in God's presence is to live a divine life in accord with the will of God (Rom 8).
To live a divine life in accord with the will of God means those men in heaven must have an impeccable will in union with God.
Yet we are told by God that men on earth sin and thereby are not impeccable.
And we are told that the just man sins seven times a day (Prov 24:16).
Hence men on earth sin regularly and thereby do not have impeccable wills.

 For man to attain the vision of God requires that man die in the state of grace.
For a man to die in the state of grace means at death, mans will is ordered by grace towards God as the object of man's supernatural love.
Yet man's loving of God as the supernatural last end of man does not exclude minor sins which can be committed, but which do not remove man being directed towards his true final end.
For some sins do and other sins do not bring death (1 John 5:16).
Those sins which bring death are mortal and cause man to love a creature over that of God (Rom 2:12).
Other sins do not bring death and thereby do not remove the love of God that dominates man's will (Prov 24:16, 1 John 5:16).

 Sins which do not bring death are venial sins.
Sins which do bring death are mortal sins.
Men who die in the state of grace may have committed venial sins and thereby have a human will directed towards God as the true final end of man.
Yet venial sin committed by men in the state of grace infers mans will in the state of grace is not impeccable.
Yet final beatitude is for men who have an impeccable will in union with the Father (1 John 3:1-3).
Hence men who die in the state of grace but have committed venial sins are unfit for heaven, for the human will is not impeccable.
Also men who die in the state of grace but have committed venial sins are unfit for hell, for the human will is correctly directed towards God as the ultimate love of man.
And hell is the false ultimate end of men who falsely love a creature above that of God when mortal sin is committed.
Hence the man who dies in the state of grace, is directed towards God, but without an impeccable will, whereby man cannot enter either heaven or hell.

 Hence there is a third state, whereby those men who die in the state of grace, without an impeccable will, in which God removes any imperfection in men to prepare men for heaven.
For men who die in the state of grace are directed by God to the true ultimate end, though imperfectly with venial sin.
And God does not act to frustrate His work of grace in men, but always acts to fulfill His work of grace.
For God does not contradict His own work in men.
Hence the work of grace performed by God, commenced when men are alive, will continue in men unto the perfection of impeccability.
Thereby men who die in the state of grace are ordered towards heaven, but because of venial sin, are not fit for heaven, but are further perfected by God in a third state that is neither heaven, nor hell.
The third state after death is rightly called purgatory by the Catechism of the Catholic Church.
Hence from the nature of man, venial sin and the impeccability of heaven, purgatory must exist.
Hence purgatory exists.

JM