Let’s deal with the specific question first: The word purgatory is not found in the Bible, true; neither is the word Trinity and yet orthodox Christians use the word Trinity to describe the three Persons in one God. Analogously, the concept that there is a cleansing between death and heavenly glory is one that is indeed taught in the Bible (Matt. 12:32, 1 Cor. 3:15, 1 Pet. 3:18-20).
Now let’s look at the more general question of how we can know a doctrine is true even if the name of the doctrine is not explicitly mentioned in the Bible.
The primary means of handing on the Christian faith has been sacred Tradition. Indeed, in one sense, the Bible itself can be considered a subset of sacred Tradition because it is the Church’s written Tradition. We can know a Christian doctrine is true if it has been continuously been taught since the time of Christ and the apostles because Christ commissioned the apostles and their successors to pass on the Christian faith whole and intact until the end of time. The Holy Spirit ensures that they do not fail in their commission from Christ.
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