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An argument called ‘proof by contradiction’ seals the deal on papal infallibility.
Many non-Catholic Christians make the claim that if papal infallibility were true, Scripture would explicitly teach it. Because Scripture fails to make an explicit pronouncement, they argue, it must have been foreign to the biblical authors. Although papal infallibility is in fact not explicit in Scripture, the biblical and historical data we do have make the infallibility of the pope quite clear.
To provide this proof, I will need to establish three points of argumentation.
Premise 1: The gates of hell will not prevail against the Church Jesus established.
Perhaps the most controversial part of this proof, we need to recognize that the Church Jesus established is infallible. This part of the argument does not actually require that the Church Jesus established be the Catholic Church. It simply requires that whatever Church he established be protected. This can be established through several Bible verses. The most common would be Matthew 16:18. We could also reference other Bible verses, such as 1 Tim. 3:15. If the Church is a pillar for truth, it must have some protections always to be such a pillar. Finally, Jesus tells his apostles in Luke 10:16, “He who hears you hears me, and he who rejects you rejects me, and he who rejects me rejects him who sent me.” This suggests that the apostles have the authority of Jesus while he is in heaven. If this is the case, the Church has certain protections from error.
Premise 2: Peter was the leader of the early Church during the time of the apostles.
Continued below.

The Pope Is Not Not Infallible
One way to prove papal infallibility is called a ‘proof by contradiction.’ We just need to establish three premises, and the logic flows from there.