Above is a drawing of Pope Pius IX, the Pope who presided over the 1st Council of Vatican in 1870. Below is my attempt to give, very succinctly, the rationale or dialectical thought from where the idea of Papal infallibility comes from.
Christ promised us that the gates of hell should never prevail against the Church (Matt 16:18); likewise, that He would “be with” the Church “unto the end of the age” (ibid 28:18), and that the Church is the pillar and bulwark of the truth.
(1) Thus, we have immediate grounds for inferring the *indefectibility* of the Church. In other words, she cannot defect from this function since it is rendered necessary (by way of supposition) in our Lord’s promise.
But, since the Church is a doctrinal missionary society *by nature* (i.e. go and preach the gospel to all creatures), her *indefectibility* must imply her *infallibility*, or more thoroughly put, the inability to teach something which would forfeit the soul’s ability to reach the destiny of salvation (i.e. a heresy so great to impugn the process of salvation)
(2) Thus, the 2nd inference would be that the society of the Church is *infallible*.
In addition, we know that the Church is, in principle, a visible society, not least because the holy fathers of the 4th century added into the Creed that the Church has four marks, one of which is Apostolicity. This does not merely mean her doctrine is the same as that of the Apostles, but also that her very being is an “Apostle”, which means her successive growth in the world is done by way of “sending”, a prior “sender” that is. This is kept by what the New Testament called the “laying on of hands by the presbyerate” , and what was later understood as the sacramental succession from the Apostles; hand-to-body.
But since this visible Church, as doctrinal missionary society, is also hierarchically ordered with a Papal head as holding supreme authority (see here – here – and here ) in all matters pertaining to that very doctrine which she is entrusted to preach to all men, both her *indefectibility* and her *infallibility*, as well as her *visibility*, must imply Papal infallibility – – – at least in some sense, and under some condition.
(3) Thus, the 3rd inference is Papal infallibility.
But since the tradition has never posited that a Pope is generally immune from making an error, or even negating a dogma in his ordinary teaching, it must be implied that his infallibility has special and extraordinary conditions.
(4) Thus, the 4th inference is a conditional Papal infallibility.
But since a conditional Papal infallibility would be defined by an intention on teaching the whole church, since it is she who cannot be destroyed, it is implied by that fact that Papal infallibility is invoked and operative only under this specific condition.
Thus, you have the rationale for the teaching of Vatican I – An indefectible & infallible Church whose doctrinal voice is concentrated in the person of the Pope when she defines the Apostolic faith to be held by the whole Church.