No other papal claimant in history has ever manifested such inexcusable pertinacity in explicit heresy as Jorge "Francis" Bergoglio; who not only professes heretical doctrines like the conciliar popes have done, but explicitly, unequivocally, and adamantly, not merely contradicts, but outright rejects dogma. This point will be explained at length in vol. 2 of my book. As Bellarmine, Ballerini, and Cappellari (Gregory XVI), explain, such a one would fall from office, and manifest by his obstinate heresy, that he had (in Ballerini's words) "in some manner abdicated" the supreme pontificate. This doctrine of tacit abdication was incorporated into the 1917 Code of Canon Law, and remains essentially unchanged in the 1983 Code. Such loss of office takes place ipso facto, i.e. automatically; ipso jure, i.e. by operation of the law itself; and therefore, sine ulla declaratione, without any judgment pronounced by competent authority -- as set forth in canon 188 n. 4.
Canon 188.4 can only be applied analogically to the case of a heretic pope. For a more authoritative explanation of the loss of papal office for heresy, I recommend an article by Fr. Girlanda, the former rector of the Gregorian University, who holds a Ph.D in canon law, which he taught for at various faculties of the Gregorian for four decades.
The article titled, which is title "The Vacancy of the Roman See", appeared in Civilta Cattolica in 2013, and was discussed by Roberto de Mattie at this years' Catholic Family News conference. Professor de Mattie noted that Fr. Ghirlanda is neither a traditionalist nor a progressive, but is simply a scholar who had gathered and studied the past thousand years of canonical tradition concerning the loss of papal office. In the article, Fr. Ghirlanda explains that there are four ways the papal see can become vacant. He writes:
Fr. Ghirlanda: “The vacancy of the Roman See occurs in case of the cessation of the office on the part of the Roman Pontiff, which happens for four reasons: 1) Death, 2) Sure and perpetual insanity or complete mental infirmity; 3) Notorious apostasy, heresy, schism; 4) Resignation. In the first case, the Apostolic See is vacant from the moment of death of the Roman Pontiff; in the second and in the third from the moment of the declaration on the part of the cardinals; in the fourth from the moment of the renunciation."
Concerning numbers 2 and 3, he goes on to explain that because 'the first see is judged by no one', the Cardinals would not depose the Pope, but would only declare the fact of his heresy or insanity, at which time the see would become vacant.
Fr. Ghirlanda: “There is the case, admitted by doctrine, of notorious apostasy, heresy and schism, into which the Roman Pontiff could fall, but only when teaching as a “private doctor,” who does not demand the assent of the faithful ... However, in such cases, because ‘the first see is judged by no one’ (Canon 1404) no one could depose the Roman Pontiff, but only a declaration of the fact would be had, which would have to be done by the Cardinals, at least of those present in Rome. Such an eventuality, however, although foreseen in doctrine, is held to be totally unlikely (…) The certainty and the perpetuity of madness, like the totality of insanity, must be ascertained through accurate medical reports. The cessation from the primatial office would only be declared by the Cardinals, at least those present in Rome; therefore also in this case there would be no deposition.”
What Fr. Ghirlanda is saying is that if a pope falls into heresy or insanity, the Church does not remove him from office by deposing him. Instead, the Cardinals ascertain and declare the fact of heresy or insanity, at which time the office becomes vacant ipso facto. The loss of office does not occur until the Cardinals investigate and declare the fact.
It is also worth noting that an ipso facto deprivation differs from an ipso facto excommunication, since the latter does not require human judgment or a declaration, whereas the former does. Cardinal Tommaso de Vio explains this distinction, and ovserves that it is the canonists.
“The power of jurisdiction is by man’s appointment: both giving it and taking it away belong to human judgment. I said ‘much less’ because more is required to incur deprivation ipso facto than to incur excommunication, since incurring the censure does not require a declaration, whereas incurring deprivation does, according to the canonists.”