Canonist Raymond Leo Cardinal Burke (Prefect of the Apostolic Signatura, the Church's "supreme court", 2008-2014) quoted in "The Canonical Nullity of the Marriage: Process as the Search for the Truth" (ch. 9 of Remaining in the Truth of Christ), § "Judicial Process and Pastoral Charity", John Paul II's

1990 annual address to the Roman Rota, in which he had noted that those who approach the tribunal in order to clarify their situation in the Church have a right to the truth, declaring:

[Ecclesiastical authority] thus takes note, on the one hand, of the great difficulties facing persons and families involved in unhappy conjugal living situations and recognizes their right to be objects of special pastoral concern. But it does not forget, on the other hand, that these people also have the right not to be deceived by a sentence of nullity which is in contrast to the existence of a true marriage. Such an unjust declaration of nullity would find no legitimate support in appealing to love or mercy, for love and mercy cannot put aside the demands of truth. A valid marriage, even one marked by serious difficulties, could not be considered invalid without doing violence to the truth and undermining thereby the only solid foundation which can support personal, marital and social life. A judge, therefore, must always be on guard against the risk of misplaced compassion, which would degenerate into sentimentality, itself only pastoral in appearance. The roads leading away from justice and truth end up in serving to distance people from God, thus yielding the opposite result from that which was sought in good faith.⁸

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^{8. Ioannes Paulus PP. II, Allocutio, “Ad Romanae Rotae Praelatos, auditores, officiales et advocatos anno iudiciali ineunte”, Ianuarii 18, 1990, AAS 82 (1990): 875, no. 5. (Hereafter, Allocutio 1990.) English translation: Woestman, Papal Allocutions, p. 211, no. 5.}

Moral certititude

He continues ("Nature of the Process for the Declaration of Nullity of Marriage", § "2. Judgment as a Declaration regarding the Truth of a Claim of Marriage Nullity"):

The college of judges or the single judge has no power to dissolve a valid marriage, but only to search for the truth about a particular marriage and declare authoritatively either that, with moral certitude, the truth of the nullity of the marriage has been established or proven (constat de nullitate) or that such moral certitude has not been reached (non constat de nullitate).

[1983] Can. 1608 [=1917 can. 1869] §1. For the pronouncement of any sentence, the judge must have moral certitude about the matter to be decided by the sentence.

"Moral certitude" is, according to Fr. John Hardon's Modern Catholic Dictionary:

Confident assent concerning human conduct based on people's normal and predictable responses to certain needs, abilities, and motivations.

McHugh, O.P. & Callan, O.P., Moral Theology:

643. Kinds of Certitude.—Judgments may be certain in a greater or less degree.
          (a) They are metaphysically certain, when error is absolutely impossible, the opposite of what is held by the mind being a contradiction in terms which omnipotence itself could not make true. […]
          (b) Judgments are physically certain, when error is impossible according to the laws of nature, the opposite of what is held by the mind being unrealizable except through intervention of another cause. […]
          (c) Judgments are morally certain, when error is impossible according to what is customary among mankind, the opposite of what is held by the mind being so unlikely that it would be imprudent to be moved by it.
     Examples: One is morally certain that what a reputedly truthful and competent person relates to one is true. A person is morally certain that a conclusion he has drawn about his duty in a particular instance is correct, if he believes that he has overlooked no means of reaching the truth. Testimony and inference, since they come from free and fallible agencies, may lead into error; but, when they appear to have the requisite qualities indicative of truth, they are for the most part reliable and in practical life have to be considered as such.

Dishonest judges

Cdl. Burke also notes that (ibid.)

one cannot exclude the possibility that there are those who consciously and explicitly reject the Church’s doctrine on marriage and yet accept and exercise an office in a tribunal in a manner that betrays their oath of office,³⁰

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^{30. See CIC, can. 1454 ["Can. 1454 All who constitute a tribunal or assist it must take an oath to carry out their function correctly and faithfully."] [=1917 can. 1621]; CCEO, can. 1112.}

^{(my emphases throughout)}