In common speech
Definitions of 'infallible' differ widely. In common speech, 'infallibility' can refer to a person (or a group of persons), to an act of teaching by these persons, or to the information being taught. Infallibility can refer to the both 'absence of error' or to the 'inability to err'. Although these definitions are similar, they are philosophically distinct; it is theoretically possible for a person to live their entire life without erring even though they had the ability (and potential) to do that. A person who never commits an immoral act or speaks a false sentence by choice would thus qualify for 'absence of error' without being qualified for 'inability to err'.
Infallibility is sometimes used to refer to someone's ability to 'learn' something with certainty. For example, a careful researcher might study a hundred books, each of which contains a few errors, and after carefully judging the statements in these books might deduce the complete, error-free truth. This is referred to as 'learning infallibly' or 'knowing infallibly'. However, this meaning is rarely used.
Epistemology, a branch of philosophy, is concerned with the question of what, if anything, humans can know. The answer to the issue of whether or not a human can be infallible depends on which philosophical school is receiving the question.
- Advocates of philosophical skepticism claim that one cannot know anything with certainty, much less be infallible. Fallibilists hold a similar position, but claim that what is true today could be wrong tomorrow.[dubious ]
- Advocates of subjectivism claim that there is no objective reality or truth, and therefore anyone can be considered infallible, since whatever is within a person's consciousness is considered the real and the true.
- Advocates of reason and rationality claim that one can gain certainty of knowledge, through a process of extreme refinement measures unlikely to be perfected enough for someone to assurably say "certainty of this knowledge is absolute", yet also assume by chance that one could land on the objective without the knowledge being confidently described as "universally certain", thus as a result, advocates tend to avoid this altogether and instead rely upon Occam's Razor as a suitable means for obtaining knowledge.
Human involvement in the revelatory process is very important because God has no choice but to communicate in human language through human agents. But if this must be so, God can at least be expected to prevent human weakness and shortcoming from marring this divine process. If our sources of knowledge are not infallible, then who is going to decide what to accept and what to reject?  The Oxford Dictionary of the Christian Church defines infallibility as "Inability to err in teaching revealed truth". Roman Catholic and Eastern Orthodox Church theology claim that the Church is infallible, but disagree as to where infallibility exists, whether in doctrines, scripture, or church authorities.
In contrast, Protestant and non-denominational Christian churches believe that the Christian Church is indeed fallible—as evidenced by the requirement of Christ's sacrifice on the cross to pay for the sins of the world, including those of his Church—and that only God's word in Scripture is infallible. They also completely reject the Roman Catholic claim regarding Papal Infallibility, citing not just scriptural reasons, but also the many times popes have contradicted each other and the history of mistakes committed by many popes throughout Roman Catholic Church history.
Because of the complexity in defining infallibility, some Protestant and non-denominational views confuse infallibility and impeccability, as if the Pope were immune from sin, but that is not the Roman Catholic Church's doctrine. In Roman Catholic theology, only the actual act of teaching may properly be called "infallible". According to the First Vatican Council (1869–71) and as reaffirmed at Vatican II (1962–1965) the Pope is infallible when speaking ex cathedra on matters of faith and morals.
Some who reject infallibility cite the Talmud, Pesachim 94b:
The sages of Israel say: "The sphere (Earth) remains fixed and the constellations revolve," while the sages of the nations say: "The sphere revolves and the constellations remain fixed."...the sages of Israel say: "during the day the sun moves below the canopy (sky) and at night above the canopy," while the sages of the nations say: "during the day the sun moves below the canopy and at night below the ground." Rebbi said: "Their words seem more correct than ours..."
The words of the Mishna are commented on by numerous commentators, and Yehuda Levi argues that evidence mounts that the Geonim and the Rambam perceived that the sages of the Talmud "erred in a matter of astronomy. The Rambam wrote that the great sages are not expected to advocate positions perfectly in-line with modern science because they were "scholars of that generation," often basing their assessments of what "they learned from the scholars of the era."
Additional Shi'a teachings
In Shi'a theology, the belief is that the Ahl al-Bayt, including Muhammad, his daughter Fatima Zahra and Shi'a Imams are all infallible and do not make mistakes. It is believed that they are infallible in the sense that all statements or teachings made by them can be relied on to be certainly true, that all information believed by themselves is true, and that they have complete knowledge about right and wrong and never intend to disobey God, in a sense, perfect creation. It is also held by Shi'as that there were 124,000 Prophets, beginning with Adam and ending with Muhammad - with all, including the latter, being infallible in the same sense as the Ahl al-Bayt. However, for information about whether or not Islam states that Moḥammad and other Messengers or Prophets were always infallible, or unquestionable for any of their acts, see the Qur'an (5: 116) (11: 36 - 37, 40 - 47) (37: 139 - 142) (66: 1).
In the Bhagavad Gita, Krishna explains to Arjuna (Bg 15.16 to 15.20) "There are two classes of beings, the fallible and the infallible. In the material world every living entity is fallible, and in the spiritual world every living entity is called infallible. Besides these two, there is the greatest living personality, the Supreme Soul, the imperishable Lord Himself, who has entered the three worlds and is maintaining them. Because I am transcendental, beyond both the fallible and the infallible, and because I am the greatest, I am celebrated both in the world and in the Vedas as that Supreme Person. Whoever knows Me as the Supreme Personality of Godhead, without doubting, is the knower of everything. He therefore engages himself in full devotional service to Me, O son of Bharata. This is the most confidential part of the Vedic scriptures, O sinless one, and it is disclosed now by Me. Whoever understands this will become wise, and his endeavors will know perfection."
- Daniel,W.Brown, "Rethinking tradition in modern Islamic thought", p61. Cambridge University Press, Cambridge, 1999. ISBN 0521653940
- Cross, F.L. and Livingstone, E.A. (eds), "infallibility" in The Oxford Dictionary of the Christian Church, p831. Oxford University Press, Oxford, 1997. ISBN 0-19-211655-X
- Levi, Yehuda, The Science in Torah, Feldheim Publishers 2004, p. 92.
- Maimonidies, The Guide for the Perplexed part 3, chapter 14.
- Sources about infallibility of Tzaddikim
- "Shia News:Infallibility of the Prophets". Retrieved 2007-05-14.
- "Answering-Ansar.org - The Khalifatullah in Shi'a Belief". Retrieved 2007-05-14.
- Quran 5:116
- Quran 11:36–47
- Quran 37:139–142
- Quran 66:1
- Bhagavad Gita
- The dictionary definition of infallibility at Wiktionary