There are two passages from the Talmud Bavli that, perhaps more than any other, are taken to be fundamental to our understanding of the legal process of Halakhah. The first is the story known as Tanur Shel Akhnai, found in Baba Metsia.
It has been taught: On that day R. Eliezer brought forward every imaginable argument , but they did not accept them. Said he to them: ‘If the halakhah agrees with me, let this carob-tree prove it!’ Thereupon the carob-tree was torn a hundred cubits out of its place — others affirm, four hundred cubits. ‘No proof can be brought from a carob-tree,’ they retorted. Again he said to them: ‘If the halakhah agrees with me, let the stream of water prove it!’ Whereupon the stream of water flowed backwards — ‘No proof can be brought from a stream of water,’ they rejoined. Again he urged: ‘If the halakhah agrees with me, let the walls of the schoolhouse prove it,’ whereupon the walls inclined to fall. But R. Joshua rebuked them, saying: ‘When scholars are engaged in a halakhic dispute, what have ye to interfere?’ Hence they did not fall, in honor of R. Joshua, nor did they resume the upright, in honor of R. Eliezer; and they are still standing thus inclined. Again he said to them: ‘If the halakhah agrees with me, let it be proved from Heaven!’ Whereupon a Heavenly Voice cried out: ‘Why do ye dispute with R. Eliezer, seeing that in all matters the halakhah agrees with him!’ But R. Joshua arose and exclaimed: ‘It is not in heaven.’ What did he mean by this? — Said R. Jeremiah: That the Torah had already been given at Mount Sinai; we pay no attention to a Heavenly Voice, because Thou hast long since written in the Torah at Mount Sinai, After the majority must one incline.
This passage depicts a debate between R. Eliezer and the rest of the sages, wherein R. Eliezer calls on increasingly impressive miracles as verification for his argument, culminating in a heavenly voice that proclaims explicitly that R. Eliezer is correct. To this R. Yehoshua responds that the Torah itself says we do not care what revelation has to say about halakhah, and instead we follow the Torah’s own law that the majority vote is decisive. Halakhah was given to the Israelites to decide; It is in human, not Divine, hands.
The second gemara is revolves around yet another heavenly voice, this one in regard to a dispute between Bet Hillel and Bet Shammai, recorded in Masekhet Eruvin.
Abba stated in the name of Samuel: For three years there was a dispute between Beth Shammai and Beth Hillel, the former asserting, ‘The halakhah is in agreement with our views’ and the latter contending, ‘The halakhah is in agreement with our views’. Then a Heavenly Voice issued announcing, ‘[The opinions of] both these and these are the words of the living God, but the halakhah is in agreement with the rulings of Beth Hillel’. Since, however, both are the words of the living God’ what was it that entitled Beth Hillel to have the halakhah fixed in agreement with their rulings? Because they were kindly and modest, they studied their own rulings and those of Beth Shammai, and were even so [humble] as to mention the actions of Beth Shammai before theirs.
This passage depicts a debate that seems to have been unresolvable. It dragged on for three years, until such time as a heavenly voice came forth to decide it, with a surprising and radical conclusion. Instead of declaring which opinion is correct, the heavenly voice declares that both opinions are correct, and it is only because practice cannot simultaneously uphold both opinions that the voice must declare which one is “right.” But it is technical “right,” a practical necessity rather than an ontological fact. The halakhic system is such that there are issues regarding which reasonable people might disagree, in which case both of them are essentially correct, even if the demands of practical law are not so forgiving.
While these two passages are often touted for giving basic principles of the Oral Law and the Jewish Tradition, there is an essential contradiction between them that is often overlooked. The essential point of the first passage is that we do not listen to revelation when it comes to deciding halakhah, with the particular form of revelation under discussion being a heavenly voice. Yet the second passage hinges on the information learned from a heavenly voice. These two passages cannot coexist. Of course, numerous commentators have gone to lengths to show how they can and in fact do coexist, usually by putting conditions on R. Yehoshua’s blanket statement that we do not listen to revelation.
The Talmud Bavli itself, however, maintains that these two passages are indeed contradictory, and that one should not attempt to resolve them by limiting one of them.
For it has been taught: ‘The halakhah is always in accordance with the ruling of Beth Hillel. Nevertheless one who desires to adopt the view of Beth Shammai may do so, and one who desires to adopt the view of Beth Hillel may do so. One who adopts the view of Beth Shammai only when they incline to leniency, and likewise the view of Beth Hillel only when they incline to leniency, is a wicked person. One who adopts the view of Beth Shammai only when they incline to strictness and likewise the view of Beth Hillel only when they incline to strictness, [is a fool and] to such a one applies the verse: But the fool walks in darkness. But one must either adopt the view of Beth Shammai in all cases, whether they incline to leniency or strictness, or the view of Beth Hillel in all cases, whether they incline to leniency or strictness’. Now is not this statement self-contradictory? At first it says: ‘The halakhah is always in accordance with the ruling of Beth Hillel’, and immediately after it says: ‘Nevertheless one who desires to adopt the view of Beth Shammai may do so’? — This is no difficulty. The latter statement relates to the practice before the Heavenly Voice was heard, whilst the former states the law as it is after the Heavenly Voice was heard. Or, you may even say that the latter statement too was made after the Heavenly Voice was heard. [and yet there is no contradiction], for that statement is the view of R. Joshua who exclaimed: We pay no attention to a Heavenly Voice!
This third passage says that the ruling of the heavenly voice, that the halakhah is like Bet Hillel, was considered legally binding, except by R. Yehoshua who did not accept revelation as legally binding.
This passage leaves one thing unclear, however. While R. Yehoshua is explicitly said to be rejecting the decision that the practical law is always like Bet Hillel, it is unclear to how he treats the first half of the heavenly voices declaration. Does R. Yehoshua agree that “both these and these are the words of the living God”? There are arguments in either direction. On the one hand, this information is received via revelation, something R. Yehoshua apparently rejects. On the other hand, the information is not halakhically decisive, and it is possible R. Yehoshua only denies revelation a decisive legal capacity, rather than the ability to convey information about halakhah more generally, in which case he would only reject the second half of the heavenly voice’s statement, but not the first.
While it’s impossible to know for certain, it may be fruitful to speculate a little. If we go back to the original statement of the heavenly voice in the second passage we looked at, the two halves of the proclamation are connected. The first half tells us that more than one halakhic opinions can be essentially correct. While this clarifies why the debate under discussion in that passage had raged on for three years, it presents an immediate practical problem. Once you know for certain that two contradictory opinions are correct, how do you decide the practical law? No matter which one you choose you will know that you are violating a different, but also correct, opinion. Therefore the proclamation has a second half, where it gives a decisive ruling as to what should be done practically. No matter what is essentially correct, there is a different practical measure of correctness, and what is most correct in that sense is the opinion of Bet Hillel.
As connected as the two halves of the heavenly voices proclamation are, it is entirely possible that R. Yehoshua not only rejects both the second and first halves of the proclamation, but rejects the second half because he rejects the first half. The heavenly voice appears in the middle of a debate which apparently could not be resolved through the normal methods of halakhic jurisprudence. Given a Divine source of information, it is easy to determine who is ultimately correct and to then establish practice accordingly (notwithstanding a surprising turn of events wherein both opinions are deemed to be essentially correct). R. Yehoshua does not have this option. Instead, he sees that there are times when the halakhic discussion will have no single, obviously correct, answer. In such cases the answer is not to declare all answers equally correct or incorrect, but to simply allow every capable person to attempt to determine what seems to them to be correct in that situation. While R. Yehoshua’s entirely human conception of the halakhic system leads him to reject R. Eliezer’s opinion by Tanur Shel Akhnai, it leads to certain cases where the only option is not to be so decisive. Thus R. Yehoshua’s different conception of viable sources of halakhic information also leads, in some cases, not only to different halakhic conclusion, but to different types of conclusion, to conclusions that are very human attempts at arriving at halakhic correctness, as opposed to divine concessions to practicality.
 TB Baba Metsia 59b. Translations from halakhah.com, with some adaptation for ease of reading and clarity.
 TB Eruvin 13b.
 TB Hulin 43b-44a.