We begin the fifteenth chapter of Maseches Yevamos, Mishnah 1. The Mishnah deals with the question of a woman's credibility when she testifies that her husband has died, and with the conditions under which the court relies on her testimony.
The first case - peace between them and peace in the world:
"HaIshah shehalchah hi uva'alah lemedinas hayam, shalom beino l'veinah v'shalom ba'olam, uva'as v'amrah meis ba'ali - tinasei" - a woman who went overseas together with her husband at a time when there was harmony between them and also peace in the world, meaning that there was no war that they were caught up in, and who returned and said that her husband died - she is believed and may remarry.
The basis of this ruling is what we already learned in the tenth chapter: a single witness suffices, and even the woman herself, because "ishah dayka u'minsava" - a woman investigates carefully to make certain that her husband really died, out of fear of the penalty that will fall upon her if it later turns out that her husband is alive.
Not only that, but "meis ba'ali - tisyabem": if her husband had no children, she is believed even for the purpose of entering into yibum with her brother-in-law. In all of these cases we rely on the fact that the woman checked the matter thoroughly.
Two cases in which she is not believed:
"Shalom beino l'veinah u'milchamah ba'olam" - all is well in the relationship between husband and wife, but there is a war raging in the world. Here we are concerned that the woman drew her own conclusion, based on what she saw, that her husband died from a wound or something similar, while in reality he may not have died at all.
"Ketatah beino l'veinah v'shalom ba'olam" - there is strife between the husband and the wife. The Gemara explains that this refers to a case in which the woman claimed to her husband that he had divorced her in front of these two witnesses, and when the witnesses were questioned they answered, "This never happened," and no divorce had taken place at all. The very fact that she claimed to have been divorced, even though it was proven untrue, clearly indicates that there is significant strife between them.
In a situation of strife we are concerned that the woman is willing to take a risk upon herself and say that her husband died even though he did not, because she desperately wants out of this marriage, and she is prepared to gamble on the matter becoming complicated later.
Therefore, in both of these cases - whether there is peace between them and war in the world, or strife between them and peace in the world - "uva'as v'amrah meis ba'ali - einah ne'emenes".
The dispute between Rabbi Yehudah and the Sages:
Rabbi Yehudah says: "L'olam einah ne'emenes, ela im ken ba'as bochah u'vegadeha keru'im" - she is not believed unless she comes in tears and with torn garments, which testify to the genuine distress she was in.
The Sages said to him: "Achas kach v'achas kach tinasei" - whether she comes with signs of grief or not, either way she may remarry.
The Gemara cites a beraisa in which the Sages answered Rabbi Yehudah: according to the logic of your position, that she must come with signs of distress, the result is that the clever and sophisticated woman - who knows how to put on an act and arrive in tears with torn garments - will be permitted to remarry, while the simple woman, who is not shrewd enough to think of putting on an act, will not be permitted to remarry. It is not reasonable to distinguish between them.
In summary: in this Mishnah we learned that a woman who says "my husband died" is believed both to remarry and even to enter into yibum, on the strength of the principle "ishah dayka u'minsava", provided that there was peace between them and peace in the world. But where there is war in the world - perhaps she drew her own conclusion that he died - and where there is strife between them - perhaps she took a risk in order to free herself from the marriage - she is not believed. We also examined the dispute between Rabbi Yehudah and the Sages regarding the signs of grief, and the argument of the Sages that her credibility should not depend on how clever she is.