You have to wonder sometimes about the strange relationships between parents and children. While parents may not always know where their children are, and while they may at times mix up the names of their children, it is hard to imagine a parent not recalling exactly whom their children may have married. Hard, but apparently not impossible.
“I betrothed my daughter but I do not know to whom I betrothed her[1].” (Kiddushin 63b) Not only is this beyond strange, it creates a nightmare halachic situation. The daughter, now married but unaware of who her husband is, is effectively an agunah, a legally married woman living without a husband. The only way out is to somehow discover who the mysterious husband might be. The Mishna continues “If one comes and says it was I who married her he is believed.” Just as we believed the father despite a lack of evidence we can believe the suitor who claims it is he to whom the father married off his daughter. 
Yet to what extent we believe the prospective husband is a matter of dispute. Rav is of the view that we allow him to give the woman a getthereby freeing her from her agunah status but we do not actually believe they are married. We believe they are married for the purposes of divorce only[2]. Of course unless the husband is a kohen he can “remarry” her after they divorce but they are not allowed to enter nissuin, the second and final stage of marriage based on his word alone. 
As to why this is so the Talmud explains that we can believe him to give a get as “one does not sin where there is no [benefit] to him.” If the man is lying he is the cause of the woman engaging in an adulterous union, whose offspring would be mamzerim. Even though others may never know this he does. We are not worried that he will cause one to sin so greatly - adultery does carry in theory a death penalty - with no actual benefit to himself[3]. Thus if he claims he is the husband, he may divorce her so that she may remarry. Yet paradoxically we do not allow them to marry “perhaps his desires overcome him” causing him to lie so that he may marry the girl. Allowing divorce and not marriage has the practical impact of removing much of the incentive of lying. 
Rav Ashi disagrees allowing the suitor to actually marry the woman despite no evidence that kiddushin had taken place. Unlike with Rav the Gemara does not explain his reasoning. Perhaps he is not concerned that one will lie even if his desires are strong for this woman. Yet that is not how Rashi understands his view. We must fear that one overcome by desire will lie. However he is afraid to do in this case. Despite the father’s announcement that he does not know to whom he married his daughter, it is hard to take that at 100% face value. He may not remember exactly to whom he betrothed his daughter but there is a good chance he would remember that it is not this fellow. The suitor is afraid to lie lest he get caught and it is that fear allows him to marry the girl. 
However Rav Ashi would agree that in the case where it is the girl who says “I was betrothed but I do not recall from whom I accepted the betrothal” the would-be husband is not believed when he says it was him[4]. This is because we assume the wife will cover up and protect her new husband. Even if the real husband shall appear the wife will deny he is the husband. Ironically it is her loyalty to her present would be husband that prevents her from marrying him. 
On the other hand even Rav would agree that in the case where two men claim to be the husband “if they want one can give a get and the other can marry” her. On the surface this makes no sense. If we don’t believe the one suitor where there is no reason to suspect he is lying, beyond a generic fear that his desire for her may cause him to lie, how can we allow marriage with two suitors? In this case there is a 100% chance that at least one of them is lying and possibly both may be. 
Let’s examine the dynamics of lying. The one who is lying must fear that the other person is telling the truth and knows that he is lying. And the liar fears that if in fact the other suitor is telling the truth the father may have his memory jogged and reveal him to be lying. Thus at the end of the day we are confident that despite his low moral character his fear of getting caught will prevent him from actually going ahead with marriage. He will be the one to give the divorce letting the other suitor marry the girl. And in the case where both are lying each suitor still must fear the other is telling the truth and here each will refrain from marrying the girl. We will then have a situation where 'they do not want' to marry and both will give a get
I find this Gemara - which could be used in an introductory course in psychology - fascinating. Sadly, for many lying is not a moral question but a practical one. Can I get away with it? 
The fear of getting caught lying plays a crucial role in halacha. It is this fear that allows a woman to remarry based on the testimony of only one witness that her husband is dead (see here). People just don’t lie if they fear they will get caught red-handed[5]. Sadly absent this fear, we must fear that people are less than truthful. This does not necessarily reflect evil but rather “their desires overcome them.” The realism of our Sages allows us to take comfort that our moral failing is part of the human condition allowing us to grow higher. 
[1] Jewish law gives a father a right to marry off his minor daughter. While this is most foreign to us - and hence its practice has been practically suspended for many centuries - in the context of Biblical society such made much sense. With the father acting in the best interest of the child no evidence beyond his word was needed to declare the daughter married.
[2] The idea that we can believe a claim for one purpose but not another, while slightly  illogical, is true in almost every legal system. A person may be found not guilty of a criminal offence but guilty for that same offence in regards to monetary payment. Rules of evidence are not the same across the board.  
[3] While I may have feared a person might be lying to free an agunah that too is not a direct benefit and while the motive may be sincere the Talmud at least did not suspect one of lying and thereby creating mamzerim. Whether this would still hold true today is something interesting to think about.
[4] If it is difficult to imagine a case where the father does not recall whom he chose as a son-in-law it is all but impossible to imagine a girl not knowing from whom she accepted a proposal. While this may be a case of using an extreme, unrealistic and unreal case to prove a point it may - and this is pure conjecture on my part - reflect the horrific conditions in existence after the Hadrianic persecutions. Poverty, death and despair ruled and in such circumstances the survivors may accept kiddushin eager to rebuild and not fully recall from whom.
[5] As we noted after the Cuban missile crisis Rav Soloveitchik questioned this assumption. Despite the fact that pictures demonstrated (there was no Photoshop in the 1960’s) the Russian lie such had little impact.