There were three laws that were said by Rabbi Yehoshua: two in his name and one in his name. A woman is allowed to go outside into the public domain on Shabbat with a city of gold, which is a crown made to look like the city of Yerushalayim. We are not concerned that she is going to remove it in the Reshut HaRabim, the public domain, which would be a violation of carrying on Shabbat. With all other jewelry, we are usually concerned. We are nervous that a woman is going to take off her jewelry. She wants to show it off to her friends. She wants to display what she is wearing. She may end up carrying it for amot (cubits) in the Reshut HaRabim, which is a biblical violation.
But what it really means is that we are not concerned about the city of gold because only very rich women would wear it. Rich women would not have to show off their jewelry in public. They did not do that. It was considered not classy to show off your jewelry. So rich women would not do that. Therefore, it is not a problem for them to wear it. Halakha (Jewish law) is different nowadays, when women are also not likely to take off their jewelry and show it to their friends. Therefore, jewelry nowadays would also become permissible.
He also taught that someone who races pigeons for money is pasul le'edut, is not allowed to give testimony in the Beit Din (rabbinical court), because he is considered to be a thief on a deRabbanan (rabbinic) level. When somebody enters a bet, they never think they are going to lose. Therefore, if they do lose, they are very reluctant to give up their money. When the other person takes it from them, he is like a thief. Therefore, this person is like a thief on a deRabbanan level, and he is not kosher. But he is reluctant to give edut, testimony.
The one law that was said in the name of Rabbi Yehoshua concerned a case of a dead sheretz, a type of insect that transmitted tum'ah (ritual impurity) when it was dead. It was in the mouth of a chuldah (weasel). The weasel was walking on loaves of Terumah (priestly offering). There is a doubt whether the sheretz touched the loaves and then made them tamei (ritually impure), or whether the loaves were not touched. We say that the Halakha, regarding the doubt about these loaves, is that they are tahor, they remain pure. A dead sheretz is being moved from place to place while it is in the mouth of the weasel. And the Halakha is that if there is a safek, a doubt, about a moving item transmitting tum'ah, we assume that it did not transmit tum'ah.