One would not expect to connect Miriam’s speaking negatively against Moshe, read (at least in the Diaspora) this past Shabbat, with the laws of a goring ox. When studying the story of Miriam we tend to focus, as we should, on the nature of lashon hara, her tragic punishment, the role of Aaron and who exactly was this Cushite woman who Moshe took. But it is her punishment that is of major significance well beyond the story itself, impacting on a goring ox.
“And the Lord said unto Moses: 'If her father had but spit in her face, should she not hide in shame seven days? Let her be shut up without the camp seven days, and after that she shall be brought in again.'” (Bamidbar 12:14) Miriam would be forced to spend seven days in isolation for her role in inappropriately speaking against Moshe, the servant of G-d. 
“An ox that causes damage in the property of others, how do we assess liability? If it gored, pushed, bit, lay on top of, kicked in public property it [the owner] pays half damage, in the property of the plaintiff Rabbi Tarfon says full damages and the Sages say half damages.” (Bava Kamma 24b)
It is rare for an ox to gore and thus the Torah prescribes that the owner only need to pay half damages the first three times it gores. After that point the animal can no longer be considered a tam, an innocent animal, and is to be considered a muad, an animal that regularly causes damages requiring the owner to pay full damages.  
Rabbi Tarfon argues that such is only true when the ox attacks in a public area but if it were to do so in private property payment of full damages would always be required.  
Rabbi Tarfon’s argument is based on the logical argument of kal vachomer, a fortiori. Our Sages divided all damages caused by an ox into one of three categories; shein, eating, regel, walking and keren, goring. With eating and walking being normal activities for animals - including homo sapiens - these categories have both a leniency and stringency. As animals, at least in the pre-modern world, had every right to walk in public property it was the responsibility of people to ensure neither they nor their property get in the way of the animal. And if they for example leave some food on the street and an animal eats the food the owner is exempt from paying for the food. On the other hand because animals do regularly cause damage, when one enters the domain of another person one is immediately responsible to pay the full extent of any damage caused.
However a goring animal is a rarity and people cannot be expected to be on the lookout for it nor can the owner be held fully responsible when the ox does gore. As a form of compromise when an ox gores in public property the owner is required to pay only half damages. With a requirement to pay half damages in public property Rabbi Tarfon argues that surely in private property it must pay full damages. After all if shein and regel which are exempt from payment in the public square must pay full damages in private property, in a case of keren where half damages must be paid in a public area surely full damages are to be paid in private property. 
It is here where Miriam’s criticism of Moshe becomes relevant. Upon speaking lashon hara against her brother she was punished with tzaarat and thus forced to leave the camp of Israel. In determining the length of time she was to be exiled G-d says that if her father had been angry with her it would make sense to be banished for seven days, seven reflecting a complete cycle. Thus if G-d is “angry” and banishes someone it should be for no less than 14 days. Yet the Torah says she was only to be banished from the camp for seven days teaching us the principle of dayu leva min hadin leheyot kanidon, we only derive the equivalent of what we are deriving from. In other words while it may be logical to say G-d’s punishment should be more than seven days, when making a kal vachomer we can only say that just as by man it's seven days so surely by G-d it’s seven days. We have no right to make the jump to 14 days, logical as it may be to do so. So too in our case if one pays half damages for keren in the public square then the kal vachomer can only allow us to say that in a private area one must pay half damages. But one cannot use the kal vachomer to derive full damages from a case of half damages.
With this principle rooted in the Biblical text the Gemara queries how Rabbi Tarfon can argue with the Sages. The Gemara explains that Rabbi Tarfon does accept the principle of dayu in general though not in this particular case. By the case of Miriam there is no direct mention of G-d’s seven day punishment and that is what the kal vachomer teaches; Miriam should be banned for seven days but the principle of dayo limits the punishment to no more than seven days. However by the case of the ox half damages are already mentioned in the Torah and thus the kal vachomer must come to add an additional half damages. The Sages claim one can find a reference to G-d’s punishment in the text and nonetheless we say dayo. The kal vachomer here serves to confirm the written text.  
From a logical point of view it seems that the Sages focus on the half payment that one is exempt from while Rabbi Tarfon focuses on the half damages one must pay. The Sages argue that as keren is unexpected one cannot hold the owner of the animal responsible. By right there is no obligation to pay anything but as a deterrent we make him pay half damages. However there is little reason to differentiate between a public and private domain. 
Rabbi Tarfon, on the other hand, focuses on the fact that the owner must pay half damages. By right the owner should pay full damages but being somewhat unexpected we allow payment of half damages the first three times. Yet this is true only in the public square where the animal has every right to be. However once one enters the private property of another it is quite reasonable to expect the owner will be extra careful in guarding his animal. If not we have little choice but to charge full damages. 
On the surface there is little to connect a goring ox and Miriam’s slip of the tongue other than the application of a textual rule derived in one area and applied to another. Yet there often are thematic connections one can find beneath the surface of technical rules. While one can appreciate the damage caused by a rampaging ox we often fail to realize the damage caused by a loose tongue. Yet as we all know emotional hurt caused by lashon hara lingers long after the physical hurt is forgotten. With laws of payment of an ox[1] derived from Miriam’s speech we are reminded of the centrality of lashon hara and how the punishment does not always fit the crime. Miriam’s punishment was reduced from 14 to 7 days and as a general rule there is no earthly punishment for lashon hara. Yet that matters little in assessing this most severe of infractions.
[1] As we have previously noted these laws apply to all animals with an ox only being the most common example of the ancient world.