In what way is the kal vochomer concerning tahara bemikve BETTER ("odif" - Ran line 8))than the kal vochomer which R' Eliezer makes concerning haforas nedorim, which the Ran describes (on line 9) as "kolish tfei" Surely the two are kal-vochomers are parrallel and Chachamim are simply illustrating their point by comparison. In what way is their Kal-vochomer BETTER or STRONGER?
Refoel Salzer, Gateshead UK
The point of the Gemara is that according to the third Girsa in the Ran (75b DH Ta), the Kal v'Chomer of the Chachamim has "double power," while Rebbi Eliezer's "Kal v'Chomer" is only a single Kal v'Chomer.
The Kal v'Chomer of Rebbi Eliezer only mentions one subject, i.e. Nedarim. To prove that this Kal v'Chomer is not valid it would have been sufficient to show that the other Kal v'Chomer that also mentions only one subject, i.e. Mikvah, is also invalid. However what the Chachamim did in fact do is to bring a Kal v'Chomer involving two subjects, i.e. both "Adam" (the person who swallowed the rings) and Mikvah.
When the Ran writes that Chachamim's Kal v'Chomer is "Adif Tefei," this is because it is a double Kal v'Chomer, while Rebbi Eliezer's Kal v'Chomer is "Kalish Tefei" because it only involves a single subject.
In other words, Rebbi Eliezer's Kal v'Chomer is that if the husband can annul a Neder which already took effect, then he can certainly annul a Neder which has not yet taken effect. To disprove this it would be sufficient to say that we see from Mikvah that this is not a good Kal v'Chomer, because even though a Mikvah removes Tum'ah that already has taken effect it cannot prevent new Tum'ah from taking effect.
Instead, Chachamim asked an even stronger question through a different Kal v'Chomer. The Ran writes that this Kal v'Chomer "rides on two horses" (see Rashi to Kesuvos 55b DH Matnos, who explains these words to mean that he gave it two powers). Even though an Adam cannot make something Tamei into something Tahor (i.e. if he swallows a Tamei ring and then vomits it up it does not become Tahor merely by being inside him), nevertheless he does prevent something Tahor from becoming Tamei (i.e. when he enters the tent of the corpse he prevents the Tahor ring he had previously swallowed from becoming Tamei), then certainly, by way of a two-way Kal v'Chomer, if Adam can prevent Tum'ah applying even though he cannot remove the Tum'ah once it does apply, then a Mikvah which can remove the Tum'ah should certainly be able to prevent it applying.
Since the Mikvah certainly does not prevent Tum'ah from applying, it can be shown that the Kal v'Chomer is not a valid one.