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The Gemara (Pesachim 66a and others) write that one can deduce a Kal VaChomer on his own (and doesn't require a tradition).

On the other hand, there's a rule that the Kal of a Kal VaChomer cannot be stronger than the Chamur ( If A > B && B == 1 , all you can prove is that A >= 1, not that A == 2 ). This rule is called the Dio.

There is, however, one exception to this rule (at least according to some Tanaim): If the Dio nullifies the Kal VaChomer, the Kal is expounded to be stronger than the Chomer (as what's the point of the Kal VaChomer in this case?)

How does this work from a logical perspective? Who says there has to be a point to the Kal VaChomer?

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    consider quoting the relevant passage which motivated this question. – mevaqesh Mar 30 '17 at 1:34
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    dayo is a machloket tanayim there are 3 shitot – kouty Mar 30 '17 at 4:07
  • contrarily o formal logic in drashot we try always extrapolation, based on stylistic. the human language can be explorated beyond his minal formal appearence – kouty Mar 30 '17 at 4:09
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Disclaimer: I have no specific source that says what I am about to say. This is just my understanding of the sugya in a way which I think explains how it is logical.

Let's start from the beginning so we can see how this makes sense. The Mishnah in Bava Kamma 24b states:

שור המזיק ברשות הניזק כיצד נגח נגף נשך רבץ בעט ברשות הרבים משלם חצי נזק ברשות הניזק רבי טרפון אומר נזק שלם וחכמים אומרים חצי נזק אמר להם רבי טרפון ומה במקום שהקל על השן ועל הרגל ברשות הרבים שהוא פטור החמיר עליהן ברשות הניזק לשלם נזק שלם מקום שהחמיר על הקרן ברה"ר לשלם חצי נזק אינו דין שנחמיר עליו ברשות הניזק לשלם נזק שלם אמרו לו דיו לבא מן הדין להיות כנדון מה ברה"ר חצי נזק אף ברשות הניזק חצי נזק אמר להם אף אני לא לא אדון קרן מקרן אני אדון קרן מרגל ומה במקום שהקל על השן ועל הרגל ברה"ר החמיר בקרן מקום שהחמיר על השן ועל הרגל ברשות הניזק אינו דין שנחמיר בקרן אמרו לו דיו לבא מן הדין להיות כנדון מה ברה"ר חצי נזק אף ברשות הניזק חצי נזק

The halacha is that keren is chayev chatzi nezek in reshus harabim. The halacha is that shen and regel are patur in reshus harabim. The halacha is that shen and regel are chayev nezek shalem in reshus hanizik. The unknown is keren in reshus hanizik. R. Tarfon argues that it should be chayev nezek shalem based on a kal vachomer. The Chachamim reject this because of dayo.

The Gemara questions this. How can R. Tarfon make such a kal vachomer if the concept of dayo is d'oraisa. The Gemara answers that R. Tarfon agrees that there is a concept of dayo, but holds that it is only applicable when the dayo does not utterly repudiate the kal vachomer. But if the dayo does utterly repudiate the kal vachomer then he allows the kal vachomer to stand despite the dayo. (The Gemara also explains how each party understands the verses from which the concept of dayo is derived.)

ור"ט לית ליה דיו והא דיו דאורייתא הוא דתניא מדין ק"ו כיצד ויאמר ה' אל משה ואביה ירק ירק בפניה הלא תכלם שבעת ימים ק"ו לשכינה ארבעה עשר יום אלא דיו לבא מן הדין להיות כנדון כי לית ליה דיו היכא דמפריך ק"ו היכא דלא מפריך ק"ו אית ליה דיו התם שבעה דשכינה לא כתיבי אתא ק"ו אייתי ארבסר אתא דיו אפיק שבעה ואוקי שבעה אבל הכא חצי נזק כתיב ואתא ק"ו ואייתי חצי נזק אחרינא ונעשה נזק שלם אי דרשת דיו אפריך ליה ק"ו ורבנן שבעה דשכינה כתיבי תסגר שבעת ימים ור"ט ההוא תסגר דדרשינן דיו הוא ורבנן כתיב קרא אחרינא ותסגר מרים ור"ט ההוא דאפי' בעלמא דרשינן דיו ולא תאמר הכא משום כבודו של משה אבל בעלמא לא קמ"ל

On the face of it, R. Tarfon's position seems to be illogical. If he grants that there is a problem called dayo, how does the fact that it would ruin the kal vachomer affect anything? There is no mitzvah to have a particular kal vachomer. So who cares if dayo would not allow the kal vachomer at all? Just don't make the kal vachomer then!

However, I think that R. Tarfon's position is actually the logical one, and if anything the Chachamim's position is illogical. Let's examine R. Tarfon's argument more closely.

As originally formulated in the Mishnah, R. Tarfon's kal vachomer seems to be that if reshus hanizik is more chamur than reshus harabim (as evidenced by the fact that shen and regel are chayev nezek shalem in the former yet completely patur in the latter) and keren is chayev chatzi nezek in reshus harabim, then keren should be chayev nezek shalem in reshus hanizik. The Chachamim correctly respond that this is dayo. All you can derive from here is that keren should be at least chayev chatzi nezek in reshus hanizik. You can't expand it to nezek shalem. R. Tarfon responds that they misunderstood him. He was not making the above kal vachomer. Rather he was making a different kal vachomer using the same premises. He was saying that if keren is more chamur than shen and regel (as evidenced by the fact that the former is chayev chatzi nezek in reshus harabim yet the latter is completely patur in reshus harabim) and shen and regel are chayev nezek shalem in reshus hanizik, then keren must at least be chayev nezek shalem in reshus hanizik. To this the Chachamim reassert that the problem of dayo exists because you are going from keren being chayev chatzi nezek in reshus harabim to nezek shalem in reshus hanizik.

Now we can see that instead of the Chachamim logically rejecting a kal vachomer for exceeding its source and R. Tarfon asserting some "chok" that dayo is inapplicable since it would ruin the entire kal vachomer, it is in fact just the opposite. R. Tarfon's position is in fact logical. The reason why dayo is inapplicable when it removes the kal vachomer is precisely because it would be illogical. If we have proven (within the Talmudic system of kal vachomer, as opposed to an actual proof) that keren is more chamur than shen and regel then it has to at least have the same level of obligation as shen and regel in reshus hanizik, which is nezek shalem. To say that keren should only be chayev chatzi nezek in reshus haznizik (because of dayo that it can't exceed the chatzi nezek of keren in reshus harabim) would defy the logic of the kal vachomer. Thus the kal vachomer compels us to ignore the fact that it exceeds keren in reshus harabim because this fact is demonstrably proven from the kal vachomer. The Chachamim, on the other hand, insist on a "chok" that it cannot exceed its source no matter what, even when the kal vachomer actually proves that it should exceed the source.

Thus, to answer the original question, the opinion of R. Tarfon that we still make a kal vachomer in a case where the dayo would entirely remove the kal vachomer, is entirely logical, and if anything it is the opposing opinion that is illogical.

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