I can understand how a gezeira shava would work when the word used is similar, like "kicha kicha m'sdei Ephron" in the beginning of Maseches Kiddushin.

But logically speaking, how can we have a 4 way gezeira shava like "tachas, nesina, yeshaleim, kesef"? (apologies- I don't remember where in shas this is)

Namely, why are different words allowed to be used for that particular gezeira shava but not others?

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    ובא הכהן ושב הכהן זו היא ביאה זו היא שיבה in 1000 places in Shas, the meaning is the same, chachamim make gzera shava – kouty Jan 24 '19 at 15:34
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    It's meditation indeed, but often the masora is not all. See shorashim Rambam and ramban about to count a mitsva that is learned by gzera shava – kouty Jan 24 '19 at 16:02
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    The one you cite is from BK 5a. See Rashi there - they’re not connecting the word תחת to נתינה directly, but rather connecting תחת תחת, and one of those also uses the term נתינה, allowing another connection נתינה נתינה, etc. See Rashi there. – DonielF Jan 24 '19 at 17:20
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    I wrote see both Rambam and ramban – kouty Jan 24 '19 at 19:46

See Ramban on Rambam's Sefer Hamitsvot shoresh 2 (the context of the quoted text is argument against Rambam who stated that a gzera shava is a din midivre sofrim except where the Talmud says that this din is din Tora)

והוי יודע שזה שאמרו שאין אדם דן גזירה שוה מעצמו אין כונתם לומר שכל גזירה שוה מבוארת להם מסיני ונמסרת להם מפי משה רבינו תלמדו מלה פלונית ממלה שבפסוק פלוני ותשוו דין שניהם לענין פלוני, אין הדבר כן שהרי מצינו אותם חולקין תמיד בהרבה מקומות בענין הזה

Despite that we have a principle that one cannot rule a gzera shava by its own reasoning, this doesn't mean that each of gzera shava was transmitted from Moshe Rabeinu with all its technical details. We show in numerous passages of the Talmud controversials about gzera shava...[the Ramban gives as example Gemara chulin 85a.

What is the reason for Rabbi Meir's view with regard to the law of 'It and its young'? - Rabbi Joshua Ben Levi answered: He derives it by an inference made from the term 'slaughtering', used both here and in connection with the slaughtering of consecrated animals outside [the Sanctuary]; as in the latter case a slaughtering which does not render [the animal] fit for food is deemed a slaughtering, so here [in connection with It and its young] a slaughtering which does not render [the animal] fit for food is deemed a slaughtering. And what is the reason for Rabbi Simeon's view? - Rabbi Mani Bar Pattish answered: He derives it by analogy from the verse: And slay the beasts and prepare the meat,; as there the slaughtering rendered [the animals] fit for food, so here the slaughtering must render [the animal] fit for food. Why does not Rabbi Meir infer it by analogy from 'And slay the beasts'? - One may infer 'slaughtering' from 'slaughtering', but one may not infer 'slaughtering' from 'slaying'. But what does this [variation] matter? Was it not taught in the school of Rabbi Ishmael that in the verse: And the priest shall come again. And the priest shall come in, the expression 'coming again' and 'coming in' have the same import [for purposes of deduction]? - This [variation] is [of no consequence] only where there is no alternative analogy based on identical expressions, but where there is an alternative analogy based on identical expressions we must then make the inference from the identical expressions. And why does not Rabbi Simeon infer it by analogy from the law of consecrated animals slaughtered outside the Sanctuary? - One may infer by analogy unconsecrated animals from unconsecrated animals, but not unconsecrated from consecrated. And [is this not an objection against] Rabbi Meir? - [No, for] does not the law of 'It and its young' apply also to consecrated animals?...

And Ramban concludes that if gzera shava was entirely transmitted, there was no such machloket.

So chachamim had partial knowledge about gzera shava and eventually did gzera shava choosing words not entirely similar.

I warmly recommend you to learn this Ramban you can find here The book of Ramban on shorashim from Rambam is really a fundamental.

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It appears to me that the answer is simple, taking into consideration the fact the, traditionally, Gzera Shava is de-jure learning, stemming from Moses, not a logical conclusion.

Therefore we accept it "as is", whether the words are exactly the same, of the same root, of the same meaning or even of a similar domain. That what we allegedly received from Sinai.

As far as I remember (Nidda 19b) Kal Vachomer is the only one from the 13 Midos that can be concluded logically.

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