καί

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330. The uses of καί are in the main the same in all periods of Greek. It is

  1. A copulative conjunction, conveying the idea of addition to what has preceded

    Ζηνὶ φόως ἐρέουσα καὶ ἄλλοις
    to Zeus and the others besides

    ὥς ἄρʼ ἔφη καὶ κτλ
    thus he spoke and thereupon, etc.

    and

  2. A strengthening or emphasizing particle meaning also, even, just.

    Il. 1.63 ἢ καὶ ὀνειροπόλον
                or even a dream-prophet

    Il. 3.176 τὸ καὶ κλαίουσα τέτηκα
                  which is the very reason that I am wasted
                  with weeping

It is especially used with words that imply comparison, increase or diminution, extension of time or the reverse, etc., as

καὶ ἄλλος
another (not this only)

καὶ αὐτός
himself (as well as others)

καὶ πάλαι
long ago (not merely nοw)

καὶ αὖθις
another time (if not now)

καὶ μάλα, καὶ λίην
(in a high degree, not merely in an ordinary degree)

so with comparatives, καὶ μεῖζον, καὶ ῥίγιον, etc. Both terms of a comparison may be strengthened in this way.

Il. 1.81 εἴ περ γάρ τε χόλον γε καὶ αὐτῆμαρ καταπέψῃ,
            ἀλλά τε καὶ μετόπισθεν κτλ.

Notice, too, the use at the beginning of an apodosis, esp. with adverbs of time.

Il. 1.477 ἦμος δʼ ἠριγένεια φάνη ῥοδοδάκτυλος Ἠώς,
              καὶ τότʼ ἔπειτʼ κτλ.

καί precedes the word which it emphasizes, but is sometimes separated from it by other particles, enclitic pronouns, etc.

Il. 1.213 καί ποτέ τοι τρὶς τόσσα
               (not merely compensation but) three times as much

Il. 2.292 καὶ γάρ τίς θʼ ἕνα μῆνα μένων
               a man who stays even one month

So Il. 7.281 καὶ ἴδμεν ἅπαντες (= ἴσμεν καὶ πάντες).

καὶ εἰ and εἰ καί. The combination καὶ εἰ indicates that the whole condition is an extreme one: even on the supposition that.

But with the order εἰ καί the καί emphasizes particular words.

εἰ καὶ μάλα καρτερός ἐστι
even if he is (I will go so far as to say) very strong

Hence εἰ καί usually implies that the supposition is more or less true.