Proctor, Mark. "It's probably untrue, but it wouldn't matter anyway": εἰ καὶ conditions in the Greek
New Testament." Neotestamentica 53:3, 2019, 437-458.
Proctor evaluates the "second class" condition with εἰ καὶ in the protasis. This condition occurs 22 times in the New Testament (Proctor 2019, 437). This type of condition regualarly marks the apodosis as untrue. However, Proctor argues the presence of the καὶ in the protasis strongly signals that the author or speaker considers the protasis to be untrue. Much scholarship, however, considers the protasis to admit to truth.
Proctor describes the conditional sentences, observing that the dependent clause (protasis) signals what may come in the independent clause (apodosis) (Proctor 2019, 438). There are generally considered to be four different classes of conditions. Proctor describes the first two.
In the first, the εἰ (if) is followed by a "then clause, not introduced by anything, and proving to be true, dependent on the protasis (Proctor 2019, 439). 306 first class conditions have been identified in the New Testament (Proctor 2019, 441). Proctor advocates translations of these conditions using "if" rather than "since" or "when."
The second class condition is similar in form, using the plain "if" in the protasis. However, here there is an indicator that the protasis is false, since the apodosis proves contrary to fact (Proctor 2019, 441). This is done normally with the particle ἄν, the use of an aorist, imperect, or pluperfect tense in both parts of the sentence, or the word μὴ for "not" (Proctor 2019, 442). Proctor goes on to briefly discuss the meanings and uses of καὶ (and), noting that it has a wide range of nuance but can generally be summed up as a conjunction showing additive relations or an adverb showing emphasis (Proctor 2019, 444). If in fact there are not a pair of words joined by the καὶ, it must be used adverbially. Proctor notes also that as an adverb the word is normally not first in its clause (Proctor 2019, 445). This leads Proctor to the condition using εἰ καὶ, where καὶ is adverbial. He suggests that the amplification used in the protasis flags it as "most likely untrue" (Proctor 2019, 446). Unfortunately, in Prctor's opinion, translators have routinely used "even though" rather than "even if." This assumes the protasis is accurate, rather than inaccurate.
Proctor continues by surveying the New Revised Standard Version's practice. He identifies 21 or 22 conditions which use εἰ καὶ (Galatians 3:4 is a disputed reading). He considers it to have translated the condition correctly six times (Proctor 2019, 44). Proctor describes these in some detail. He then describes the thirteen passages which, in his opinion, need improvement (Proctor 2019, 449ff). His descriptions effectively illustrate how "even though" is an inferior translation when compared to "even if."
Next, Proctor reviews the statements of 2 Cor. 7:12 and 11:15, where "it is clear from their surrounding contexts that Paul would grant them only begrudgingly" (Proctor 2019, 456). The protasis is actually true, but Paul hesitates to admit it. Again, he uses the εἰ καὶ construction.
Proctor concludes that while the ἄν suggests a denial of the claim in an apodosis, the εἰ καὶ suggests a hesitancy about the protasis. This was demonstrated by his evaluation of its 22 instances in the New Testament (Proctor 2019, 456).