At first sight, the text passage does not seem particularly difficult, but entangling its intricacies has kept me busy for a long time —and the results are still not satisfying. The main problems are:

1. How exactly can one formalise a case of arthāpatti as a kevalavyatirekin anumāna? What are the two absences at stake?
2. Does Śālikanātha accept or reject the kevalavyatirekin anumāna in general?
3. If he does not reject it, what is the problem in the formalisation of arthāpatti as kevalavyatirekin anumāna?

The standard case of arthāpatti being discussed is the following one:

Devadatta, who is known to be alive, is not home. Thus, he is outside.

The text passage starts by presenting an opponent, who maintains that gṛhābhāva `the absence from home’ is the trigger for the bahirbhāva `being outside’, because it is not otherwise possible (anyathānupapadyamāna). Here the problem seems to be just that the absence from home alone (i.e., without causing one to doubt about whether Devadatta is alive) is not enough to be a trigger. There is still no mention of the kevalavyatirekin anumāna.

The siddhāntin answers that this analysis does not apply, because it is not true that the gṛhābhāva is not otherwise possible (if this were an inference, we would say that the hetu is asiddha `not established’). Why not? Here the discussion turns into a discussion of whether the arthāpatti can be read as a kevalavyatirekin anumāna since the siddhāntin assumes that gṛhābhāva could be a trigger only if it could be verified that it is otherwise impossible and this could only be verified through a kevalavyatirekin anumāna. The siddhāntin thus states that gṛhābhāva is not otherwise impossible because it is impossible to verify an absence in all possible loci of concomitant absence (vipakṣa) of hetu and sādhya and the fixed relation between hetu and sādhya needs to be first ascertained through their concomitant presence (anvaya). This seems to be a critique of the kevalavyatirekin anumāna, insofar as the vipakṣas are endless, and I would be happy with the idea that the arthāpatti cannot be described as a kevalavyatirekin, because a kevalavyatirekin is not a valid anumāna, since one never achieves certainty and at most high probability. But the problem remains, that is, what are the absences at stake? Does this amount to say that we cannot check whether the absence from home lacks consistency in all possible cases but the one of being outside? Or that we cannot check whether the absence from home is always concomitant with the absence of something else? If the latter, what could be this something else?

However, Śālikanātha then goes on explaining that, if the fixed relationship has been ascertained, then the fixed absence of the hetu is ascertained in relation to the sādhya‘s absence (i.e., it has been ascertained that the absence of the hetu necessarily leads to the absence of the sādhya):

avadhārite hi tasminn arthāpattyā sādhyābhāve hetvabhāvaniyamo ‘vasīyate.

This would not make sense if we were to conceive the ascertainment mentioned at the beginning of the sentence as something deemed to occur through anvaya. In fact, once the niyama has been established through anvaya, we just have a normal anumāna and do not need arthāpatti at all. Thus, the sentence must rather mean “In fact, if the relation has been ascertained, then it is through arthāpatti that the fixed absence of the hetu is ascertained in reference to the absence of the sādhya. Going back to our example, what are the two absences at stake? The absence of gṛhābhāvopapatti leads to the absence from home, i.e., to the being outside? Or is the first absence the gṛhābhāva itself? Or is Śālikanātha not really concerned with that since his main concern is instead to say that arthāpatti is not an anumāna, because kevalavyatirekin anumānas are not valid (since ultimately they only work if the connection between sādhya and hetu has already been established through an anvaya) and arthāpatti is not identical with the straight anvaya-based anumāna?

For further posts on arthāpatti, see here.