Arthāpatti is the postulation of the only possible solution out of a seeming contradiction, e.g. “Devadatta is alive” and “Devadatta is not in the place where we usually see him” (this is expressed in Sanskrit epistemology by “Devadatta is not at home”, in contemporary terms we could think of something like “What happened of Jim? He is not in his office!”). Different contemporary scholars have tried to discuss the relation of arthāpatti with ‘presumption’ and ‘inference to the best explanation’.

The following is a translation of the beginning of the arthāpatti section in the Mānameyodaya:

1. DEFINITION OF ARTHĀPATTI

“Arthāpatti is the postulation of [something] producing [a solution] when there is a logical impossibility”: this is the definition explained in the Śābara Bhāṣya || 128 ||

As for this definition, the “logical impossibility” is said to be the contradiction between two [types* of] instruments of knowledge. Therefore, the following definition [of the arthāpatti] should be taught:

Arthāpatti is defined as the cognition of something non-contradictory caused by the contradiction between instruments of knowledge about general topics, and ones about a specific topic || 129 ||
For instance, the postulation of being outside of home due to the contradiction, which is instrumental [for the arousal of postulation], between the knowledge of his not being at home and the knowledge of his being alive || 130 ||

One understands in general that Devadatta is alive, either at home or outside, due to an inference based on astrology (i.e., because out of astrological calculations one knows that he will live a long life). There being a contradiction with the fact that he is not at home, one hypothesises —for the sake of the non-contradiction— that he must be out. And this is an arthāpatti-cognition whose instrument is the contradiction between the two [types] of instruments of knowledge.

2. NAIYĀYIKA POSITION: THE ARTHĀPATTI IS A FORM OF ANUMĀNA

However, the experts of logic (i.e., the Naiyāyikas), thinking that this (arthapatti) is a form of inference, say that |
“there is no contradiction between the two [types of] instruments of knowledge. And this (non-contradiction) is the same in the case of everything well-known” (NKu 3.11)|| 131 ||

[Nai:] To begin with, it is impossible that there is a contradiction between instruments of knowledge, since there would be the undesirable consequence that one of the two is not an instrument of knowledge, as in the case of “this is silver, this is not silver” (where one of the two ends up being recognised as not valid).

[Obj.] But nonetheless, one does see a contradiction between the two knowledges about which we spoke before (in the case of Devadatta)!

[Naiyāyika:] This is just a wish! In fact, in the case at hand, the room for doubt** regarding the specific place, namely ‘is he at home or outside?’ is blocked by the knowledge of his not being at home.

*dvaya must refer to two types of instrument of knowledge, and not just to two instruments of knowledge, given that the next verse explains that the contradiction may be between a specific knowledge and several general ones (sādharaṇapramāṇānām). Thanks to Andrew Ollett for having discussed this issue with me.
**The Naiyāyika is here speaking of doubt instead of logical inconsistency, probably because the former, unlike the latter, is among the padārthas his school accepts.