Like the closely related notion of 'subject', the notion of 'object' is an ancient one in the Western linguistic tradition, and in modern Western linguistics 'object', like 'subject', remains an important descriptive notion. In theoretical approaches, the status of the notion object largely stands or falls with that of subject: subject and object may be derivative notions, definable in terms of e.g. phrase structural configuration, or they may be treated as basic categories of grammar.
Unlike the notion of subject, that of object remains relatively understudied and under-defined. The lack of detailed and specific treatments of object and objecthood was noted by Börjars and Vincent (2008), and this relative neglect has continued, their own brief paper the only notable exception since Plank (1984).
Part of the problem in studying the category of object is the apparent lack of absolutely clear definitional criteria. While it has been shown that a number of syntactic operations target subjects specifically, or can be used to distinguish subjects from other grammatical functions, the situation is rather more problematic in the case of objects. As discussed by Börjars and Vincent (2008), the role of the object in passivization - namely, that the object in the active becomes the subject in the passive - is in some respects the most obvious and most robust criterion, but in fact even this is inadequate for defining the notion of object, since on the one hand there are objects - e.g. secondary objects - which cannot become subjects in the passive, and on the other, in certain circumstances non-object oblique arguments may become subjects in the passive.
Börjars and Vincent (2008) discuss a number of other issues in the definition of objecthood, including the relation between 'core objects' and secondary objects, between objects and clausal complements, the notions of cognate objects, pseudo-objects, and object expletives, and the relation between the notion of object and the notion of Theme, which they take to be extremely close.
2. The Indian tradition: the karman
In the ancient Indian grammatical tradition, the closest notion corresponding to the Western 'object' is the category of karman. This is one of the kārakas, abstract argument structure categories discussed in detail in 'Argument structure'.
As one of the kāraka labels, karman applies at a level of abstract representation which does not necessarily map one-to-one to the surface structure. So, in a prototypical active sentence, the karman does indeed correspond to the 'object', and the argument concerned is marked with the characteristically objective case-marking, accusative; but in the corresponding passive sentence the same argument retains the label karman, even though it surfaces in the nominative case, that is from a Western perspective as the subject. On some level, then, karman corresponds to what is sometimes called the 'logical object' (that is, roughly, the argument which would be the object in a prototypical active usage).
In Pāṇini's Aṣṭādhyāyī there are three main rules defining the kāraka karman:
Aṣṭ. 1.4.49 kartur īpsitatamaṃ karma 'The term karman denotes that which is most desired by the agent.'
Aṣṭ. 1.4.50: tathāyuktaṃ cānīpsitam '(The term karman) denotes what is likewise connected even when not desired.'
Aṣṭ. 1.4.51: akathitaṃ ca '(The term karman) also applies to something to which no other kāraka designation is given.'
The first two rules attempt a semantic definition of the role of the karman. The first appears to have the status of the prototypical case, where the karman represents the argument which the agent most desires (to obtain, create, reach, affect) through the action in question. This relation is prototypical insofar as it is taken as the basis for the extension, by Aṣṭ. 1.4.50, to cases where the argument in question is not desired (in whatever way) by the agent. The obvious difficulty in providing a clear semantic definition of the karman reflects the difficulty for modern Western linguistic theory of defining the semantic role of Theme, which underlies a part of the semantic range which Pāṇini was here trying to capture.
The semantic definition, and semantic range, of the karman became a more extensive topic of discussion in the later grammatical tradition, as discussed below. But the third rule in (1), akathitaṃ ca, provides more of a syntactic extension of the scope of karman, which complicates its definition in a similar way to the problems discussed above with the Western notion of object. The traditional interpretation takes this rule as referring to secondary objects in double object constructions, such as:
|‘Devadatta milks milk (from) the cow.’
In this sentence, what Devadatta most desires through his action is the milk, so the label karman applies to the word payaḥ by Aṣṭ. 1.4.49. Here, the cow bears no other distinct kāraka relation to the action of the verb, and so falls under the scope of Aṣṭ. 1.4.51, and is also given the label karman. This ultimately accounts for the accusative case marking on both words. The 'akathita' karman, here the cow, is understood as being apradhāna 'non-primary', while the karman which receives its label from the earlier rule is the pradhāna 'primary' karman. This appears to be close to the distinction between core object and secondary object in Western linguistic theory. However, the distinction here is primarily semantic; crucially, it is not the case that the pradhāna-karman is syntactically primary. For most ditransitive verbs, in fact, it is the non-primary karman which becomes the subject in the passive. Only four of the sixteen verbs traditionally listed as ditransitive by the tradition show promotion of the pradhāna-karman to subject in the passive (Deshpande 1991).
Whatever the differences between the notions of primary vs. non-primary karman and core vs. secondary object, the designation of two arguments of a ditransitive verb as karman raises the same problem encountered with the definition of object in the Western tradition: promotion to the subject in the passive (or, in the Indian context, being co-referenced by the verbal ending in the passive) is no longer a definitional feature of being a karman/object.
In other respects, though, the assignment of the label karman is fundamentally associated with the ability of an argument to become the subject in the passive. It is also associated, though less absolutely, with assignment of accusative case, the grammatical object case in Sanskrit. For example, Aṣṭ. 1.4.52 applies to the causative of some verbs, assigning the role karman to the argument that was the kartṛ in the corresponding non-causative. This rule ultimately licenses the accusative case marking on the causee with causatives of intransitive and some other verbs, and also licenses the 'promotion to subject' of the causee in the passives of such causatives.
Other rules assign the term karman to arguments of certain verbs which would otherwise be given a different kāraka label. For example:
Aṣṭ. 1.4.37: krudhadruherṣyāsūyārthānāṃ yaṃ prati kopaḥ '(The term sampradāna) denotes one towards whom anger is felt with verbs with the meanings 'feel angry', 'injure', 'envy', 'find fault'.'
Aṣṭ. 1.4.38: krudhadruhor upasṛṣṭayoḥ karma 'The term karman denotes one towards whom anger is felt with the verbs krudh 'feel angry' and druh 'injure' when co-occurring with preverbs.'
According to the first rule in (3), the argument of a verb like krudh 'feel angry' which expresses the object of anger is given the kāraka label sampradāna; this label is linked with the dative case. According to the second rule in (3), on the other hand, with a preverb-verb complex like abhi-krudh 'feel angry towards' the same argument will get the label karman. This means it will appear in the accusative case:
|'Devadatta feels angry towards Yajñadatta.'
|'Devadatta feels angry towards Yajñadatta.'
The classification of the source of anger as sampradāna with the basic verb but karman with the prefixed verb predicts also that this argument could become the passive subject only with the prefixed verb.
Further on the category karman in the Aṣṭādhyāyī, see the discussion of karmavadbhāva under 'Subjecthood'.
The term karman, then, does not correspond precisely to the notion of object familiar in the Western linguistic tradition. It is also not the same as semantic role notions like Patient or Theme, although (as for 'object') all or most Patient or Theme arguments will get the label karman. It is closest to the descriptive notion of 'logical object', and shares with this the association with the subject role in the active, but it also encompasses secondary objects and, as discussed, includes primary and secondary objects which cannot become passive subjects.
3. Types of karman
Building on the partially semantic definition of the karman in the Aṣṭādhyāyī, the great grammarian-philosopher Bhartṛhari, in Vākyapadīya 3.7.45-46, developed a seven-fold categorization of the karman which became authoritative in the Pāṇinian tradition. The following discussion is based on the later presentation by Kauṇḍabhaṭṭa, in the Subarthanirṇaya of his Vaiyākaraṇabhūṣaṇasāra.
The seven types of karman are: nirvartya 'to be brought about', vikārya 'to be altered', prāpya 'to be obtained', audāsīnyena yat prāpyam 'what is obtained through indifference', yat kartur anīpsitam 'what is not desired by the agent', saṃjñāntarair anākhyātam 'what is not explained by another label', anyapūrvaka 'that which was previously something else'.
The first three of these are subdivisions of Pāṇini's īpsitatama 'most desired' type, that is the prototypical karman defined in Aṣṭ. 1.4.49 (see ex. 1). The action of an agent can be intended to bring about the creation of a thing, e.g. ghaṭaṃ karoti 'he makes a pot', where the pot is the nirvartya 'to be brought about' karman. The action of an agent can also be intended to bring about the alteration of one material into another; this is the vikārya 'to be altered' karman. Kauṇḍabhaṭṭa recognizes two variants of this type: first, where something comes about through elimination of another material, and second where we are dealing with alteration of attributes. Examples of each respectively are kāṣṭhaṃ bhasma karoti 'he makes the firewood into ashes' and suvarṇaṃ kuṇḍalaṃ karoti 'he makes the gold into an earring'.
The third type of the prototypical karman is the prāpya 'to be obtained', e.g. ghaṭaṃ paśyati 'he sees the pot'. While the nirvartya and vikārya karmans are close to the semantic role of Patient, the prāpya-karman appears to be closest to the semantic role of Theme: this subtype of karman is crucially unaffected by the action of the verb.
The fourth and fifth types of karman correspond to the anīpsita 'undesired' karman of Pāṇini's Aṣṭ. 1.4.50. The division recognizes that the attitude of an agent towards a patient/theme-like argument, where that attitude is not one of desire (to obtain, reach etc.) may be one of either indifference or active aversion. The former is the karman which is audāsīnyena yat prāpyam 'what is obtained through indifference' (alternatively called udāsīna 'indifferent'), e.g. tṛṇaṃ spṛśati 'he touches the blade of grass'. The latter is the karman which is yat kartur anīpsitam 'what is not desired by the agent' (also called dveṣya 'hated'), e.g. viṣaṃ bhuṅkte 'he eats poison'.
The secondary karman, Pāṇini's akathita type defined in 1.4.51, remains a distinct type in this classification too: saṃjñāntarair anākhyātam 'what is not explained by another label', e.g. gāṃ dogdhi 'he milks the cow'.
The seventh type recognizes as distinct those cases where the label karman is assigned to arguments of specific verbs or preverb-verb complexes where we would otherwise expect a different kāraka label, such as the case of abhi-krudh discussed above. Kauṇḍabhaṭṭa's term, anyapūrvaka 'that which was previously something else' interestingly acknowledges a kind of derivational process here; his example parallels our example above: krūram abhikrudhyati 'he is angry with the cruel man'.
The karman defined in Aṣṭ. 1.4.52, that is where the kartṛ of a non-causative verb is labelled karman in the corresponding causative, is not counted as a type of karman, underlyingly, but as a type of kartṛ, the so-called karmakartṛ.
4. The semantic basis of the karman
Alongside deeper consideration of the categorization of types of karman, the later tradition also developed theories of the semantic basis of karman-hood. The following discussion is again based on Kauṇḍabhaṭṭa's Vaiyākaraṇabhūṣaṇasāra, Subarthanirṇaya, and see further Sarma and Grimal (2013: 3-7).
A verbal root fundamentally expresses some kind of kriyā 'action'. Kriyā consists of two parts: vyāpāra 'activity', and phala 'result'. These two aspects of kriyā reside in particular substrata, and in the case of an ordinary transitive verb, the vyāpāra resides in the kartṛ (≈ the agent or active subject), while the phala resides in the karman.
The karman, then, can be defined as the kriyājanyaphalāśraya 'the substratum of the result brought about by the action'. For example, in the sentence odanaṃ pacati 'he cooks rice', the rice is the karman by virtue of its being the location of the result of the action, namely the softening which is the intended result of the cooking.
This definition is not quite sufficient, in fact, because in the case of intransitive verbs the vyāpāra 'activity' and phala 'result' are understood both to reside in the agent, e.g. Devadattas tiṣṭhati 'Devadatta stands', where Devadatta is the one to whom the result of his action applies. In addition, in some cases the phala 'result' can be understood as residing in both the kartṛ and the karman, e.g. caitro grāmaṃ gacchati 'Caitra goes to the village', where the co-location which is the result of the act of going applies equally to Caitra and the village. For both cases there are two possible solutions: we could add a specification to the effect that the karman is not the substratum of the vyāpāra 'activity' of the action (as well as being the substratum of the result). Alternatively, and more simply, one can understand a kind of blocking relation: the label karman applies to the substratum of the result of an action, but where the label kartṛ would also apply (as it does to the substratum of the vyāpāra), the latter trumps the former.
The notion of transitivity can then be defined in these terms (Joshi 1960): either as svārthaphalavyadhikaraṇavyāpāravācitva 'the state of denoting activity which has a different locus from the result which is the meaning of the root' or svārthavyāpāravyadhikaraṇaphalavācakatva 'the state of denoting the result which has a different locus from the activity which is the meaning of the root'.
The phala of an action can be of two types: bhāva 'state' or kriyā 'action'; the difference between these is understood by the tradition in terms of whether there is or is not some kind of movement. This means that there can be two types of karman, one which is the substratum of a phala which is a bhāva, and one which is the substratum of a phala which is a kriyā. For example, in the sentence ghaṭaṃ pacati 'he bakes the pot', the phala resulting from the action of baking is the change of colour of the pot; such a use of pac 'cook' is called karmasthabhāvaka 'denoting a state residing in the karman'. In contrast, in the sentence kāṣṭhaṃ bhinatti 'he splits the logs', the result of the action of splitting involves a movement, a kriyā 'action', on the part of the logs; such a use of bhid is called karmasthakriyaka 'denoting an activity residing in the karman'.
 The classic argument for the existence of grammatical relations like subject and object as base categories of grammar is by Perlmutter and Postal (1977); in one form or another the argument was accepted in grammatical theories like Lexical-Functional Grammar, Head-driven Phrase Structure Grammar, and Relational Grammar, but was rejected by Chomsky (1981: 123-124). See further Pollard and Sag (1994: 118-123).
 See in particular Deshpande (1991), and also on ditransitive verbs Joshi and Roodbergen (1975), and Hock (1985). An alternative interpretation of this rule is given by Kiparsky (2009).
 It can alternatively be interpreted as a source, and given the kāraka label apādāna; this would result in ablative case marking.
 Deshpande surveys the various attempts within the tradition to resolve the lack of consistency in which of the karmans is targeted in the passive, but the tradition presents no final solution.
 Although in fact, a passive of krudh does not appear to be attested.
 See also Vergiani (2013), who summarizes Bhartṛhari's categorization and discusses its adaptation in the Tamil grammatical tradition and some other later authors.