Pariṇāma-vāda and atyanta-abheda

The Maharashtrian encyclopedic writer Bhāskararāya Makhin, flourishing in Tamil-Nadu in the first half of XVIII c., was a prominent figure of the Tantrik śākta school commonly known as Śrīvidyā. Highly learned and prolific, calling himself genuinely Tantrik (tāntrika) and fully Vedic (vaidika) at once, Bhāskararāya relentlessly pursued his effort to harmonize the main tenets of the Śrīvidyā school with the most authoritative sources of the brahmanical milieu—the Vedic (śruti) and post-Vedic (smṛti) corpora, first and foremost, as well as his repeatedly declared affiliation with the advaita-vedānta tradition. In Varivasyārahasya 3 (VVR 3), along with his Prakāśa auto-commentary (VVR-P 3), Bhāskararāya describes the nature of mahat-prakāśa, the great radiance.Footnote 1 Once that light (jyotis)—which is brahman and unobstructed (anāvṛta) ātman—is known, every other thing is known as well.Footnote 2 This alone is the true fundament (adhiṣṭhāna). That ultimate (carama-vṛtti), indeterminate (nirvikalpa), transformation of cognitive motion will definitely obliterate (nāśyatva) all further content. The phenomenal object (dṛśya)—denotable (gamya) by means of the demonstrative pronoun ‘this’, idam—is but a real transformation (pariṇāma) of ‘that [sole reality]’ (tat = ahaṃtā-rūpa-śakti-viśiṣṭa-brahman).Footnote 3 That is, between ‘power’ (śakti) and ‘power-owner’ (śaktimat) and between the material cause (upādāna) and what is caused (upādeya), there is absolute non-difference (atyanta-abheda).Footnote 4 Bhāskararāya highlights that all Vedic passages (śruti)—the peak (mūrdhanyā) of all means of true cognition (pramāṇa)—concerning non-duality and every Tantra consistent with the former agree on this issue (abhiprāya).Footnote 5Brahman, you see, is this whole world” (Chāndogya Up. 3.14.1: 209): syntactical homogeneity (sāmānādhikaraṇya), claims Bhāskara, expresses non-difference (abheda) and the absence of any contradiction (bādha).Footnote 6 While the two ‘differential counterparts’ (bhedāṃśa) effect (kārya) and cause (kāraṇa) manifest as constructs (kalpita), the entire phenomenal extension (prapañca) does not.Footnote 7 Negation concerns only the facet (aṃśa) of difference. Passages such as “there is nothing diverse at all here!” (Bṛhadāraṇyaka Up., 4.4.19: 125; Kaṭha Up., 4.11: 395) or “one only, without a second one” (Chāndogya Up., 6.2.1: 247) refer to the absence (abhāva) of that which possesses difference (bhedavat) by reason of the lack (abhāva) of any qualifier (viśeṣaṇa).Footnote 8 The features of Bhāskara’s thesis are therefore: full agreement with the śruti, absolute non-difference (linguistically expressed by syntactic homogeneity), radical negation of any difference but not of what is differentiated (that is, phenomenal manifestation), and the absence of any qualifier to differentiate real transformation. Bhāskara supports his thesis by referring to the section of Vācaspati Miśra’s Bhāmatī (IX-X c.; hereafter VM-B) dedicated to the ‘golden crown’ (hāṭaka-makuṭa).Footnote 9 “The property of possessing a [degree of] reality (sattākatva) that is [ontologically] inferior (nyūna) to gold is said solely of difference, not of the crown; since [what results from] transformation (pariṇāma) possesses by necessity (avaśyakatva) the very same (samāna) [degree of] reality as that which has transformed (pariṇāmin)”.Footnote 10

Needless to say, “the main concern of the Advaitin is to establish non-duality (advaitasiddhi). Of course advaitabrahman is always a self-established reality (svataḥsiddha), auto-luminous (svaprakāśa), pure consciousness (śuddhacaitanya), so no proof is necessary to establish it. As consciousness requires no proof, [usually it is first and foremost] the falsity of the world alone [that] is to be established. Once established the falsity of the world [and this is the keystone of the argument], the non-duality becomes automatically established” (Pellegrini 2014, pp. 3–4).Footnote 11 The strategy advanced by Bhāskara appears instead to be symmetrically opposite. Here, the aim is to harmonise, in strictly non-dual terms, the brahman-ātman auto-evidence with the full reality of manifestation. In other words, the advaitin’s first move usually consists in proving the falsity (mithyātva) of the world (prapañca, or jagat) in order to validate the reality and unity of brahman: where I saw a snake, there is but a rope. So, where is the difference between snake and rope, if there is no snake? In this case, however, Bhāskara draws on Vācaspati to overturn the question. It is no longer a matter of denying the reality of the world, but rather of denying only the reality of difference—and this becomes the keystone of his argument. How to fully conceive the difference, if there is one, between a crown and the gold of which it is made? If there is no difference, where is the difference between jagat and brahman?Footnote 12

If “Advaitins [undoubtedly] place the stream of arguments that refute difference at the core of their logical investigation”, in so doing, “they first utilize the categorical analysis found in Nyāya” (Timalsina 2009, p. 86; cf. also Ganeri 2011, pp. 223–236), just as this paper is methodologically proposing to do. Indeed, the highly refined language and techniques of Navya-Nyāya—along with the formalistic methodology derived therefrom, named ‘Navya-Nyāya Formal Language’ (NL; cf. infra)—will be here programmatically adopted in order to describe in detail a non-dualistic argumentative architecture. Clearly, this does not imply that the Naiyāyikas’ account, conceived in its own prerogatives, will be considered interchangeable or confusedly intermingled with the Advaitins’ one. On the contrary, the philosophical claims of NL qua hermeneutical device methodologically stop just before being committed to the various and different theoretical frameworks NL purposes to analyse (cf. Anrò, forthcoming). This therefore means that, despite the respective deep structural differences, the Nyāya machinery—envisioned, in accordance with a well-established tradition, as a ‘lingua franca for intellectual exchange’ (Ganeri 2011, p. 223)—will be here methodologically put at the service of Vācaspati’s reasoning, in its turn viewed through the lens of the issue Bhāskara raised.

Syntactic Homogeneity and Coreferentiality

What is this golden bracelet? Undoubtedly, it is gold. It is, in this perspective, non-distinct (a-bhinna) from its cause (kāraṇa) because, as stated above, the bracelet is golden. Nevertheless, it is also a bracelet, and not another ornament such as an earring or crown. Indeed, the bracelet is distinct as an effect (kārya) exactly because it is a bracelet and not an earring. It seems, thereby, to appear as simultaneously distinct and non-distinct.Footnote 13

The notion (pratyaya) of sāmānādhikaraṇya indicates syntactical homogeneity on the linguistic and grammatical level, and coreferentiality on the ontological one, both at the same time. Using this notion to express the relationship between earring and gold clearly exhibits the simultaneous occurrence of difference and non-difference (bhedābheda).Footnote 14 In Vācaspati’s view, the sāmānādhikaraṇya relation acts as the ratio cognoscendi with respect to the a-bheda relation, for its part the ratio essendi of the former. This relationship of sāmānādhikaraṇya between two terms in a Sanskrit sentence—terms which share the same grammatical ending (say, nominative or first ending) and the same referent, here generically named A and B—can be expressed in the following mannerFootnote 15:

[1] A1B1,x is y’ (e.g. kuṇḍalaṃ suvarṇam, ‘The earring is gold’).

Vācaspati points out that sentences such as [1] are not in any way reducible to the substratum-superstratum relation (ādhāra-ādheya-bhāva): if A7B1 (‘B on/in A’), e.g. kuṇḍe badaram (‘A jujube in a bowl’), this does not imply that the fruit is the bowl (na hi bhavati kuṇḍaṃ badaram iti); or to the relation of ‘residing in one locus’ (ekāśrayatva): C7A1B1 (‘A and B on/in C’). If ‘Caitra and Maitra [are dwelling] on the same seat’ (ekāsane caitramaitrau) it does not follow that ‘Caitra is Maitra’ (cf. fn. 14).

Two possible interpretations of [1] are then formulated in Vācaspati’s analysis. [a] The relation of sāmānādhikaraṇya can point to an absolute non-difference (ātyantika-abheda) according to which in [1] A = B; e.g. ‘The earring is gold’, that is, ‘earring = gold’. However, based on this premise, what will occur is the undesired outcome (prasaṅga) of a double occurrence (dviravabhāsa) of the term itself: if A = B, then A = A or B = B. Thereby, if ‘earring = gold’, then: ‘earring = earring’; or ‘gold = gold’.Footnote 16 [b] In the case where, in order to avoid the doble occurrence at point [a], the total difference (ātyantika-bheda) between the two terms in [1] is stressed, then A ≠ B, with the likewise undesired consequence that any form of sāmānādhikaraṇya relation would be then denied—as in the case of go ≠ aśva (cow ≠ horse). Thus, if A ≠ B, then: earring ≠ gold ≠ horse.

Still claiming [1], it is therefore not possible to conclude either that the erring is gold, without falling into [a], nor that the earring is not gold, without falling into [b]. The relation of sāmānādhikaraṇya—while being unobstructed (abādhita), indubitable (asaṃdigdha) and universal (sarvajanīna)—ends up determining (vyavasthā) both the difference and the non-difference between the effect (earring) and its cause (the gold of which it is made), simultaneously.Footnote 17

Cognition as a Relation

If the relation of difference (bheda or dvaita; A ≠ B, e.g. go ≠ aśva) does not seem to present any difficulty, what kind of a relation is there between the two terms of a non-difference? Since non-difference (hereafter expressed by the strikethrough cypher ‘2’, i.e. advaita, lit. ‘non-two’) cannot be reduced to an equality or identity relation (A = B), how can these two terms be simultaneously equal and different, as explicitly claimed by Vācaspati?

Some formal tools are required to perform the analysis in NL (Navya-Nyāya Formal Language).Footnote 18 Let the notation ‘_t’ be here the abstraction functor, capable of expressing the Sanskrit suffix -tva or -tā.Footnote 19 So, for instance, if the primitive term g (small italics) is a single pot (ghaṭa),Footnote 20 then ‘gt’ = ‘the property of being g’ or ‘g-hood’ (i.e. ghaṭatva, ‘pot-hood’) whose extension corresponds to the set ‘pots’ G (capital), to obtain |gt| = G. According to what could be called the Axiom of Possession or Tadvattva-Nyāya (TvN), the element g is said to belong to the set G because gt-possessing (viz., qualified by the property g-hood = pot-hood). Thus ghaṭo ghaṭatvavān, ‘a pot [is a pot because it is] in possession of pot-hood’, lest it be not the pot it is. More generally, TvN: tadvattvam (in extended form, taddharmavattvam or tattvavattvam) tad eva, ‘What possesses the property of being that, is that’.Footnote 21

That premised, the crown (m) is surely gold (h). What would be left, indeed, if the gold of which the crown is made were subtracted? Thus, m = h. Nevertheless, m  h because the crown is not only gold, it is a crown as well. Saying that ‘the crown is gold’ implies two distinct properties: the abstract properties ‘gold-ness’, hāṭakatva (ht), and ‘crown-hood’, mukuṭatva (mt), in reference to the two distinct sets M (the set Crowns; for m∈M and |mt| = M) and H (the set Gold; for h∈H and |ht| = H).

Clearly, these two properties could be structured around three possibilities: golden crowns, golden bracelets and iron crowns could exist.Footnote 22 In current notation:

  1. [a]

    (∃x) (Mx ∧ Hx)—hāṭakaṃ mukuṭam (A1B1; ‘A golden crown’); in case both qualificans and qualificandum are present: ubhaya-bhāva-prayuktaviśiṣta-bhāva.

  2. [b]

    (∃x) (~ Mx ∧ Hx)—hāṭakaṃ na mukuṭam or hāṭakaṃ makuṭānyatvam (A1 ~ B1; ‘Gold which is not a crown’); in case the qualificandum is absent and qualificans present: viśeṣyābhāva-prayuktaviśiṣṭābhāva, viśeṣanabhāva-prayuktaviśiṣṭa-bhāva.Footnote 23

  3. [c]

    (∃x) (Mx ∧ ~ Hx)—ahāṭakaṃ mukuṭam (~ A1B1; ‘A non-golden crown’); in case the qualificandum is present and qualificans absent: viśeṣya-bhāva-prayuktaviśiṣṭa-bhāva, viśeṣanābhāva-prayuktaviśiṣṭābhāva.

Nevertheless, the conjunction expressed in standard notation for assertion [a] cannot be considered fully proper from a Naiyāyika’s perspective. “Nyāya develops a language which can perhaps be given the appellation of a ‘property-location language’ […]. The model sentence of such a language contains the introduction of general concepts and ‘the indication of their incidence’. Under this interpretation, the qualifier can be viewed as the feature-universal […], and the qualificand can be viewed as the locus where the qualifier is said to occur” (Matilal 1968: 16).Footnote 24 For a Naiyāyika, a golden crown is a ‘qualified entity’ and, bizarre though it may seem at first glance, ‘A golden crown is a crown’ just as ‘A blue pot is a pot’.Footnote 25

In compliance with assertion [a], the statement ‘A blue pot’ can be plainly described, in standard notation and according to a predicative account, through the linear string (∃x) (Gx ∧ Nx), true iff ‘There does exist a variable x’, ‘This variable is a pot’ (Gx), and ‘This variable is blue’ (Nx).Footnote 26 According to the Nyāya-property-location language (implying TvN), the attribution of these properties would be better described not by the coordination of a double predication, but by a relational structure whose fulcrum is a primitive term and not an existing variable. In dealing with such a sentence, ‘A blue pot’, it must first be noted that the element under discussion here is relations, not predications.Footnote 27 In general terms, this case of coreference could be seen as a viśeṣaṇa-viśeṣya-bhāva, i.e. a qualifier-qualified relation, conceived as a form of determined cognition (savikalpa or viśiṣṭa jñāna).Footnote 28 The Nyāya relation-based analysis cannot therefore be directly reduced to predication, and any attempt to force the Nyāya account into this grid seems doomed to failure. If the first inaccuracy is thinking in terms of predication, the second is confusing the connective ‘and’ (‘∧’; which in the theory of sets corresponds to intersection, ‘’), with the qualifier-qualified relation.Footnote 29

The abstract property gt (ghaṭatva, pot-hood; cf. supra Gx) has as its locus the primitive term g—that is, an actual pot—while the further abstract property nt (nīlatva, blue-ness; cf. supra Nx) occurs in an instance of blue (n), which is in turn located in ‘a pot locus of pot-hood’.Footnote 30 If the property gt (whose reference set is G) is referred to its locus g (ghaṭa-niṣṭha-ghaṭatva), then this property will be the prakāra or mukhya-viśeṣaṇa (chief or root qualifier) and the primitive term g the mukhya-viśeṣya (chief or root qualificand).Footnote 31 Yet, the root-property gt is in turn the locus of a colocated (samānādhikaraṇa) second-order property nt. In other words, nt (blue-ness) occurs in gt (pot-ness), referred to the primitive term g (an actual pot). Thereby, the colocated second-order property nt turns out to be dependent on the first-order property gt, the mukhya-viśeṣaṇa. Blue pots are thereby pots because blue-ness is dependent on pot-ness—which sounds quite striking if not wholly false. How could such a claim be justified? More generally, how could such a relation be conceived?

“Relation (sambandha) is what, though distinct (bhinnatva) from the relata (sambandhin), in them occurs (āśrita). […] So, [for instance] contact (saṃyoga) [is the relation between] pot and ground; and direct contact (saṃnikarṣa), in the case of perception, between sense organ and the [perceived] object” (NK, p. 920).Footnote 32 Similarly, in set theory, a “pairing functionFootnote 33 or “relation is a set of ordered pairs” without any further restrictions: “any set of ordered pairs is some relation, even if a peculiar one” (Enderton 1977, p. 40).Footnote 34 To put it another way, given two generic sets or classes A and B, for x∈A and y∈B, the relation R is their Cartesian product (A × B)—written xRy or 〈x, y〉∈R, in which x stands in the relation R to y. Conversely, any subset of ordered pairs, an element of the power set A × B, is some sort of relation.Footnote 35 “The domain of R (domR), the range of R (ranR), and the field of R (fldR) [are defined] by: (x ∈ domR) ↔ (∃y) (〈x, y〉 ∈ R) [i.e., x belongs to the domain of R iff there exists at least an y, such that x stands in relation R with y], (x ∈ ranR) ↔ (∃t) (〈t, x〉 ∈ R), and fldR = (dom R ∪ ranR) [i.e., the union of the two]” (Enderton 1977, p. 40). Consequently, R is a relation from A (set of departure) to B (the set of destination) iff: R is a relation, domR ⊆ A, and ranR ⊆ B. In other words, R maps the image set of the domain in A into B (R: A ↦ B), since the image set of the domain is equal to or a subset of the set of destination.Footnote 36

Now, what could possibly be meant by the qualifier-qualified relation? ‘‘A qualifier (viśeṣaṇa) [is known as such because it is] in possession of the property qualifier-ness (viśeṣaṇatā). […] In the case of [a cognition such as] ‘A blue pot’, etc., the property qualifier-ness [finds his limitor] in the property blue-ness. […] The limitor (avacchedaka) of the qualifier-ness in the qualifier is the qualifier itself. Accordingly, in the example ‘A man with a staff’, the property staff-hood [operates] as the limitor of [this] qualifier-ness’’ (NK, pp. 788–789).Footnote 37 In parallel, “it is said qualified (viśiṣṭa) a qualificandum (viśeṣya) possessing a qualifier (viśeṣaṇa). Therefore, a substance (dravya) [e.g., a pot] possessing a quality (guṇa) [e.g., blueness] is a substance qualified (viśiṣṭa) by that quality” (NK, p. 779).Footnote 38 Linking the previous two notions, it could be stated that “a qualified-qualifier cognition (viśiṣṭa-viśeṣaṇaka-jñāna) has as its content (viṣaya) a property (vaiśiṣṭya) [occurring] in a subject (dharmin); [in particular, it is a cognition] of a qualificand in possession of a qualificans. So, [e.g.] it is the cognition [concerning] ‘A man with a staff’. […] [In the same way], it becomes evident that the qualifier [i.e. the staff] of a certain qualified [e.g. the man] is [in turn] qualified by another qualifier (viśeṣaṇāntara) [i.e. the staff-hood]. In such a cognition, by virtue of the property qualifier-ness (viśeṣaṇatā), the staff appears as the qualifier on the man’s side, and the property staff-hood as the qualifier of the staff. In such a cognition, on the man’s side (āṃśa), [the qualifier is] the staff, [but] on the staff side what appears is the staff-ness, by virtue of the relational abstract qualifier-ness: staff-hood must not be conceived on the man’s side indeed, because it [only operates] as the limitor (avacchedaka) of qualifier-ness. It must be understood, in this regard, that a distinct (viśṛṅkhala) object of cognition (upasthiti) is the eliciting factor (prayojikā)” (NK, p. 780).Footnote 39 Indeed, man-hood qualifying men is completely independent from staff-hood qualifying staffs. Nevertheless, in the context of this particular qualified-qualifier cognition, staff-hood is the limitor of qualifier-ness, occurring in this particular staff qualifying this man.

And again in NK, “the qualifier-qualified relation (viśeṣaṇa-viśeṣya-bhāva) is a specific (viśeṣa) objectivity (viṣayatā). Consequently, in the verbal cognition (śābdabodha) of [the expression] ‘staff holder’, the relation qualifier-qualified [itself is the very object of cognition, and that conceived] between staff and man. […] The qualifier-ness and the qualified-ness, both stand (āpanna) in a conditioned-conditioner (or restricted-restrictor) relation (nirūpya-nirūpaka-bhāva)” (NK, p. 789).Footnote 40 Although this last sentence may appear straightforward, it deserves a glossa. On the surface—in an initial broad sense which ignores the word-order asymmetry in the text—this could generically refer to the relata mutual dependence within the given relation: which is certainly true, but not very informative. The latter definition (nirūpya-nirūpaka) should thus be taken as a mere rephrasing of the former (viśeṣaṇa-viśeṣya): the qualifier (viśeṣaṇa) is the conditioner or restrictor (nirūpaka) and the qualified (viśeṣya) is what is conditioned or restricted (nirūpya). Taking more seriously the inversion of the word-order symmetry in NK text (nirūpya-nirūpaka vs. viśeṣaṇa-viśeṣya), however, the extended copulative structure (ca) and abstracting forms (-tva), there is also a potential second sense: both qualifier-ness and qualified-ness could equally and complementary acquire the status of conditioner or conditioned. The first case has already been discussed: the qualifier is the conditioner and the qualified is the conditioned. The second appears much more striking, however: the qualifier would be the conditioned and the qualified the conditioner. Thereby, in the context of a qualifier-qualified relation, the qualifier could be conceived as what is conditioned, thereby becoming a conditioned qualifier; and the qualified as the conditioner or restrictor, acting as a qualified conditioner (or conditioning qualified)—paradoxical though it may sound (cf., end of § 4.).Footnote 41

Relations in NL & the Colocated Qualification Principle (SVN)

We can now return to the case of the golden crown. Following the NL formalisation method, let crown-hood (mukuṭatva, mt) be the root-property (mukhya-viśeṣaṇa; cf. supra), for |mt| = M and m∈M; and gold-ness (hāṭakatva, ht) a second-level colocated property, for |ht| = H and h∈H. Furthermore, let (italic bold capital) be the relational abstract ‘coreferentiality’ (sāmānādhikaraṇyatā) referring to the binary relation (italic capital; sāmānādhikaraṇya, ‘coreference’ or ‘syntactic homogeneity’).Footnote 42 In parallel, be V (viśeṣya-viśeṣaṇa-saṃsargatā, or viśeṣaṇatā) the relational abstract of relation V (viśeṣya-viśeṣaṇa-bhāva-sambandha), the relation qualifier-qualificand as viśiṣṭa-jñāna (cf. supra). Let ‘⌝’ (top left corner) be the avacchedaka operator, so that ‘b + top left corner + relational abstract’ (i.e., bR) would mean ‘b operates as the avacchedaka of the relational abstract R’ (for 〈a, b〉 ∈ R). In parallel, be ‘⌞’ (bottom right corner) the nirūpaka operator, so that ‘relational abstract + bottom right corner + a’ (i.e., Ra) would mean ‘a is the nirūpaka of R’. A basic relation would thus appear in NL as: bRa, ‘The relation R is conditioned by a (the relational adjunct, or pratiyogin) and limited by b (the relational subjunct, or anuyogin)’. We are now in a condition to analyse the assertion ‘mukuṭaṃ hāṭakam’ (‘A golden crown’)Footnote 43 in NL as:

[2] h. Ṇm

yā sāmānādhikaraṇyatā hāṭaka-niṣṭhā sā mukuṭa-nirūpitā; ‘The relational abstract of coreferentiality or syntactic homogeneity, conditioned (nirūpita) by a crown (m), occurs (niṣṭha) in an instance of gold (h; viz., it refers to this gold as its locus)’; iff h ∈ (|ht| = H) (‘Being an instance of the property gold-ness, a specimen of gold belongs to the set What is golden, that is, the set Gold’), m ∈ (|mt| = M) (‘Being an instance of the property crown-ness, a crown belongs to the set Crowns’), (h ∈ |m|) (‘A specimen of gold belongs to the set What is coreferential with a crown’), that is, 〈m, h〉 ∈ (‘A crown and an instance of gold are an ordered couple belonging to the relation x is coreferential/syntactically homogeneous to y’). In standard notation: (∃x) (Hx ∧ Mx) (‘There do exist an x which is gold and a crown’), for H∩M ≠ ∅ (‘The intersection of the set Gold and the set Crown is not empty’).

Be noted here the niṣṭha operator (‘.’; a dot instead of ‘⌝’), connecting a property with a primitive term conceived as its locus.Footnote 44 The relation [2] can then be further specified, for TvN, as:

[2a] (h . ht)⌝ ⌞(m . mt)

yā sāmānādhikaraṇyatā hāṭaka-niṣṭha-hāṭakatvāvacchinnā sā mukuṭa-niṣṭha-mukuṭatva-nirūpitā; ‘The relational abstract of coreferentiality, conditioned by the property crown-hood referring to a crown, is limited (avacchinna)Footnote 45 by the property gold-ness occurring in an instance of gold’—the purport (tātparya, henceforward (t)) of which is (t)‘Gold-ness in a specimen of gold occurring in a crown qua instance of crown-hood’.

The relation [2a] can now be interpreted and rephrased in terms of the qualifier-qualified relation (V). The crown is () gold because it is qualified (V) by gold:

[3] (h . ht)⌝ V(Ṇ) ⌞(m . mt)

yā viśeṣaṇatā hāṭaka-niṣṭha-hāṭakatvāvacchinnā sā mukuṭa-niṣṭha-mukuṭatva-nirūpitā; ‘The relational abstract qualifier-ness, conditioned by the property crown-hood, referring to a crown, is limited by the property gold-ness occurring in an instance of gold’. Iff h ∈ (|ht| = H); m ∈ (|mt| = M); h ∈ |V(Ṇ)m| (‘A specimen of gold belongs to the set [Coreferential] Qualifiers of a crown’).Footnote 46 Note here V(Ṇ), that is, ‘Ṇ interpreted as V, salva veritate’.

The root property crown-hood (mukuṭatva, mt)—adjunct of the relational abstract V(Ṇ) ( as V)—binds the dominion of the relation to M (the set Crowns), thereby effectively ruling out the complement set of M (i.e. \(\bar{\rm M}\), the set Everything which is not a crown). Consequently, if coordination [a] is true for M∩H ≠ ∅, relation [3] from the set of departure M de facto excludes possibilities [b] and [c]. In other words, it obliterates \(\bar{\rm M}\)—consequently, H∩ \(\bar{\rm M}\) as well, viz. ‘Everything which is gold but not a crown’—and it is true for Hsub[3] ⊆ V(Ṇ)[M] (i.e., given [3], we are dealing only with gold coreferential to crowns; for domV(Ṇ) ⊆ M and ranV(Ṇ) ⊆ V(Ṇ)[M]). Stemming from the fact that we are talking about the properties of a crown, gold-ness ends up being gold-ness in crowns and thereby included in the set Coreferential properties of crown-ness (V(Ṇ)[M]).Footnote 47

Since it concerns a pair of coreferential (samānādhikaraṇa) locatees occurring in the very same locus, relation [3] is describable by what I will hereafter call Samānādhikaraṇa-Viśiṣṭatva-Nyāya (SVN, ‘Principle of Coreferential Qualification’). In case of coreference, SVN, following a strictly relational logic, can bind all further properties to a chief or root one (mukhya-viśeṣaṇa). According to SVN, the qualifier (viśeṣaṇa) corresponds—under the condition of relation V(Ṇ)—to the image of the qualificandum (viśeṣya); this is in turn already qualified (i.e. it is a crown and not a bucket) and alone defines, as the root-property, the relational dominion. Thereby, gold-nesssub[3] ends up being a subset of properties of crowns because, having considered the viśeṣya primarily as a crown, no further cognition can avoid this basic qualification any longer. The qualificans gold-ness, occurring in the qualifier and referring to a crown, corresponds to the image of crown-hood under relation V(Ṇ), which consequently has as its elements the instances of gold-ness solely in crowns because it is conditioned by crown-ness (htV(Ṇ), hāṭakatvāvacchinna-viśeṣaṇatā, ‘Gold-ness as qualifier’—as a consequence, we are not primarily talking about gold, which is only what qualifies something else; V(Ṇ)mt, mukuṭatva-nirūpita-viśeṣaṇatā, ‘Crown-hood as qualified’, that is, what we are talking about). It goes without saying that SVN applies only in coreference cases (i.e. as V). If a blue pot is a pot ( as V), a man with a stick is not a stick (V only)—even though the man is qualified by his stick.

A relation can be grasped more effectively if topologically displayed in a Cartesian coordinate system. Ordered pairs on the plane make pictorially evident the fact that the first and foremost concern of Nyāya account is relations. To provide a first example, be given a general relation different from V. Let L be the relation ‘locus of’ and L its relational abstract ‘locus-hood’ (āśrayatā). An instance of smoke (d, dhūma) on a mountain (p, parvata) could be thus expressed in NL as: p. Ld, yā āśrayatā parvata-niṣṭhā sā dhūma-nirūpitā, true for p ∈ |Ld|, viz. ‘A mountain belongs to the set Loci of an instance of smoke’. Because the relation is 〈p, d〉 ∈ L (or ‘p is the locus of d’), it follows that on the Cartesian plane L identifies the ordered pair ‘smoke’ (in abscissa) and ‘mountain’ (in ordinate). This latter is a member of the set ‘Loci of a smoke’ along with e.g. ‘a portion of space’, ‘a fire’, etc. Mountain and smoke are obviously distinct objects, with different qualifying properties (for TvN) and different reference sets. Nevertheless, bound by the relation ‘locus of’ under the condition ‘smoke’, this mountain ends up belonging to the set ‘Loci of a certain smoke’. This implies that the main element of interest is neither the mountain nor the smoke. As topologically made evident in the Cartesian plane, what is at stake here is the property locus-hood with respect to smoke; a property occurring in this mountain along with others that are completely different in nature (e.g. ‘a fire’). Clearly, SVN cannot apply.

Let us now focus on the specific relation as V. So, let be in abscissa the set ‘Triangles’ (T) and in ordinates the set ‘Coreferential properties of triangles’ (V(Ṇ)[T]). This latter includes all the properties referable to triangles, such as ‘having the sum of internal angles equal to 180°’ (p1), ‘possessing a right angle’ (p2), ‘possessing equal sides’ (p3), etc. (i.e., p1, ……, pn). If p1 is a property possessed by all instances of triangles, p3 (itself a subset of the set in ordinates) it will on the contrary be referable only to a subset of T in abscissa: by definition, referable only to equilateral triangles. Thereby the relation 〈t, p3〉 ∈ V(Ṇ) (‘t is qualified by p3’, for V(Ṇ) ⊆ T × V(Ṇ)[T]) will define the portion of the plane identifying equilateral triangles. The dominion of the relation plainly claims that only triangles are under discussion here: an equilateral triangle—qualified via 〈t, p3〉 ∈ V(Ṇ)—is but a triangle, for: (domV(Ṇ) ⊆ T) ∧ (|p sub_domV(Ṇ)3 | ⊆ V(Ṇ)[T]).Footnote 48 However, as V by definition imposes that V(Ṇ)[T] refer to T; consequently, both domV(Ṇ) and ranV(Ṇ) are equal to or a subset of T, for V(Ṇ): T ↦ V(Ṇ)[T] and V(Ṇ)[T] ⊆ T. In general, “for a relation R, a class A is said to be R-closed, or closed under R, if whenever x ∈ A and xRy then also y ∈ A (i.e., R[A] ⊆ A)” (Levy 1979, p. 61). Therefore, the relation as V under examination is revealed to be an instance of closure: the set Coreferential properties of triangles is T-closed under the relation as V.Footnote 49

The same applies to the case of golden crowns and blue pots. Indeed, the relation is presented as ordered pairs with crowns or pots in abscissa (for M, the set Crowns; and G, the set Pots), and Properties of crowns or Properties of pots in ordinate. It follows that in [3]: (h ∈ (|ht| = H)) ∈ |V(Ṇ)m|, i.e. an instance of the property gold-ness belongs to the set What qualifies a crown (or Properties of a crown)—along with many others, such as heaviness, brightness, etc. The set Hsub[3] (qua Hsub[3] ⊆ V(Ṇ)[M]) is thus M-closed under the formula [3], for V(Ṇ)[M] ⊆ M. The relation as V is in fact a mapping of M (for domV(Ṇ) ⊆ M, the set Crowns as set of departure) onto the set Properties of crowns (for ranV(Ṇ) ⊆ V(Ṇ)[M], the set of destination); that is, V(Ṇ): M ↦ V(Ṇ)[M]. In other words, the relation as V defines the image of Crowns in Properties of crowns through the medium of a particular property, here gold-ness; for this reason, the property gold-nesssub[3] is but a sub-set of Properties of crowns. Clearly, the properties involved—gold-ness and crown-ness—are reciprocally unrelated (viśṛṅkhala) (cf. fn. 39) because the former is certainly not a subset of the latter; at most, the intersection of their two domains might be non-empty. However, here hāṭakatva plays the role of coreferential viśeṣaṇa (qualifier) of a particular viśeṣya (qualified), in turn qualified by the property mukuṭatva—and this root-qualification cannot simply be dismissed. A golden crown is a crown because the viśeṣya itself (the crown) in relation as V is already qualified by crown-hood: SVN in [3] identifies gold-ness as a property occurring in a crown—precisely, a golden one—and not the set of all golden things. For the same reason ‘A blue pot is a pot’.

Let us now proceed by adopting a different approach to demonstrate SVN in terms of limiting properties only (avacchedaka). It has been shown (cf. fn. 47) that the V-relational subjunct (viśeṣaṇatā-saṃsargīya-anuyogin, a) or limitor (vV, viśeṣaṇāvacchinna-viśeṣaṇatā) is always the V-qualifier (viśeṣaṇa, v; thus: v = a) because it is what expresses the quality (viśeṣa). In the example, gold-ness in gold is the V-limitor. It should be recalled that the relational abstract V reverses the terms of relation V (i.e. 〈viśeṣya, viśeṣaṇa〉∈V; or in short and for v1 = viśeṣya: 〈v1, v〉∈V), making explicit the fact that V refers to the viśeṣaṇa only under the condition of the viśeṣya.Footnote 50 In the case of a golden crown ( as V), gold-ness in gold is the qualifier (v) of a crown (v1): mukuṭa-viśeṣaṇaṃ hāṭaka-niṣṭha-hāṭakatvam. Thereby, hāṭakatvāvacchinna-viśeṣaṇatā, ‘The qualifier-ness (V(Ṇ)) is limited by gold-ness’ (cf. vV). In general:

[4] vV(Ṇ)v1

yā samānādhikaraṇa-viśeṣaṇatā viśeṣaṇāvacchinnā sā viśeṣya-nirūpitā; ‘Coreferential () qualifier-hood (V(Ṇ)), conditioned by the qualified (v1), is limited by the qualifier (v)’.

As a general scheme, ‘The relational abstract subjunct-ness (A), limited by the relational subjunct (a), is limited by the relational abstract coreferential qualifier-ness (V(Ṇ)) limited by the qualifier (v), expressing the ascribed quality (viśeṣa; e.g. gold-ness in gold)’, that is:

[5] vV(Ṇ)aA

viśeṣaṇa-avacchinna-samānādhikaraṇa-viśeṣaṇatā-avacchedaka-avacchinna-anuyogy-avacchinna-saṃsargīyānuyogitā. More straightforwardly: (t)‘The qualifier (v) is always the relational subjunct (a) in V(Ṇ)’.Footnote 51

Conversely, the relational pratiyogin (a1; i.e. the qualified, viśeṣya, v1) operates in V(Ṇ) as a dominion conditioner (nirūpaka): mukuṭatva-nirūpitaṃ hāṭaka-niṣṭha-hāṭakatvam, ‘Gold-ness in gold conditioned (i.e., under the dominion restriction imposed) by crown-ness’. At the same time, the crown is the qualified which is qualified by gold-ness: hāṭakatvena viśiṣṭaṃ viśeṣyaṃ mukuṭam. And yet the viśeṣya (v1)—being that which is qualified, as well as the adjunct (pratiyogin) and conditioner (nirūpaka) in V(Ṇ) (cf. [4])—is in as V (cf. supra: 〈v1, vV(Ṇ)) the limitor (avacchedaka) of the attributed property (viśeṣa, ś). Indeed, the property occurs in what is qualified: v1ś, viśeṣyāvacchinna-viśeṣaḥ (‘The quality limited by the qualified’). What does gold-ness refer to? The only viable answer is obviously the crown. Thus: mtht, mukuṭatvāvacchinna-hāṭakatvam. In general, substituting [4] and [5] in 〈v1, vV(Ṇ):

[6] 〈 (v1V(Ṇ)1a1A1), (vV(Ṇ)aA) 〉 ∈ V(Ṇ)

viśeṣaṇa-avacchinna-samānādhikaraṇa-viśeṣaṇatā-avacchedaka-avacchinna-anuyogy-avacchinna-saṃsargīyānuyogitā-viśiṣṭa-viśeṣya-avacchinna-samānādhikaraṇa-viśeṣyatā-avacchedaka-avacchinna-pratiyogy-avacchinna-saṃsargīya-pratiyogitā; ‘The relational abstract adjunct-ness (A1), limited by the relational adjunct (a1), is limited by the relational abstract coreferential qualified-ness (V(Ṇ)1) limited by what is qualified (v1); this compound is in turn qualified (viśiṣṭa; in bold) by relational abstract subjunct-ness (A), limited by the subjunct (a), limited by the relational abstract coreferential qualifier-ness (V(Ṇ)) limited by the qualifier (v)’.Footnote 52

Roughly speaking, if ‘x is qualified by y’ (〈x, y〉∈V(Ṇ)), what is x? The pratiyogitā in the pratiyogin occurring in the qualified-ness in the qualified. And what is y? The anuyogitā in the anuyogin occurring in the qualifier-ness in the qualifier. However, it has been shown that the qualifying property (ś) occurs in the qualified (v1ś) and it goes without saying that the qualifying property is nothing but the qualifier (v = ś); thus, in composing the above partial formulas, we can bring together [4] and [6] in a description such as [7]:

[7] v1V(Ṇ)1a1A1vV(Ṇ)aA

viśeṣya-avacchinna-samānādhikaraṇa-viśeṣyatā-avacchedaka-avacchinna-pratiyogy-avacchinna-saṃsargīya-pratiyogitā-avacchinna-viśeṣaṇa-avacchinna-samānādhikaraṇa-viśeṣaṇatā-avacchedaka-avacchinna-anuyogy-avacchinna-saṃsargīyānuyogitā; ‘The relational abstract subjunct-ness (A), limited by the subjunct (a), limited by the relational abstract coreferential qualifier-ness (V(Ṇ)) limited by the qualifier (v), whose limitor is the relational abstract adjunct-ness (A1), limited by the relational adjunct (a1), limited by the relational abstract coreferential qualified-ness (V(Ṇ)1) limited by what is qualified (v1)’. Roughly speaking, (t)‘That which is the anuyogin in V(Ṇ) (i.e. the qualifier) occurs in the pratiyogin (i.e. the qualified).

In light of the above, however, given [3] h.htV(Ṇ)m.mt (implying mt = v1 and ht = v), then:

[7a] m.mtV(Ṇ)1a1A1htV(Ṇ)aA

mukuṭa-niṣṭha-mukuṭatva-avacchinna-samānādhikaraṇa-viśeṣyatā-avacchedaka-avacchinna-pratiyogy-avacchinna-saṃsargīya-pratiyogitā-avacchedaka-avacchinna-hāṭakatva-avacchedaka-avacchinna-samānādhikaraṇa-viśeṣaṇatā-avacchedaka-avacchinna-anuyogy-avacchinna-saṃsargīyānuyogitā; ‘The relational abstract subjunct-ness (A), limited by the subjunct (a), limited by the relational abstract coreferential qualifier-ness (V(Ṇ)) limited by gold-ness (ht), whose limitor is the relational abstract adjunct-ness (A1), limited by the relational adjunct (a1), limited by the relational abstract coreferential qualified-ness (V(Ṇ)1) limited by crown-hood (mt) in a crown’.

It is thus confirmed that, if [3], then m.mtht, or mukuṭa-niṣṭha-mukuṭatvāvacchinna-hāṭakatvam (‘Gold-ness in crown-ness in a crown’). Indeed, if there is a colocated viśeṣaṇa, there must be a viśeṣya on which the former is dependent, lest it not be the qualifier it is. Therefore, gold-ness is revealed to be a colocated conditioned qualifier by virtue of its being conditioned by the domain it qualifies; and crown-ness is a qualified conditioner (or conditioning qualified), imposing the relational reference domain on the colocated qualifier that qualifies it.

SVN can conclude that, in cases of coreferentiality interpreted as a qualified-qualifier relation ( as V), whatever further colocated qualification (viśeṣa) be attributed to whatever target of qualification (viśeṣya), the former must be considered as already bound to the root-property of the latter, the relation reference domain. In other words, since as V is an instance of closure, its range must be acknowledged as a subset of the dominion. Golden crowns are crowns because the relational domain is rooted in the set Crowns. Or rather, if there are golden crowns it is because there is gold-ness in crowns. In other terms, as V is a mapping of the domain of the qualified (viśeṣya) onto the range of colocated qualifiers (viśeṣaṇa) and, in so doing, defining a subset of the range which is in turn equal to or a subset of the domain. Consequently, setting aside predication and connective ‘and’ (‘∧’), in Nyāya relational account a golden crown is a crown because the set Crowns is the starting and arrival point—a set which stands alone, along with its image under the condition ‘gold-property’ as a subset of itself. In relation as V in [3], when talking about gold-ness we are talking about nothing but crowns. The same holds for blue pots qua pots.

At this juncture, a preliminary account of the notion of coreferentiality has been provided here, relying on the unforeseen and to some extent counterintuitive output of SVN. If that is the case, then it is clear that—being the very same being—a crown and the gold of which it is made cannot actually be said to be different tout court, e.g. the way a crown and a chair are. Nonetheless, it still remains unanswered the question regarding the relational nature of non-difference, and in particular whether this latter might be considered, or rather reduced, to simple cases of equivalence, equality, or identity. The second part of this investigation will be devoted to this issue.