Nothing But Gold. Complexities in Terms of Non-difference and Identity

Beginning from some passages by Vācaspati Miśra and Bhāskararāya Makhin discussing the relationship between a crown and the gold of which it is made, this paper investigates the complex underlying connections among difference, non-difference, coreferentiality, and qualification qua relations. Methodologically, philological care is paired with formal logical analysis on the basis of ‘Navya-Nyāya Formal Language’ premises and an axiomatic set theory-based approach. This study is intended as the first step of a broader investigation dedicated to analysing causation and transformation in non-difference.


Abbreviations a
Primitive term (lowercase italics) _ t Abstraction functor, expressing the Sanskrit suffix -tva or -tā (e.g., a t = a-hood) A Set A (capital) |a t | Extension of an abstract; |a t | = A R Relation R (capital italics) R Relational abstract (bold capital italics) R (R′) Relation R′ interpreted as R, salva veritate R [A] The relation R set of destination; for R: A↦B, domR⊆A, ranR⊆B, and R[A] = B ⌝ Avacchedaka operator; identifying the limitor of a relational abstract ⌞ Nirūpaka operator; identifying the conditioner of a relational abstract .
[…] A second model [is] known as 'reflectionism' (pratibimbavāda) […]. The tradition is that the Bhāmatī school propounds limitationism and the Vivaraṅa school reflectionism […]". For a detailed analysis of Vācaspati's main philosophical tenets, cf. also Ram-Prasad (2002, pp. 95-132). Regarding contrasting models within Advaita, cf. also: Timalsina (2006, pp. 21-24). For a general survey on Ś aṅkara and his Brahmasūtrabhāṣya, cf. Potter (1981, pp. 115-179). For a well-documented survey of the complex mutual interchange between the Advaita and Nyāya schools, cf. Phillips (1997). 10 VVR-P 3: ata eva bhāmatyāṃ hāṭakamakuṭagranthe bhedasyaiva hāṭakanyūnasattākatvaṃ na makuṭasyoktam, pariṇāmasya pariṇāmisamānasattākatvāvaśyakatvāt | "māyāmātram idaṃ dvaitam" (Gauḍapādīya-kārikā, 1.17) ity atrāpi dvaitaśabdena bhedasyaiva mithyātvam ucyate, na punar bhedavataḥ |. 11 A quote analysing a passage of Madhusūdana Sarasvatī's Advaitasiddhi (cf. AS 1997, p. 8; tatrādvaitasiddher dvaitamithyātvasiddhipūrvakatvād dvaitamithyāvam eva prathamam upapādanīyam); additions in square brackets are mine. Cf. also Chakraborty (1967, p. 41): "Ś aṁ kara interprets Brahma Sūtra in his own way and shows that the consistency of the upaniṣadic texts can alone be maintained on the admission of the sole reality of consciousness and falsity of the world". On this point, see also Timalsina (2009, p. 85): "There are two ways Advaita can be established: by confirming the existence of a singular reality, or by rejecting the existence of duality". which it is made? If there is no difference, where is the difference between jagat and brahman? 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 machineryenvisioned, 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, nondistinct (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. 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 nondifference (bhedābheda). 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 manner 15 : [1] A 1 B 1, 'x is y' (e.g. kuṇḍalaṃ suvarṇam, 'The earring is gold').
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'. 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. 17
Some formal tools are required to perform the analysis in NL (Navya-Nyāya Formal Language). 18 Let the notation '_ t ' be here the abstraction functor, capable of expressing the Sanskrit suffix -tva or -tā. 19 So, for instance, if the primitive term g (small italics) is a single pot (ghaṭa), 20 then 'g t ' = 'the property of being g' or 'ghood' (i.e. ghaṭatva, 'pot-hood') whose extension corresponds to the set 'pots' G (capital), to obtain |g t | = 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 g t -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'. 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 (h t ), and 'crown-hood', mukuṭatva (m t ), in reference to the two distinct sets M (the set Crowns; for m∈M and |m t | = M) and H (the set Gold; for h∈H and |h t | = H).
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). 24 For a 22 I ignore as being superfluous to my argument here the fourth possibility in which both properties are negated: ubhayābhāva-prayuktaviśiṣtābhāva, 'absence qualified by the absence of both qualificandum and qualificans'. Being not crowns or gold, there would be, for instance, gingerbread or whatever else. I refer here to the inferential taxonomy displayed by Viśvanātha's Muktāvalī when discussing the "governing factors" (prayojaka) for "qualified negation" (viśiṣṭābhāva)" as conditions for inferential subjectness (pakṣatā). I add here a fourth possibility (case [a]) not present in the original text: ubhaya-bhāva, or double presence. Cf. NSM (1988: 496-506) and Rai (1995, pp. 6-7), also quoted by Pellegrini (2014, p. 12). 23 Instead of the perhaps more common '¬', I will use the tilde operator (∼) as negation functor here to avoid any confusion with the NL operator top left corner (⌝); cf. infra fn. 51-52. 24 Cf. also Matilal (1968, p. 32): Naiyāyikas "tended […] to speak in terms of dharma (property) and dharmin (property possessor or locus of property"; and Matilal (2001, p. 202): "A simple qualificative cognitive state is one where the cognizer cognizes something (or some place or some locus, as we will have to call it) as qualified by a property or qualifier. It is claimed by most Sanskrit writers that to say that something or some place is qualified by a qualifier is equivalent to saying that it is a locus of some property or locatable". The concept of 'property' is used in this paper according to a principle of ontological parsimony, which naiyāyikas name lāghava: "It seems that Nyāya tries in the main to avoid disputes about ontology, and develops a theoretical language which can be used even by those who do not share its ontological dispositions (cf. [16, p. 66], [3, p. 201])", Ganeri (2008, p. 112), quoting B.K. Matilal and S. Bhattacharyya. Cf. also Matilal (2001, pp. 208-209): "Suppose by 'property' we mean non-universal, abstract features, or even tropes"; the thought experiment of 'ability to swim ' and 'water';and Matilal (2001, pp. 209-210) on the difficulties in translating dharma: "[…] the word dharma has a wider extension than the word 'property' […]. Dharma sometimes means not only abstract properties or universals but also concrete features, that is, the particular features of some object or locus. Dharma and dharmin constitute a pair in Sanskrit that is the equivalent to the pair 'locatee' (or the locatable) and 'locus' (location, which may be a place or a time or even an abstract object)". 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'. 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). 26 According to the Nyāya-propertylocation 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. 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). 28 25 Nīlatva-viśiṣṭa-nīla-rūpa-vān ghaṭatva-viśiṣṭaḥ ghaṭaḥ, 'a pot, possessing blue color in turn qualified by blue-ness, is qualified by pot-ness'; cf. Matilal (1968, p. 15), translation mine. The puzzle 'a pot = a blue pot' "[…] express a truism, viz., a thing is identical with itself no matter whether you refer to it in a general way (sāmānyena) by calling it a 'pot' or in a special way (viśeṣeṇa) by calling it 'a blue pot'."; Matilal (1968, p. 48). Cf. also Ingalls (1951, p. 69): "That which is expressed by 'pot' alone is the type of framework that subsists in all knowledges of pots.
[…] If one does not accept [that 'blue pot = pot'], one must admit that a blue pot is not a pot (cf. Raghunātha, Dīdhiti, 19-20;in Ingalls 1951, pp. 160-161)". Thus, viśiṣṭa (qualified) as 'accompanied by': "vaiśiṣṭyaṃ ca sāhityaṃ sāmānādhikaraṇyaṃ vā jñeyam", according to the NK definition (cf. NK, p. 779 and Ingalls 1951, p. 69, fn. 137). In this case I follow Matilal (1968, p. 48) regarding Ingalls' positions (Id. 1951, P. 69); cf. also the reviews of Ingalls (1951) in Potter (1954) and Staal (1960). 26 Quine (1981, p. 27): "To say that a city or a word has a given property, e.g. populousness or disyllabism, we attach the appropriate predicate to a name of the city or word in question". Cf. also Quine (1981: § 22. Class and Member, 119-123): "In such context ['Paris is a city'] 'is' expresses rather possession of a property, or membership in a class: Paris belongs to the class of cities […]. It is this sense of 'is' that is rendered symbolically with '∈': 'Paris ∈ city' […]". Or, to put it in its simplest terms: "Traditional [western] grammar tell us that the simplest sentences are composed of a subject and a predicate.
[…] [The subject] tell us what the sentence is about. The rest is the predicate: this tell us what is said about it"; Priest (2000, pp. 17-18). For a survey of "subject-predicate discourse" in the context of the problem of universals and realist-antirealist debate, cf. Loux (2006, pp. 21-27). 27 Regarding the primacy of proposition and predication as conceived in Western logic, and its differences with respect to the Nyāya qualifier-qualified approach, cf. Matilal (2001, pp. 201-205): "Now, in the Indian context the basic combination is not called a proposition. It is a structured whole that is grasped by an atomic cognitive event. We call it an atomic qualificative (viśiṣṭa) cognition.
[…] A qualifier and a predicate-property may not always be the same, such that we can say there is only a terminological variation". Regarding the "basic combination of predication" in propositions as the "focal point in current logic", Matilal quotes Quine (1960, p. 96) and Strawson (1974, p. 4). See also Shaw (1976 and Staal (1988, p. 63): "Western thought is inclined to analyse a close relationship in terms of subject and attribute whereas Indian thought considers the relation to the adhikaraṇa". 28 Interpreting coreference as qualification will be discussed in Part 2. On "determinate and indeterminate knowledge", cf. Ingalls (1951, pp. 39-40). Cf. also Matilal (1968, p. 13): "The content of a qualificative cognition, then, taken as a whole, is articulated in such a way that a certain feature or features of it will be emphasized as features of, or occurrent in, or related to, the remaining portion or portions of the content.
[…] Thus a qualificative cognition may be said to be an answer to question of the form: 'What is this?', 'What property does it possess?', 'When or where does it occur? '";and Id. (1968: § 3.7). See also Dalai (1992, pp. 10-13). Regarding jñāna-fundamentally as "cognition or psyche-dependent awareness"-see also Bilimoria (1985, p. 75): "we may note how jñāna is used […] sometimes to indicate 'knowing' in the sense of 'propositional attitude' [Matilal (1968, pp. 8-9)] towards beliefs, or towards what one is actually believing and judging at some time, as would occur, say, in reflective and introspective states, where there 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. 29 The abstract property g t (ghaṭatva, pot-hood; cf. supra Gx) has as its locus the primitive term g-that is, an actual pot-while the further abstract property n t (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'. 30 If the property g t (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 mukhyaviśeṣya (chief or root qualificand). 31 Yet, the root-property g t is in turn the locus of a colocated (samānādhikaraṇa) second-order property n t . In other words, n t (blueness) occurs in g t (pot-ness), referred to the primitive term g (an actual pot). Thereby, the colocated second-order property n t turns out to be dependent on the first-order property g t, the mukhya-viśeṣaṇa. Blue pots are thereby pots because blue-ness is Footnote 28 continued is affirmation of particular cognitive contents, as for example, when one becomes aware of 'table contentness' in his consciousness as his eyes fall on the large 'object' (this something) in the kitchen. The judgement is not about the 'object' as such, but it is an affirmation of his mental mode in relation to the object. However, often, too swift a move is made […; thereby] when a reflective judgement is taken to be an assertion of the truth-value of a cognition, jñāna is rendered as knowledge, implying that it is a judgement with a truth-value […]". In any case, the "significance"-in an "epistemic sense"-of a jñāna is "having contentness: viṣayatā"; Bilimoria (1985, pp. 76-77). Regarding viśeṣya-viśeṣaṇa-bhāva-sambandha cf., of the many possible sources, Gadādhara (1990, pp. 125-126): "saṃsargatayā ca samaṃ prakāratāyā viśeṣyatāyāś ca nirūpyanirūpakabhāvākhyaḥ sambandhaviśeṣo'bhyupagantavyaḥ | sa ca sambandhaḥ kāryatvakāraṇatvādheyatvādhāratvapratiyogitvānuyogitvādīnāṃ mithas tādṛśasambandha iva svarūpaviśeṣaḥ padārthāntaram eva vā, anyathā tatra tena sambandhena tat prakārakam ity etad arthasya durvacatvāt |. Translation: It is also to be admitted that there is a relation of determinerdetermined-ness between relationness, on the one hand, and modeness as well as qualificandumness, on the other. And that relation is either a particular self-linking relation or a separate ontological reality, just like the relation with cause-ness and effect-ness, superstratum-ness and substratum-ness, successor-ness and predecessor-ness. Otherwise the meaning of 'having that as a mode by that relation' cannot be explained". Regarding prakāratā, cf. also NK, p. 515. 1) prakāratā-viṣayatā | [ka] viśeṣaṇatvāparanāmā vilakṣaṇaviṣayatāviśeṣaḥ. 29 Halmos (1960, p. 12): "If A and B are sets, the intersections of A and B is the set A∩B defined by A∩B = {x∈A: x∈B}". Jech (2006, p. 8): "One consequence of the Separation Axioms is that the intersection and the difference of two sets is a set, and so we can define the operation X∩Y = {u∈X: u∈Y} and X-Y = {u∈X: u∉Y}"; Cf. also Enderton (1977, p. 21). For a plain explanation about the connections between basic operations on sets, Boolean operations, and Venn diagrams: Moschovakis (2006, pp. 2-4). Cf. also Levy (2002, pp. 244-246) on Boolean algebra; and Quine (1981, pp. 11-12) for an introduction to connective 'and'. 30 It must be noted here the absence of quantifiers, variables and operators, such as '∧' (cf. ∃x : m t (x) ∧ h t (x)). The NN logic syllabus thus basically consists of primitive terms, abstract properties, relational abstracts, and the two operators 'limitor' and 'conditioner'; cf. Anrò (forthcoming). Regarding the fact that "Indian logic has no variables" and the "strange doctrine of repeated abstraction" without quantifiers, cf. also Bochenski (1956, pp. 149-150). 31 Cf. Ingalls (1951, p. 43) and Matilal (1968, p. 15). 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?
[…] 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). 32 Similarly, in set theory, a "pairing function" 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). 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. 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 32 NK, p. 920. saṃbandham-1[ka] saṃbandhibhinnatve sati saṃbandhyāśritaḥ | [kha] […] yathā ghaṭabhūtalayoḥ saṃyogaḥ | yathā vā pratyakṣasthale indryārthasaṃnikarṣaḥ |. Regarding the six kinds of "intercourse" (saṃnikarṣa or sannikarṣa; ṣaṭsaṃnikarṣa) in perception, cf. Sihna (1934, pp. 75-85): "Perception depends upon some sort of intercourse (sannikarṣa) or dynamic communion between its object and a particular sense-organ". Contact or saṃyoga is the first saṃnikarṣa, given the case of a substance (dravya; say, a pot) in union with the visual organ. See also Shaw (1989, p. 383;2010, p. 626). A set z is said to be an ordered pair if for some x and y, z = 〈x, y〉. 6.2 Proposition 〈x, y〉 = 〈u, v〉→x = u ∧ y = v.". Cf. also: Levy (2002, p. 25): "A class S is said to be a (binary) relation if every member x of S is an ordered pair. We shall write y S z for 〈y, z〉 ∈ S. For example, if\is the natural order relation on the natural numbers (i.e., 〈x, y〉 ∈\if and only if x is less than y), then we write x\y for 〈x, y〉 ∈\.
[…] We shall sometimes say collection instead of set.
[…] In some approaches to set theory 'class' has a special technical meaning. […] Some sets are not really sets and even their names must never be mentioned. Some approaches to set theory try to soften the blow by making systematic use of such illegal sets but just not calling them sets; the customary word is 'class'.
[…] Roughly speaking, a class may be identified with a condition (sentence), or, rather, with the 'extension' of a condition"; and Enderton (1977, p. 6): "Any collection of sets will be a class. Some collection of sets […] will be sets. But some collections of sets (such as the collections of all sets not members of themselves) will be too large to allow as sets. These oversize collections will be called proper classes". Or, more formally, Moschovakis (2006, p. 27): "For every unary, definite condition P there exists a class A = {x | P(x)} (3.7), such that for every object x, x ∈ A ↔ P(x) (3.8).
[…] Every set will be a class, but because of the Russell Paradox [cf. 3.5], there must be classes which are not sets, else (3.8) leads immediately to the Russell Paradox in case P(x) ↔ Set(x) & x ∉ x.
[…] By definition, a class is either a set or a unary definite condition which is not coextensive with a set". Cf. also Levy (2004, pp. 7-11) and Russell (1919, pp. 42-51). 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. 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,. 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). 38 Linking the previous two notions, it could be stated that "a qualified-qualifier cognition (viśiṣṭa-viśeṣaṇaka-jñāna) has as its content (viṣaya)  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 36 Enderton (1977, p. 40): "For example, let ℝ be the set of all real numbers […] an suppose that R⊆ℝ ℝ. Then R is a subset of the coordinate plane. The projection of R onto the horizontal plane axis is dom(R), and the projection onto the vertical axis is ran(R)". Smullyan (1996, p. 23): "By the domain, dom(R), of a relation R is meant the class of all x such that 〈x, y〉∈R for at least one y. By the range of R, ran(R), is meant the class of all y such that 〈x, y〉∈R for at least one x.
[…] We note that R⊆(dom(R) ran(R)). We say that a relation R is on a class A if dom(R) and ran(R) are both subclasses of A. (This is equivalent to saying that R is a subclass of the Cartesian product A A)". Cf. also Levy (2002, p. 26). Cf. also: Halmos (1960, p. 27): "If R is the relation of marriage, so that xRy means that x is a man, y is a woman, and x and y are married to one another, then dom(R) is the set of married man and ran(R) is the set of married women". In referring to relations here, I use a lexicon commonly proper only to functions (mapping, image, etc.) on the account which defines a binary relation as a multi-valued function: "This term [multi-valued function] is generic; it indicates that we are not solely concerned with 'single-valued' functions. In fact, convention forces us to use different terms, following the preoccupations of different authors: we speak of a multi-valued mapping whenever we study properties concerned with linearity or continuity; we speak of a binary relation whenever we study certain structural properties (order, equivalence, etc.); we speak of an oriented graph whenever we study combinatorial properties"; Berge (1963, p. v). Cf. also Berge (1963: ch. II 'Mapping one set into another', § 1 'Single-valued, semi-single-valued and multi-valued mappings', 20-22): "Let X and Y be two sets. If with each element x of X we associate a subset Γ (x) of Y, we say that the correspondence x→ Γ (x) is a mapping of X into Y; the set Γ (x) is called the image of x under the mapping Γ". 37 NK, viśeṣaṇatāvat | […] nīlo ghaṭa ity ādau nīlatve viśeṣaṇatā | […] viśeṣaṇatāvacchedakaṃ tu viśeṣaṇe yad viśeṣaṇaṃ tat | yathā daṇḍavān puruṣa ity atra daṇḍatvaṃ viśeṣaṇatāvacchedakaṃ iti |. Regarding self-linking relations (svarūpasambandha) and viśeṣaṇatā, cf. Matilal (1968, pp. 40-44): "an absence of something [e.g.] is looked upon as the qualifier of the locus […]. Nyāya calls such relations relations of qualifier-ness. This is a merely stylistic method Nyāya adopts to describe such a 'supposed' relation without committing itself to the reality of such a relation as a separate entity over and beyond the data". Along the same lines, cf. also Matilal (1968, pp. 69-70, 142 (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 qualifierness. It must be understood, in this regard, that a distinct (viśṛṅkhala) object of cognition (upasthiti) is the eliciting factor (prayojikā)" (NK, p. 780). 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 conditionedconditioner (or restricted-restrictor) relation (nirūpya-nirūpaka-bhāva)" (NK, p. 789). 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 qualifiedness 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. i.e. the property of being a relation of qualification], too, is a special kind of objectivity; hence, the qualification [saṃsarga], too, has the objectivity of qualified cognition. Explanation: The object of qualified cognition is a relational complex having three elements-a qualificandum, a qualifier and a relation between them. Now, since the entire relational complex is what is cognized, and, according to Nyāya, the relational complex is not an ontological entity over and above the three elements, all three of the elements have to be accorded a different type of objectivity". Square brackets are mine. Cf. also, NK, p. 935, saṃsargaḥ-1 [ka] saṃbandhaḥ |. Regarding the notion of 'restriction' cf. Anrò (forthcoming: § 3.2). 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.). 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, m t ) be the root-property (mukhya-viśeṣaṇa; cf. supra), for |m t | = M and m∈M; and gold-ness (hāṭakatva, h t ) a second-level colocated property, for |h t | = H and h∈H. Furthermore, let Ṇ (italic bold capital) be the relational abstract 'coreferentiality' (sāmānādhikaraṇyatā) referring to the binary relation Ṇ (italic capital; sāmānādhikaraṇya, 'coreference' or 'syntactic homogeneity'). 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., b⌝R) 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., R⌞a) would mean 'a is the nirūpaka of R'. A basic relation would thus appear in NL as: b⌝R⌞a, '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') 43 in NL as: 41 Cf. NK, p. 432: nirūpyatvam-nirūpitatvam |. NK, p. 432. nirūpitatvam-svarūpasaṃbandhaviśeṣaḥ | yathā rājñaḥ puruṣa ity adau puruṣaniṣṭhasvatve rājaniṣṭhasvāmitvanirūpitatvam | śiṣṭaṃ tu nirūpakatvaśabde draṣṭavyam |. 'The property of being conditioned (or restricted) is a peculiar self-linking relation. In sentences such as 'The servant of the king', the property of being conditioned by the ownership (svāmitva) occurring in the king [must be sought] in the possess-ness (svatva) occurring in the man. What remains must be seen sub voce nirūpakatva (being a conditioner)'. In this example, the qualifier (the ownership occurring in the king) is also the conditioner, precisely because the property of being conditioned (i.e. the qualified-ness) of the qualified (the possess-ness occurring in the man) is in question here. However, the relation could easily be reversed: the qualifier-ness occurring in the qualifier (the ownership occurring in the king) could be conditioned (nirūpita) by the qualified-ness occurring in the qualified (the possess-ness occurring in the man), which consequently becomes the conditioner. QED. 42 For a discussion of ordered pairs formalisation (albeit limited to vṛtti-niyāmaka relations only), cf. also Staal (1973, p. 152 ff). In his plain notation: A(x, y). Cf. also Bhattacharyya (1987, p. 174): "To distinguish this sense of 'property' […], we shall write 'property (N-N)'". 43 For the purposes of this analysis, it is not paramount whether the description is definite or indefinite; let us assume here that it is indefinite and non-generic: 'A golden crown', expressed by m as a primitive term. For these particular examples, cf. Ganeri (2006, pp. 10-11). Cf. also Matilal (1968, § 9.7, 78-79); Ingalls (1951, p. 50): "Navya-Nyāya regularly expresses its universal statements and knowledges not by quantification but by means of abstract properties"; Ganeri (2008, pp. 110, 118): "The Nyāya authors themselves do not […] show much interest in the problems of scope ambiguity […]. And often the language is used in only a semiformal way, especially when used by non-Nyāya authors". Russell (1919, pp. 167-180): "An indefinite description is a phrase of the form 'a so-and-so', and a definite description is a phrase of the form 'the so-and-so' (in the singular)". For a complete introductory survey of generic (or definite) and generic (or non-generic) descriptions, cf. Ludlow (2018).
[2] h. Ṇ ⌞m 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 ∈ (|h t | = 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 ∈ (|m t | = M) ('Being an instance of the property crown-ness, a crown belongs to the set Crowns'), (h ∈ |Ṇ⌞m|) ('A specimen of gold belongs to the set What is coreferential with a crown'), that is, 〈m, h〉 ∈ Ṇ ('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. 44 The relation [2] can then be further specified, for TvN, as: yā sāmānādhikaraṇyatā hāṭaka-niṣṭha-hāṭakatvāvacchinnā sā mukuṭa-niṣṭhamukuṭatva-nirūpitā; 'The relational abstract of coreferentiality, conditioned by the property crown-hood referring to a crown, is limited (avacchinna) 45 by the property gold-ness occurring in an instance of gold'-the purport (tātparya, 44 Regarding the niṣṭha operator, cf. Anrò (forthcoming). A primitive term is always on the operator's left side, while a property is always on its right. Thereby, for a generic primitive term 'a' and a generic property 'a t ', the expression 'a. a t ' will mean 'a-hood occurring in a'. 45 Ganeri (2008, pp. 109, 115): "So a conditioner maps to an existential quantifier, whose domain is restricted to the class assigned to the conditioner, and which binds the second place of a dyadic predicate. Similarly, a delimitor maps to a universal quantifier, whose domain is restricted to the class assigned to it, and which binds the first place of a dyadic predicate.
[…] The universal quantifier corresponding to the limitor always has wider scope than the existential quantifier corresponding to the conditioner". Cf. also Ingalls (1951, p. 48): "The relational abstracts […] are limited by the qualifiers of the entities in which they reside. Technically these abstract are said to be limited through a relation of residency (niṣṭhatvasambandhāvacchinna; cf. Śiv. Miśra, 22.8)". Ingalls (1951, p. 49): "No one method can be followed for reducing expressions employing 'limited' to the terms of Western logic". henceforward (t) ) of which is (t) 'Gold-ness in a specimen of gold occurring in a crown qua instance of crown-hood'.
The relation [2 a ] can now be interpreted and rephrased in terms of the qualifierqualified relation (V). The crown is (Ṇ) gold because it is qualified (V) by gold: yā viśeṣaṇatā hāṭaka-niṣṭha-hāṭakatvāvacchinnā sā mukuṭa-niṣṭha-mukuṭatvanirūpitā; '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 ∈ (|h t |=H); m ∈ (|m t |=M); h ∈ |V (Ṇ) ⌞m| ('A specimen of gold belongs to the set [Coreferential] Qualifiers of a crown'). 46 Note here V (Ṇ) , that is, 'Ṇ interpreted as V, salva veritate'.
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-ness sub[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 (h t ⌝V (Ṇ) , hāṭakatvāvacchinnaviśeṣaṇatā, 'Gold-ness as qualifier'-as a consequence, we are not primarily talking about gold, which is only what qualifies something else; V (Ṇ) ⌞m t, mukuṭatvanirūpita-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. Ṇ as V). If a blue pot is a pot (Ṇ 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. L⌞d, yā āśrayatā parvata-niṣṭhā sā dhūma-nirūpitā, true for p ∈ |L⌞d|, 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 Ṇ 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°' (p 1 ), 'possessing a right angle' (p 2 ), 'possessing equal sides' (p 3 ), etc. (i.e., p 1 , ……, p n ). If p 1 is a property possessed by all instances of triangles, p 3 (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, p 3 〉 ∈ V (Ṇ) ('t is qualified by p 3 ', 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, p 3 〉 ∈ V (Ṇ) -is but a triangle, for: (domV (Ṇ) ⊆T) ∧ (|p 3 sub_domV(Ṇ) |⊆V (Ṇ) [T]). 48 However, Ṇ 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 Ṇ as V under examination is revealed to be an instance of closure: the set Coreferential properties of triangles is T-closed under the relation Ṇ as V. 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 ∈ (|h t |=H)) ∈ |V (Ṇ) ⌞m|, i.e. an instance of the property goldness belongs to the set What qualifies a crown (or Properties of a crown)-along with many others, such as heaviness, brightness, etc. The set H sub [3] (qua H sub . In other words, the relation Ṇ 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-ness sub[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 Ṇ as V is already qualified by crown-hood: 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, Ṇ 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 Ṇ as V in [3], when talking about goldness 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.