Two Indian dialectical logics: saptabha ng	 and catus.kot.i ∗ Fabien Schang December 3, 2010 Abstract. A rational interpretation is proposed for two ancient Indian logics: the Jaina saptabha ng	, and the M	adhyamika catus.kot.i. It is argued that the irrationality currently imputed to these logics relies upon some philosophical preconceptions inherited from Aristotelian metaphysics. This misunderstanding can be corrected in two steps: by recalling their assumptions about truth; by reconstructing their ensuing theory of judgment within a common conceptual framework. Contents 1 Two logics? 2 2 Two opposite logics? 4 3 Two many-valued logics? 6 4 Jaina's theory of seven-fold predication 8 5 N	ag	arjuna's Principle of Four-Fold Negation 17 6 Two contrary logics? 21 7 Conclusion 26 ∗I want to thank Jonardon Ganeri for a fruitful informal correspondence with him, as well as Shrikant Joshi for his educated support about sanskrit and pali expressions and the anyonymous referee for his/her helpful requirements. 1 1 Two logics? A note on Indian logics is in order, to begin with. By a logic, it is ordinarily meant a specic set of consequence relations between a set of premises Γ and a conclusion B such that, for every formula A ∈ Γ, if A is true then so is B. Formally: if v(A) = T then v(B) = T, where v is a valuation function from a set of formulas to a set of truth-values. But such a modern denition of logic as a set of rules for truth preservation cannot be properly applied to ancient logics, including those from India. Rather, ancient and medieval logics include epistemology in the scope of the formal discipline: how to assess the content of a judgment isn't separable in Aristotle's Organon or the Port-Royal Logic, for instance, and Indian logics are not an exception. The epistemological import of Indian logics largely accounts for their peculiar content; the metaphysical assumptions that underlie these Indian schools of philosophy also results in specic theories of truth, and the main aim of the present paper will be to give a formal presentation of the ways to produce a judgment or predication such as S is P or S is not P (where S is the subject-term and P the predicate-term). As a matter of rule, Indian logics are about judgments and not about the sentences expressing them; we will restrict our attention to two such cases: the Jaina saptabha ng	; and the catus.kot.i from the Buddhist school of M	adhyamaka (literally, Middle Way). As a general rule, the logics emerging from the Jaina and M	adhyamika schools include both a theory of knowledge (about how to come to know something) and a complementary theory of judgment (about how to express this something known). Concerning the theory of knowledge, the nayav	ada is a Jaina theory (v	ada) of standpoints (nayas) that includes seven kinds of justication for the truth of a sentence.1 Furthermore, a set of seven (sapta) distinct judgments (bha ng	) can be made about a given topic. There is no causal relation between the number of standpoints and judgments, however. After all, the Greek skeptic Agrippa proposed ve kinds of justication while 1The seven kinds of justication (nayas) include metaphysical, physical and grammatical features. These are the following: naigama-naya (non-distinguished standpoint); sam. graha-naya (collective standpoint); vyavah	ara-naya (particular standpoint); r. ju-s	utranaya (momentary viewpoint); ±abda-naya (synonym viewpoint); samabhir	ud. ha-naya (etymological viewpoint); and, nally, evam. bh	uta-naya (momentary etymological viewpoint). For instance, the existence of an entity such as a pot, depends upon its being a particular substance (an earth-substance), upon its being located in a particular space, upon its being in a particular time, and also upon its having some particular (say, dark) feature. With respect to a water-substance, it would be non-existent, and the same with respect of another spatial location, another time (when and where it was non-existent), and another (say, red) feature. It seems to me that the indexicality of the determinants of existence is being emphasized here. ([12], p. 132). 2 sticking to an Aristotelian or bivalent view of judgments: either S is P or S is not P, period. Rather, the number of the Jaina judgments is due to their endorsement of a metaphysical pluralism according to which reality is many-faceted and cannot be restricted to a unique predication. As to the M	adhyamika school and its founder N	ag	arjuna (' 100 C.E.), they did not present a competing theory of knowledge but advanced four (catus.) main sorts of stances (kot.i) for any subject-matter. As noted in [15], logic is not metaphysically neutral, and the dierence between the Jaina seven and N	ag	arjuna's four judgments is due to their rival views of truth. Ganeri advances (in [6], p. 268) a relevant distinction between three semantic views of truth-assignment, namely: doctrinalism, skepticism, and pluralism. According to the doctrinalist view, it is always possible, in principle, to discover which of two inconsistent sentences is true, and which is false. This doctrine is related to Aristotle's two-valued logic, where only two judgments can be made about any subject-matter (S is P, S is not P) and only one of which comes to be accepted as true while the other is to be false. Bivalence is the logical cornerstone of such a doctrine and entails that every judgment is either a truthor a falsity-claim, i.e. a statement. Skepticism and relativism challenge this binary view in opposite directions. According to skepticism, the existence both of a reason to assert and a reason to reject a sentence itself constitutes a reason to deny that we can justiably either assert or deny the sentence, so that some sentences can be taken to be neither true nor false. Conversely, the pluralistic watchword is to nd some way conditionally to assent to each of the sentences, by recognizing that the justication of a sentence is internal to a standpoint; in this sense, one and the same sentence can be taken to be both true and false depending upon the condition under which its content is assessed. We take these three doctrines of truth-assignment to be the crucial path for a better understanding of Indian logics. While these have been dismissed by Western thinkers, as having irrational or unintelligible outlook2, we suspect this uncharitable preconception to stem from a narrow reading of bivalence that takes Frege's modern logic as a standard for any meaningful judgment. If so, the next sections insist upon the discursive and non-standard form of judgments in Jaina and M	adhyamika logics: it is still possible to preserve bivalence within these Indian theories and, thus, to preserve their intelligibility, but only if such a bivalence is not dened in Fregean terms and reformulated as a question-answer game between speakers. 2Manifoldness in this context is understood to include mutually contradictory properties. Hence on the face of it, it seems to be a direct challenge to the law of contradiction. However, this seeming challenge should not be construed as an invitation to jump into the ocean of irrationality and unintelligibility ([12], pp. 129-30). 3 2 Two opposite logics? An intriguing feature of Jaina and M	adhyamika logics concerns their attitude towards inference: the relativist doctrine of truth seems to entail a fully inconsistent logic, whereas the skeptic doctrine of truth would entail a fully incomplete logic. This means that, for any sentences A and B, B seems to be inferred from every premise A in Jaina logic (say, J): A 6|=J B (for every B); whereas no sentence B would be inferred from A in N	ag	arjuna's logic (say, N): A 6|=N B. Parsons described in [14] these cases in terms of ultimate eclecticism and complete nihilism, respectively3. Is Jaina logic a formal system of eclecticism, and N	ag	arjuna's logic a system for nihilism? This is not so, at least for one simple reason: nihilism assumes that the premise A is accepted as true, while the coming exposition of N	ag	arjuna's Principle of Four-Cornered Negation amounts to a denial of every sentence including A. As to the Jaina logic, the role of standpoints means that not every conclusion B can be inferred from A irrespective of the context in which A and B are assessed. This entails that not everything can be inferred from every given context, and Priest recalls this fact in [15] to make his own dialetheist reading of Jaina logic immune from triviality. We will return to this modern translation in Section 5. Two Sanskrit notions will be introduced now, in order to throw some light upon the Jaina and M	adhyamika ways of doing logic. The rst concept is anek	antav	ada: this term means non one-sidedness and characterizes the Jaina conditional view of truth, according to which the truth of a sentence is never one-sided (ek	anta) but always depends upon the context in which it is assessed. The second concept is prasajya pratisedha (see [5],[11],[13]); Mohanta mentions this concept in [13] as a non-relational negation which somehow corresponds to the contemporary denegation or illocutionary negation4. 3See [14], p. 141. Roughly speaking, eclecticism refers to the view that sentences of two dierent theories can be accepted consistently within a third embracing theory: T1 |= p, T2 |= q, T3 |= p and T3 |= q. This is not the point of Jainism. As to nihilism, it refers to the belief that nothing is true. This is not the point of M	adhyamaka, either. The dierence between such nihilists and the latter could be made clearer by the dierence between atheism (negative assertion about the existence of God) and agnosticism (mere denial about the existence of God). 4Illocutionary negation (denial, or denegation) has been dened by John Searle in [19]. Let the speech act F(p) = I promise that I will come, where F is the act of promise and p the sentential content I will come; then its locutionary negation F(∼p) is I promise that I will not come, while its illocutionary negation ∼(Fp) is I do not promise that I will come. Denial has been ordinarily rendered as a reversed turnstile a, in reference to Frege's turnstile of assertion, while Kei views it in [11] as a merely failed assertion 0. In both cases, denial occurs as an operator; in QAS, however, denial is an operand (a logical value: the no-answer ai = 0). 4 In contrast to the Jaina conditions for truth-assignment, the M	adhyamikas defended the view that being dependent upon anything else is a sucient ground for denying a corresponding predication: S cannot be said to be P or not to be P whenever S is not self-originated and is caused by another substance than itself. This refers to the two-truths doctrine and its distinction between absolute truth (param	artha-satya) and conventional truth (sam. vr. tisatya) in the M	adhyamika's s	unyav	ada (doctrine of emptiness); we will see how this doctrine leads to Ganeri's previous distinction between the pluralist and skeptic conditions for truth-assignment. While the Jains favor a contextual theory of armation, N	ag	arjuna endorses a peculiar use of denial which is to be rigorously distinguished from negative assertion and departs from falsity-assignment. Thus, saying that S is not P results in an ambiguous judgment between arming that the sentence S is P is false and denying that S is not-P is true. From an Aristotelian or doctrinalist approach, afrming S not to be P and denying S to be P are synonymous with each other; from a M	adhyamika or skeptic approach, however, P may be denied to be true of S without being armed to be false of S. Such a confusion amounts to a harmful confusion between two sorts of Indian negations (pratisedha), namely: the previous prasajya pratisedha and paryud	asa pratis.edha, which is a relational (see [13]) or locutionary negation used by the later Navya school. To sum up, Jaina and M	adhyamika logicians do oppose each other with respect to their underlying criterion for truth-assignment. Given two opposite sentences S is P and S is not P, how to decide on the truth of either? The main dierence between Jainas and M	adhyamikas lies in their answer to this question. Thus, Matilal claims (in [12], p. 129) that the dierence between Buddhism and Jainism in this respect lies in the fact that the former avoids by rejecting the extremes altogether, while the latter does it by accepting both with qualications and also by reconciling them. It is worthwhile to note that these opposite modes of truth-assignment also foreshadow the contemporary opposition between semantic realism and antirealism: [22] and [23] notice that the Jains countenance a correspondence theory of truth, whereas Siderits' comparison (in [21]) between N	ag	arjuna's denials and Dummett's anti-realist semantics entails that N	ag	arjuna's conception of truth doesn't transcend recognitional capacity by a given agent. Before approaching this last problem about the relations between judgments, let us consider the way to describe their various admitted judgments within a clear and uniform formal semantics. 5 3 Two many-valued logics? One of the primary aims of the paper is to insist upon the dialectical nature of Indian logics, i.e. their presentation in terms of speech-acts within an argumentative framework of questions and answers. To put it in other words, each truthor falsity-assignment proceeds by means of an intermediary act of armation and denial. Importantly, we take the asymmetry between the pairs true-false and armation-denial to be the key for a better understanding of Indian logics. A number of logical techniques have been proposed in the literature to catch the dialectical or discursive feature of Indian logics: relational or possible-world semantics ([15]), dialogics ([8],[11]), and algebraic or many-valued semantics ([6],[15],[18],[20])5. In order to give a more ne-grained description of Jaina and M	adhyamika logics, we resort here to many-valuedness. Roughly speaking, the various ways of making a judgment require the introduction of alternative logical values beyond the doctrinalist values of truth and falsity. In the case of Jaina philosophy, no judgment uniquely claims plain truth or falsity because of its underlying one-many correspondence theory of truth: a given sentence partly describes a fact following the perspective from which its content may be described.6 In the case of N	ag	arjuna's Principle of Four-Cornered Negation, it will be shown that the assumption of bivalence cannot make sense of the four negative stances together (see section 5). At the same time, the metaphysical pluralism of the Jains does not entail that new truth-values 5Gokhale rejected the many-valued interpretation of Jain logic because, according to him, a dierence is to be made between epistemological and logical values. Thus: The middle value designated by the term avaktavyam is therefore better understood as the epistemic middle rather as the logical middle. It is closer to the middle truth-value called `undeterminable' of Kleene's three-valued system than to the ukasiewiczian third truthvalue called `indeterminate'. (. . .) As a result we can say that avaktavya is not the third truth-value in the logical sense of the term, because it does not arise out of the violation of the laws of logic such as non-contradiction and excluded middle ([7], p. 75). This objection assumes that every logical value should have an ontological import, but our purely algebraic viewpoint of logic does not require this and Belnap's four-valued system is an instance where all the logical values have an epistemological import. 6Sylvan noted that Jainism apparently entailed a correspondence theory of truth (p. 62), so that the Jain values have an ontological import that diers from Belnap's four values in FDE: a sentence is true and false (in some respects), rather than told true and told false. The dierence between Jain and Aristotelian logic relies upon their underlying ontology: the latter takes a true sentence to correspond to a fact, while the former reject such a one-one correspondence between sentences of a language and states of aairs of the world. Thus Tripathi argued in [23] that Jainism is a realistic system. It not only holds that reality is pluralistic, but also that reality is many-faced (anantadharm	atmakam vastu). ([21], p. 187) The Wittgensteinian Bildtheorie should be strictly kept apart from the Jain view of reality, consequently. 6 should be devised in addition to the Aristotelian framework of bivalence. Rather, these alternative logical values are various combinations of truth and falsity inside the initial set of values T (for true) and F (for false). In particular, the Jaina theory of sevenfold predication (saptabha ng	) reminds one of Belnap's system of generalized truth-values and Shramko & Wansing's extension from 2 to n truth-values (see [3],[20]). Taking 2 = {T,F} as a basic set and its two elements of truth and falsity, an extension from 2 to 4 results from its powerset ℘(2), that is the set of the subsets of 2. Thus 4 = {{T},{F},{T,F},∅}, and Belnap symbolized the new combinations of truthvalues as {T,F} = B (for both true and false) and ∅ = N (for neither true nor false) in its four-valued logic FDE (First Degree Entailment). The same process can be applied indenitely, leading to a set of ℘(n) elements for any n-valued logic (where n ≥ 1). Another such generalized set is ℘(3), with n = 3 basic elements T, F and {T,F}. One of these generalized sets is 8 = {{T},{F},{B},{{T},{F}},{{T},{B}},{{F},{B}},{{T},{F},{B}},∅}. We will see that the latter set can be made very similar to the Jaina semantics, even though the odd number of the seven Jaina judgments may surprise at a rst blush. Moreover, Bahm rightly noted in [2] that Indian logics are not just formal combinations of truth-values but require a more comprehensive reading of their original texts. For this purpose, we propose now a conceptual framework to grasp the rationale of Indian logics: a Question-Answer Semantics (QAS) that encompasses Belnap's generalizations and helps to account for the M	adhyamika's dialectical logic of Four-Cornered Negation. DEFINITION 1. A question-answer semantics is a model QAS = 〈M,A〉 upon a sentential language L and its set of logical connectives©. It includes a logical matrix M = 〈Q,V,D〉, with: a function Q(α) = 〈q1(α), ...,qn(α)〉 that turns any sentence α of L into a specic speech-act (the sense of which is given by appropriate questions about it); a set V of logical values (where Card(V ) = mn); a subset of designated values D ⊆ V . It also includes a valuation functionA, such that the logical valueA(α) = 〈a1(α), . . . , an(α)〉 of V that characterizes a statement by giving an ordered set of m sorts of answers to each question qi in Q(α) = 〈q1(α), . . . ,qn(α)〉. This semantic framework results in a variety of logics L = 〈L , |=M〉 that include an entailment relation in a model |=M such that, for every set of premises Γ and every conclusion α in L , if A(Γ) ⊆ D then A(α) ⊆ D: Γ |=M α. A crucial dierence with the more familiar logics is the meaning of the 7 semantics values in QAS: each element {a1(α), . . . , an(α)} of A(α) is a basic answer ai(α) (where 1 ≥ i ≥ n) with the symbol 1 for armations (yesanswers) and the symbol 0 for denials (no-answers). Let us call by the general heading of logical value every such ordered set of answers, rather than the customary truth-values: these values are a combination of yesno answers to corresponding questions, whereas not every question is to be asked about the truth-value of a sentence in QAS. Once the formal structure is set out for any question-answer game, let us have a closer look at our two Indian logics at hand while attempting to reconstruct their argumentative games. 4 Jaina's theory of seven-fold predication It has been previously claimed that not everything can be derived from every premise from a Jaina perspective: meaningfulness presupposes that a restricted set of sentences can be accepted on the basis of certain premises in a given language, while the remaining sentences of the language should not be accepted. But the question is how the Jaina predications do make sense in a consistent set of statements. In particular, the Jaina theory of seven-fold predication (saptabha ng	) has been viewed as a challenge to Aristotle's logic. According to Aristotle, the Principle of Non-Contradiction (PNC) is a universal law of thought that cannot be violated without committing its opponent into plain nonsense. It is stated in [1] as follows: It is impossible for the same thing to belong and not to belong at the same time to the same thing and in the same respect. (Book IV, 1005b19-20) An instant reection suces to see that the Jains did not oppose to this principle as it stands: their semantic pluralism relies upon a doctrine of conditioned, relative or partial truth (sy	adv	ada). The Jaina philosopher V	adiveda S	uri (1086-1169 C.E.) displayed the following set of seven predications and witnessed the crucial role of sy	ad (arguably, or in some respect) in every corresponding statement, where every predication expresses a conditioned judgment about a sentence7: 7The saptabhan. g	 clearly departs from the Fregean logic of propositions, where a sentence expresses a thought and refers to a unique truth-value. To the contrary, the seven arguments of nayav	ada assume that the meaning of a sentence is context-dependent and doesn't refer to some eternal entity as the True. Thus Matilal: Realists or believers in bivalence (as Michael Dummett has put it) would rather have the proposition free from ambiguities due to the indexical elements an eternal sentence (of the kind W. V. Quine talked about) or a Thought or Gedanke (of the Fregean kind) such that it would have a value, truth or falsity eternally xed (. . .) We may assume that a proposition has an eternally xed truth-value, but it is not absolutely clear to us what kind of a proposition that 8 (1) sy	ad asty eva: arguably, it (some object) exists. (2) sy	an n	asty eva: arguably, it does not exist. (3) sy	ad asty eva sy	an n	asty eva: arguably, it exists; arguably, it does not exist. (4) sy	ad asty eva sy	ad avaktavyam eva: arguably, it exists; arguably, it is non-assertible. (5) sy	ad asty eva sy	ad avaktavyam eva: arguably, it exists; arguably, it is non-assertible. (6) sy	an n	asty eva sy	ad avaktavyam eva: arguably, it does not exist; arguably, it is non-assertible. (7) sy	ad asty eva sy	an n	asty eva sy	ad avaktavyam eva: arguably, it exists; arguably, it does not exist; arguably, it is non-assertible. Each of these predications is a combination of three basic semantic predicates (m	ulabha ngas)8, namely: assertion, or truth-claim; denial, or falsityclaim9; and a third sort of judgment that Jains called by non-assertibility (avaktavya). Before discussing the meaning of this third predicate #, it follows from their combinations that the three basic statements are very similar to the set 3 = {T,F,#} and its eight combined subsets in ℘(3) = 8 = {{T},{F},{#},{T,F},{T,#},{F,#},{T,F,#},∅}. The logical structure of QAS brings out the two main features of this sevenfold predication, where each component is to be rendered in terms of corresponding questions and answers. DEFINITION 2. A Jaina predication expresses an ordered answer A(α) = 〈a1(α), a2(α), a3(α)〉 to n = 3 basic questions Q(α) = 〈q1(α),q2(α),q3(α)〉, such that q1: Is α asserted?, q2: Is α negated?, and q3: Is α nonassertible?. There are m = 2 kinds of exclusive answers ai(α) 7→ {0, 1} to each ordered question qi, where 0 is a denial no and 1 is an armation yes. This yields the following list of mn = 23 = 8 predications and their counterparts in a Belnap-typed set 8: would be. For it remains open to us to discover some hidden, unsuspected determinants that would force us to withdraw our assent to it. ([12], p. 136) 8A judgment proceeds as a statement in which a semantic value is predicated of the sentence. Gokhale claims for this higher-order level of discourse: A sy	at-statement, in so far as it is a statement about a sense of a sentence, is a metalinguistic statement and not an object-linguistic one. ([7], p. 80). 9Jain denial corresponds to the relational negation of the realists (paryud	asa pratis.edha), by contrast to the M	adhyamika non-relational negation (prasajya pratisedha). Accordingly, the denial of the second m	ulabha ngi (2) amounts to an act of negative assertion or falsity-claim and stands for a commitment of the speaker about how the world is not, whereas every disciple of M	adhyamaka typically endorses an attitude of non-committment. 9 (1) = 〈1, 0, 0〉 for {T} (2) = 〈0, 1, 0〉 for {F} (3) = 〈1, 1, 0〉 for {{T},{F}} (4) = 〈0, 0, 1〉 for {#} (5) = 〈1, 0, 1〉 for {{T},{#}} (6) = 〈0, 1, 1〉 for {{F},{#}} (7) = 〈1, 1, 1〉 for {{T},{F},{#}} (8) = 〈0, 0, 0〉 for ∅ Each of the seven Jaina statements is an expression of single yes-answers (ai = 1) among three possible ones, while the remaining no-answers (ai = 0) are left silent by the armative nature of Jaina philosophy. The rst two statements (1) and (2) mean that every standpoint is such that it makes a given sentence true or false, respectively. (3) means that there are standpoints for asserting the truth and the falsity of the sentence, while noting that a standpoint does not make this sentence both true and false at once. The internal consistency of the standpoints is stated in terms of successive assertion and denial. (4) is the troublesome statement that the sentence is non-assertible: although this semantic predicate seems to entail merely that a given sentence cannot be asserted (made true), this should leave place for strong denial (falsity-claim); but such a translation would collapse (4) into (2), all the more that this third m	ulabha ngi is translated as a case of simultaneous assertion and denial. How can one and the same sentence be non-assertible and asserted at once? We return to this point in the next paragraph. The three remaining predications are combinations of the four preceding ones: (5) and (6) mean that there are standpoints that make the sentence true and non-assertible, or false and non-assertible. (7) is a combination of the three basic predications such that the available standpoints make the sentence true, false, and non-assertible. The ultimate subset (8) doesn't appear in the list of the Jaina predications, however; hence the odd number of 8−1 = 7 elements. A combinatorial account for this odd number of predications can be given as follows: there is an innite number of particular arguments for any predication, and all of these are classied among a set of seven general standpoints in the nayav	ada. Now since any two dierent kinds of standpoints may result in one and the same statement of the sy	adv	ada, it follows from it that every sentence is made (or claimed to be) either true, false or non-assertible by a variable set of related standpoints. Therefore, there is always at least one standpoint ai(α) = 1 for any sentence α. This entails that no sentence a can be an exception to these three basic judgments 〈a1(α), a2(α), a3(α)〉, and the answer A(α) = 〈0, 0, 0〉 is made an impossible case. As rightly noted by Priest10, no contemporary counterpart has been de10What are the semantic values of such compound sentences? Such a question is not one that Jaina logicians thought to ask themselves, as far as I know. So we are on our own here. ([15], p. 268). 10 vised for the so-called Jaina logic: the Jains have not dened any closed formal language with a set of constants (connectives) and a closed set of consequences. However, we can develop a plausible Jain logic within QAS. DEFINITION 3. Jain logic is a model J7 = 〈M, A〉 upon a sentential language L and its set of logical connectives © = {∼,∧,∨,→}. It includes a logical matrix M = 〈Q;7;D〉, with: a function Q(α) = 〈q1(α),q2(α),q3(α)〉; a set 7 of logical values; a subset of designated values D ⊆ 7. The cardinality of D and the dierent matrices for© cannot be uniquely determined without solving an intermediary problem: the meaning of the non-assertible avaktavya in q3, by contrast to the two assertible vaktavya (asti, nasti) that constitute expressible predications in q1 and q2. Each ordered answer is a logical value from our many-valued perspective, and the meaning of the semantic predicate non-assertible is crucial to determine whether a positive answer to q3(α) results in a designated or non-designated value11. For if A(α) = (4) = 〈0, 0, 1〉, then a1(α) = a2(α) = 0 and a3(α) = 1. Assuming with Priest that a semantic value is designated if it expresses truth, then a non-assertible sentence should be asserted to be at least true in order to be designated. Is it so? There are three main interpretations of avaktavya: (4.1) neither true nor false, (4.2) both true and false, (4.3) none (taking to be granted that not two of these can be accepted without extending the set of semantic predicates from 8−1 to 16−1 = 15 elements)12. Given the crucial role of the number 7, only one of these three possibilities is to be accepted as the third m	ulabha ngi. 11An alternative way consists in characterizing logical consequence in terms of an ordering relation between the elements of V , such that p |=J7 q if and only if A(p) ≤ A(q). See [3],[22] about this process. An algebraic presentation for Jain logic is also given in [20],[22] and results in a bi-anda-half-lattice (a product of two Belnap's bi-lattices) with no lower bound ∅ (〈0, 0, 0〉, in J7). But given that nothing seems to justify a specic hierarchy between the seven logical values, we stick to the view of logical consequence as preserving the designated value. 12Priest mentions the possibility of four-valued facets or m	ulabha ngi and a subsequent 15-valued logic in [15], in such a way that a sentence could be said to be either asserted or denied, or both, or neither. Some other extensions of the basic predications have been entertained in [2] for Jain logic, assuming it to be a positive counterpart of the catus.kot.i ; these yield an extension from 4to 8and 12-valued logics, where a given standpoint is more asserted (or not) than another. But such a probabilistic extension misleadingly takes the doctrine of relative truth for a logic of partial truth-values. Gokhale argues against this reading, because nayav	ada, as has generally been held, gives us a class of `partial truths', whereas sy	adv	ada gives us a class of whole truths (or the whole truth). ([7], p. 74). In other words, each sentence is plainly true (or not) from each given standpoint. 11 The rst interpretation is defended by [6], [7], and [9]; the second is urged by [4], [12], and [15]. [15] and [18] admit both interpretations, while the third interpretation is supported by [2] and [23]. Those who advocate (4.1) usually claim that the Jains always sustained internal consistency or non contradiction as an unquestionable meta-principle (paribh	as	a); this amounts to reject any case of simultaneous assertion and denial from the same standpoint. Ganeri advanced in [6] a reductio argument against the inconsistent interpretation, to the eect that admitting a simultaneous assertion and denial would reduce the logical values (5) and (6) to (4). This collapsing argument is rejected in [18]], insofar as it omits to take the dierence between the standpoints a1 and a2 into account 13. As a further argument for (4.3), Tripathi claimed that the incomplete interpretation (4.1) cannot square with the armative basis of the Jaina predications14. The latter means that any sentence can be made true from at least one standpoint, so that no sentence can be said to be neither true nor false. Assuming that armative basis essentially refers to an act of assertion (the second predication is a negative assertion), this implies that every Jaina predication asserts something about a sentence and cannot amount to a pure denial without assertive counterpart15. Conversely, Priest quotes some sources in support of (4.2) and takes them 13Ganeri's argument (see [6], p. 272) proceeds as follows: if avaktavyam means (4.2): {T,F}, then the fth and sixth predicates yield (5.2): {T,{T,F}} and (6.2): {F,{T,F}}, respectively; now (5.2) and (6.2) are logically equivalent with {T,F}, given the trivially twofold occurrence of T and F. Hence the adoption of (4.2) entails that (5) and (6) conate into (4), and the sevenfold predication is done. Ganeri's mistake is due to his set-theoretical equation between sets and subsets of elements in V : this argument seems to rely upon a conation of two distinct standpoints: to state that p is asserted from one standpoint and both asserted and denied from another standpoint doesn't entail that p is merely asserted and denied, unless the crucial sy	ad is suddenly removed from the meaning of a statement. But it could not be so, and Ganeri unduly commits the following simplication: p ∧ (p ∧ ∼p) = (p ∧ ∼p). ([18], pp. 63-4) 14To say that a thing neither exists (asti) nor does not exist (n	asti) is sheer skepticism, and the Jaina would never accept it as a bha nga (predicate), and as one of the m	ulabha ngas (primary predicates) at that. (. . .) What is worse, the interpretation of the avaktavya as neither would make it indistinguishable from the fourth kot.i (alternative viewpoint) of the M	adhyamika catus.kot.i, as also from the anirvanacan	ya (indescribable as either being or not-being) of the Ved	anta. ([21], pp. 187-8). The argument is unconvincing, however, given that the M	adhyamikas deny the neither . . . norposition and don't arm it (see Section 5); no confusion should arise from (4.1), accordingly. 15It could be objected to the view of a pure denial that any rst-order denial implicitly contains a second-order assertion. Such an objection suggests that (4.3) includes a secondorder armative basis (something like arguably, I assert that I don't assert anything about p); see Section 6 about this. 12 to mean a plausible admission of internal inconsistency16. The present paper does not purport to have the nal word, but to note two main properties of J7 that are established in [18] 17. On the one hand, the essential occurrence of standpoints gives rise to a quasi-value-functional set of logical matrices for J7 where the logical value of a complex sentence is partly determined by the value of its components18. On the other hand, the incomplete or inconsistent interpretation of avaktavya makes J7 quasi-equivalent to two famous manyvalued systems: Kleene's 3-valued logic K3 or Priest's 3-valued Logic of Paradox LP, respectively. This can be stated by the two following theorems: THEOREM 1. J7 is a paranormal logic that is either paraconsistent or paracomplete. That is: for some sentences α, β of L , either α,∼α 6|= β or 6|= α does not entail |=∼α. J7 is paracomplete and quasi-equivalent withK3 if and only if (4) is interpreted incompletely, and J7 is paraconsistent is quasiequivalent with Priest's 3-valued logic LP if and only if (4) is interpreted inconsistently. THEOREM 2. The matrices for the connectives © of J7 are invariant, irrespective of the interpretation of (4). For every connective • ∈©, A(α • β)icm = A(α•β)ics for every value of α and β including the incomplete (icm) or inconsistent (ics) reading of #. Apart from these technical results, it remains that no denite interpre16Priest adduces his usual argument for dialetheism, according to which some (but not every) contradictions are true: What should seem to be meant by two things being contradictory here is that they cannot obtain together. If [(4)] is both true and false, then [p] and [∼p] are precisely not contradictories in this sense. ([14], pp. 271-2). Does this mean that a dierence should be made between possibly true and impossibly true contradictions? A plea for possibly true contradictions has been made in [16], arguing that (4.1) could mean that some standpoint aords an evidence both for and against the truth of p. But the latter explanation does not seem to match with the denite value of a sentence in each standpoint, according to Gokhale (see note 12 above). This is why the third interpretation (4.3) will be favored in the following. 17A quantied epistemic interpretation of the standpoints has been suggested in [17]: each standpoint stands for a single belief within a community of agents, so that each Jain statement about α is translated as ∃xBx(α) and reminds us of Ja±kowski's discussive logic D2. Such a translation helps to explain the paraconsistent behavior of the Jains: a set of inconsistent standpoints does not entail the truth of everything. Nevertheless, it doesn't account for Jain realism (see note 6 above): a standpoint is not the mere epistemic expression of a belief or opinion, but the genuinely ontological expression of a facet of reality. 18Quasi-truth-functionality is due to the relative truth of standpoints. Two any sentences α and ψ can be true from two dierent standpoints; but there may be no standpoint from which α and ψ should obtain at once, according to the existential translation of a standpoint in [18]: v(∃xBx(α)) = T and v(∃xBx(ψ)) = T don't entail v(∃xBx(α ∧ ψ)) = T, but v(∃xBx(α ∧ ψ)) = T or F. On the origins of quasi-truth-functionality, see [17]. 13 tation of avaktavya occurs in the literature and thus leaves the Jaina set of logical consequences indeterminate. The next point is to see whether a meaningful interpretation can be given to the third interpretation (4.3): what can be meant by avaktavya, if it is neither both asserted and denied nor neither asserted nor denied? For even though such an alternative reading prevents Jaina logic from reducing to what Matilal called a mere facile relativism19, a formal approach hardly makes obvious any statement beyond being either true, or false, or both true and false, or neither true nor false. For one thing, Bahm takes it (in [2]) to mean something like an incomplete thought: a sentence is non-assertible whenever no property P can be completely predicated of S. But this is the essential feature of anek	antav	ada, the partial truth for every standpoint of the Jaina nayav	ada: the cornerstone of their pluralist metaphysics is that reality is an indenite collection of incomplete perspectives. Assertion and denial are not categorical or one-sided speech-acts, therefore, and the essential incompleteness of any sy	ad is likely to undermine Bahm's explanation. A more insightful reading seems to emerge in [23], where non-assertibility is synonymous with non-distinction: a sentence is non-assertible whenever its object S cannot be said to be properly P or not P. The dierence is thus made with the interpretation (4.2), in the sense that S is said to be both P and not-P by including both opposite properties from one contradictory standpoint. But again, Tripathi claims in [23] that the Jains fully subscribed to the law of non-contradiction and would have refused any self -contradictory statement20. A plausible account of being indistinguishable refers to the Hegelian view of an internal or inclusive contradiction without exclusive opposition between its terms. In support of this awkward view of contradiction, it is worthwhile to note that most of the Jaina or M	adhyamika sentences are about such metaphysical subjects as atman, Brahman and their being existent. One may be hesitant about the logical form of an expression like 	atman 19It also amounts to a view which announces that all predicates are relative to a point of view; no predicates can be absolutely true of a thing of a thing or an object in the sense that it can be applied unconditionally at all times under any circumstances. Jainas in this way becomes identied with a sort of facile relativism. ([12], p. 133). Again, the crucial role of standpoints clearly points out that the Jain logic is not a real challenge to PNC. 20No system of philosophy can aord to accept self-contradiction as valid, because if self-contradiction is accepted as valid without any qualications, then there remains no weapon for criticism, anything which is said will have to be accepted, because even self-contradictories is valid. It is certain that the Jaina does not take leave of logic and consistency; he does criticize others by pointing out self-contradiction. Every system of philosophy has its contradictory which is regarded as false. This is why when a system has to accept a synthesis of contradictories as valid, it has to invent one device or another which at least seems to take o the edge from the contradictories. ([21], p. 188). 14 is self-existent, where existence occurs as a predicate; but a more charitable reading would be to the eect that the subject-term S is elliptically said to exist or to be as falling under a certain property P. Consequently, avaktavya might mean that S is not any more P that non-P. But which sort of S could be so indistinguishable as not only to cover both P and all its complementary properties, but also to cancel any distinction between these properties? Tripathi mentions as a non-expressible sentence that which can be thought but cannot be expressed (for want of a distinguishable set of properties)21. Such a subject should be kept silent, according to the Wittgensteinian stance that the limits of language are the limits of thought. (But our former reference to Hegel should give rise to a non-Wittgensteinian relationship between language and the world.) While noting that Hegel's philosophy supported a transcendental idealism and clearly diers from the Jaina realism, a common point between Jainism and the Buddhist trend of M	adhyamikas seems to be their common rejection of logical atomism: reality is not a whole whose parts would be objects and their properties, or at least not for some extra-natural entities that transcend the empirical level of illusory data (pr	atibh	asika). This plausible account of (4.3) will be pursued in the next section, because it might make sense of N	ag	arjuna's radical skepticism. To conclude our discussion of Jaina logic, Priest uses in [15] an analogy with the cube to make sense of complete truth: every facet of reality is a side of a cube, and reality is the collection of every such facet. But Jaina cubism is such that the indenite number of facets turns the cube into a polygon even more complex than Descartes' chiliagon. Just as Picasso wanted to catch a conceptual reality by pooling dierent perspectives of a character together in one and the same prole, the Jaina philosophy relies upon a plurality of standpoints to grasp the essence of reality. A logical translation of this view is given in [4]): plain truth amounts to a complete knowledge (prama	na) whose expression in a complete judgment consists in the addition of the seven sorts of predication. Is this a right way to describe the transition from partial to complete truth22? 21Bahm's account must be distinguished from Meinong's famous example of a round square, which has frequently been mentioned as a case of impossible object and a challenge to PNC. A round square is an object that can be expressed (described) but cannot be thought (imagined, or conceived mentally). To the contrary, the third interpretation of avaktavya refers to something that can be thought but cannot be expressed. Is there such a subject S that can fulll this requirement? A Wittgensteinian reader would answer negatively to this question, assuming that whereof one cannot speak, thereof one must be silent. 22The following denition of plain truth is given in [4]: An object X can be viewed from any one of the seven standpoints. However, since the totality of all these seven possibilities comprises the pram	an. a-saptabha ng	 (complete judgment of the phenomenal world in terms 15 An alternative account would be to state that a subject is completely described when absolutely every particular standpoint is listed, rather than just the seven kinds of argument from the nayav	ada. Such an exhaustive completion is impossible, given the innite sort of standpoints that constitute the proper description of any object. A natural translation of (4) within J7 might be taken to be the twofold answer yes and no to the third basic question: a3(α) = {1, 0}. But it is not so, given that this third question is positively answered if the corresponding sentence is inexpressible. No yes-no answer occurs in the Jaina questionanswer game, consequently: two dierent questions can result in the same answer or not, but no single question can be answered oppositely by yes and no at once23. This is the gist of self-contradiction, and even the third basic predicate of inexpressibility does not state it because non-distinction does not mean an internal coexistence of opposite properties. These cannot coexist, by denition. Whatever the nal word may be about (4), we argue two things about complete truth: it does not mean for a given sentence either to be assigned a designated value (this is partial truth) or to be uniquely asserted and, therefore, be given the logical value (1) in J7 24; partial truth is a sucient of seven possibilities), the disjunction, denoted by ∧, of these seven predications should lead to a tautology. ([4], p. 186). In algebraic terms, the Jains would thus assimilate onesided truth with logical tautology and dene the latter as the union of the seven elements of V . That is: > = ((1)∪(2)∪(3)∪(4)∪(5)∪(6)∪(7)). This denition of tautology clearly diers from that of Priest's in [15] or J7 in [18]: a sentence is a tautology if it is designated from every standpoint. But this is a denition of tautology in the conventional sense of truth, by contrast to the aforementioned absolute sense of truth that uniquely leads to a pram	an. a. One could wonder another thing, with respect to this denition of one-sided tautology: does it correspond to the union of the seven kinds of standpoints or, rather, should it collect the indenitely many particular standpoints that are included in each of these seven kinds? 23Three levels of inconsistency can be graded within the framework of QAS: light inconsistency, or inconsistency from two dierent standpoints: {{T},{F}}, i.e. ai(α) = aj(∼α) = 1 (where i 6= j); mild inconsistency, or inconsistency from one and the same standpoint: {{T,F}}, i.e. ai(α) = ai(∼α) = 1; and strong inconsistency, or inconsistency in one and the same answer: {{T,∼T}}, i.e. ai(α) = ai(α) = {1,0}. The Jain anek	antav	ada embodies a logic of light inconsistency; Priest's Logic of Paradox LP argues for a mild inconsistency that corresponds to the inconsistent interpretation (4.1) of avaktavyam; but no counterpart seems to occur for the strong inconsistency of self-contradiction, going beyond the so-called impossible values of [20]. Indeed, strong inconsistency consists of non-empty subsets including an element and its complement. Such a case is impossible even in a combinatorial approach of semantic values, insofar as Priest's value {T,F} assumes that T and F are not complementary to each other. 24Returning to the comparison with Ja±kowski's Discussive logic D2, the Polish logician rendered each standpoint by the modality of possibility, ♦. Accordingly, any sentence α 16 condition of truth-assignment for the Jains, while the skeptic M	adhyamikas take complete truth to be a necessary condition for truth-assignment. Let us now consider this skeptic logic within a question-answer game of QAS. 5 N	ag	arjuna's Principle of Four-Fold Negation N	ag	arjuna's radical skepticism is summarized in hisM	ulamadhyamaka-k	arik	a, where the rst verse includes four sentences (or lemmas) that are equally denied by means of stances (dr.s. t.is, or kot.i) and result in the the so-called Principle of Four-Cornered Negation (thereafter: 4CN) or Tetralemma (catus.kot.i). Thus: (a) Does a thing or being come out itself? No. (b) Does a thing or being come out the other? No. (c) Does it come out of both itself and the other? No. (d) Does it come out of neither? No. How can N	ag	arjuna consistently deny all the four questions at once? While noting that their content refers to the M	adhyamika's doctrine of emptiness (s	unyav	ada), a problem arises about the meaning of negation in the four aforementioned answers. A tentative formalization of (a)-(d) yields the following, where a is a predication of the form S is P (with S for thing and P for coming out iself) and ∼ is classical negation: (a′) Not (S is P) = ∼(α) (b′) Not (S is not P) = ∼(∼α) (c′) Not (S is P and S is not P) = ∼(α ∧ ∼α) (d′) Not (neither S is P nor S is not P) = ∼(∼(α ∨ ∼α)) Assuming that negation is the relational paryud	asa pratis.edha, the set of four negative statements is clearly inconsistent: (b′) is equivalent with the armation α (by double negation), and this is patently contradictory with its negation in (a′). Even more than for the Jains, it is commonly acknowledged that the M	adhyamikas unexceptionably subscribed to PNC and cannot then accept both (a′) and (b′). Furthermore, (d′) occurs as a denial of the denial of the Principle of Excluded Middle (PEM), according to which every sentence or its negation is true. But it clearly appears that the double denial arising that is uniquely asserted (such that v(α) = (1)) is logically necessary because it is cannot be but asserted, and it is not possible for it to be denied or taken to be non-assertible. Thus v(α) = (1) means the same as α. This modal interpretation squares with the idea of one-sidedness; however, the Jain view of pram	an. a still goes beyond such a logical necessity (see note 22 above). 17 in (d′) does not amount to an armation of PEM, since (a′) and (b′) already reject the armation of both α and ∼α. A way to avoid the contradiction (a′)-(b′) has been urged by Horn (in [10]), who claimed that the negation of every sentential content should be rendered as a predicate-term negation rather than a predicate negation25. The distinction between predicate-term and predicate negation cannot be expressed in a modern or Fregean logic, where predicate-terms and predicates are collapsed into a unique function. By using term logic, (b′) should be read as S is not-P, the contrary opposite of (a′). The conjunction (a′)- (b′) results in a stronger relation of incompatibles, and Horn is right to say that two contraries can be consistently negated without entailing any selfcontradiction. In this respect, an application of intuitionistic negation (¬α for S is not-P) should ll the bill and be preferred to the classical negation (∼α for S is not P): ∼(∼α) becomes ∼(¬α), and the latter cannot be reduced to a by the law of double negation. Does this mean that intuitionistic logic should be seen as a proper logic for 4CN? It is not, given that the last statement (d′) leads to another contradiction. For since one of de Morgan's laws states that (∼α ∧ ∼(¬α)) is equivalent to ∼(α ∨ ¬α)), how to claim with (a′)-(b′) that S is neither P nor not-P: ∼(α ∨ ¬α)) while denying it at the same time with (d′): ∼(∼(α ∨ ¬α))↔ (α ∨ ¬α) ? The whole result turns 4CN into a case for radical skepticism: not only does the speaker N	ag	arjuna ignore whether S is P or not, but he goes on denying that he does ignore it. This troublesome stance has been noted by Raju26 and accounts for the dierence between Buddhism and nihilism, as currently urged by a number of commentators: nihilism is the armation that nothing is real or can be known to be so; whereas Buddhism argues for 25Horn claims that crucially, no distinction between contradictory and contrary negation was regularly made within classical Indian logic. ([10], p. 80) However, the contrary or contradictory feature of a negation crucially depends upon the nature of the subject in a predication: are the subjects of a Jain predication sometimes universal, sometimes particular? No denite answer seems to be available to disentangle the meaning of 4CN; it is only the later school of Navya-Ny	aya that will deal with such equivocation cases. See in this respect J. Ganeri: Towards a formal regimentation of the Navya-Ny	aya technical language (parts I,II), in Logic, Navya-Ny	aya and Applications (Homage to Bimal Krishna Matilal), M.K. Chakraborti and Löwe, B. and Mitra M.N. and Sarukkai S (eds.), College Publications, London, 2008, pp. 105-121. 26The alleged founder of 4CN, Sañjaya (' 6th century B.C.), would have inuenced the Greek philosopher Pyrrho in his radical skepticism; Raju states this point by claiming that Pyrrho maintained that `I am not only not certain of the knowledge of any object, but also not certain that I am not certain of such a knowledge'  ([16], p. 695). It is worthwhile to note that the Greek principle of indierence ou mallon (not any more than) strikingly parallels 4CN. 18 a mere denial without any positive counterpart. The positive basis of each Jaina statement included a case of negative assertion, as witnessed by the predication (2); but no such assertion arises in 4CN, where negation is pure denial. Before answering to whether there can be a negation without any positive counterpart, we suspect the core diculty with 4CN to lie in the meaning of its wide scope negation (the answer No): it is used to produce a denial, and this no-answer should nd a proper treatment within the formal framework of QAS. Unlike the Jaina statements, and following the connection established between M	adhyamika skepticism and anti-realism, we assume that each kot.i deals with the impossibility of knowledge: the human failure to catch any absolute truth (param	arthasatya) about reality is a sucient reason to deny any justiable belief and thus any truth-assignment, according to N	ag	arjuna's s	unyav	ada. If so, we introduce a four-valued logic of acceptance and rejection for 4CN. DEFINITION 4. A logic of acceptance and rejection is a model AR4 = 〈M,A〉 upon a sentential language L and its set of logical connectives© = {∼,∧,∨,→}. It includes a logical matrix M = 〈Q; 4;D〉, with : a function Q(α) = 〈q1(α),q2(α)〉; a set 4 of logical values; a subset of designated values D ⊆ 4, where D = {〈1, 0〉, 〈1, 1〉}. Q(α) is an ordered set of n = 2 questions about the sentence α, with q1: is a justiably be true? and q2: is a justiably false? 27, and n = 2 sorts of answers such that a(α) 7→ {0, 1}. It results in a set V of mn = 22 = 4 logical values, each standing for an explicit belief-attitude in 4 = {〈1, 0〉, 〈1, 1〉, 〈0, 0〉, 〈0, 1〉}. The dierence with J7 is that no third question q3 occurs here: avaktavya is not a M	adhyamika concept, so that only two basic semantic predicates or muladr.s. t.is are required in 4CN. At the same time, AR4 is a general logic of statements that could include the Jaina stances as well: the Jaina value 〈a1(α), a2(α), a3(α)〉 can be equated with the value Q(α) = 〈q1(α),q2(α) of AR4 by canceling the third bha	nga a3(α). Then 〈1, 0〉 = {〈1, 0, 1〉, 〈1, 0, 0〉}, 〈1, 1〉 = {〈1, 1, 1〉, 〈1, 1, 0〉}, and 〈0, 1〉 = {〈0, 1, 1〉, 〈0, 1, 0〉}. A relevant exception concerns the third value 〈0, 0〉 = {〈0, 0, 1〉, 〈0, 0, 0〉}, which includes the eighth forbidden value 〈0, 0, 0〉 in J7. This forbidden value is our key to a better understanding of N	ag	arjuna's four stances, with the following denition of negation and its distinction with the speech-act of denial. 27The second question Is α justiably false? is equivalent with Is ∼α justiably true?. This results in the following equation for negation in AR4: a1(∼α) = a2(α), and conversely. 19 DEFINITION 5. For every sentence α such that A(α) = 〈a1(α), a2(α)〉: A(∼α) = 〈a2(α), a1(α)〉. The import of QAS is to bring an algebraic distinction between logical negation and denial: contrary to the usual perplexing presentation of 4CN, denial should not be rendered as a connective that is part of the sentential content α; rather, a denial is a no-answer that does not stand for a function but its resulting value. Correspondingly, a proper formalization of 4CN is suggested in the following style: (a′′) a1(α) = 0 (b′′) a1(∼α) = 0 (c′′) a1(α ∧ ∼α) = 0 (d′′) a1(∼((α ∨ ∼α)) = 0 Only one valuation of AR4 accounts for the consistency of (a ′′)-(d′′), namely: A(α) = 〈0, 0〉, the forbidden value of Jaina logic. Following the denition of conjunction and disjunction in AR4 28, (a′′) and (b′′) entail that a2(α) = a1(α∧ ∼α) = a1(∼(α∨ ∼α)) = a2(α ∨ ∼α) = 0. Once again, the usual perplexity caused by N	ag	arjuna's stance is due to a confusion between the relational and non-relational reading of negation. The former negation (paryud	asa pratis.edha) is not an answer about whether the sentence α is true or false, given that it occurs within its sentential content in the whole expression ∼α; most importantly, it assumes bivalence and entails that ∼α is false whenever α is true (and conversely). Therefore, no sentence can be given a gappy value (neither true nor false) with such a relational use of negation. Furthermore, introducing the intuitionistic negation ¬ for this purpose is not the solution either: that α is said to be neither true nor false cannot explain again why this gappy solution is insucient to account for the fourth stance (d'). This leads to the conclusion that N	ag	arjuna's denial should be strictly distinguished from assertive negation and be equated with the absolutely no-answer 〈0, 0〉. Our point about logical values actually holds for every negation, in the sense that there is no functional dierence between classical and intuitionistic negation AR4. For the dierence between the two negations does not lie in the denition of their mapping from L to V but, rather, in the domain of values they range over. Given that classical negation assumes a one-one correspondence theory of truth, this entails that a sentence cannot be said 28A complete description of the semantics for AR4 is not required in the context of 4CN, but it includes maximal and minimal functions (max,min) upon the values of V , given a total ordering function < between these elements proceeds as follows: 〈0, 1〉 < 〈0, 0〉 < 〈1, 1〉 < 〈1, 0〉. Hence the following denition of the connectives of conjunction and disjunction: v(α ∧ ψ) = min(α,ψ), and v(α ∨ ψ) = max (α,ψ). 20 to be either both true and false or neither true nor false; hence a restriction of the range from V = 4 to V = 2 = {〈1, 0〉, 〈0, 1〉}. As to the intuitionistic theory of truth as justiable truth, no sentence can be said to be true unless the justication is denite and this stringent view of justication implies another restriction from V = 4 to V = 3 = {〈1, 0〉, 〈0, 0〉, 〈0, 1〉}. The Jaina case embodies a paraconsistent variant, where a sentence can be said to be both true and false but excludes the possibility that it be none; hence a corresponding restriction from V = 4 to V = 3 = {〈1, 0〉, 〈1, 1〉, 〈0, 1〉}. The relative truth of nayav	ada also accounted for the combination of such basic answers into new logical values in J7, unlike the non-relative, absolute or one-sided view of truth in the M	adhyamika school. But that is not the whole story of 4CN. Recalling a former quotation by Raju, two problems remain to be solved. Firstly: does N	ag	arjuna deny absolutely everything, including his own denials? And secondly: is the catus.kot.i a mere reversal of the saptabha ng	, i.e. the transformation of a common set of positive statements into negative statements? 6 Two contrary logics? Let us note about the rst question that a distinction can be made between two generic forms of skepticism, a moderate and a radical one. The former is closer to what the Buddhists meant by nihilism and wanted to be strictly distinguished from; it means that nothing can be known about reality, but one least thing to be known is precisely that nothing mundane can be known. In contrast to this, the radical version goes on denying any denial about our knowledge about reality: ignorance is not asserted but doubted itself. Whether or not such a distinction relates to the Greek schools of the New Academy (Arcesilas, Carneades) and Pyrrhonism (Pyrrho, Timon of Phlius) does not really matter in what follows. Rather, the point is whether N	ag	arjuna endorsed radical skepticism and what his rejection consisted in. In the light of QAS, the complete denial of 4CN means that only no-answers are given to preceding questions. As to the second question, Bahm replies in [2] that the two Indian logics cannot merely seen as mutual contraries: Jaina logic cannot be reduced to a Principle of Four-Cornered Armation. QAS already brought this point out by the cardinality of the sets of logical values, given the essential occurrence of a third question (about avaktavya) in J7. Nevertheless, there is a reason to claim that these philosophical schools are really opposite to each other in some respect. The catus.kot.i can be taken to be a reversal of saptabha ng	 only if the sentential content of a denial or an armation is of the rst order, 21 i.e. stands for a declarative sentence about reality; but the same cannot be safely said for higher-order questions about the answerer's attitudes29. Let us exemplify this symmetrical behavior by means of two Socratic dialogues, where an initial question about the atomic sentence p is accompanied with a sequence of oratory questions (the questioner expects to have a given answer) and answers. The answerer to a common questioner (the doctrinalist Aristotle) is a Jaina speaker (V	adiveda S	uri) and a M	adhyamika speaker (N	ag	arjuna), respectively. It clearly appears that the resulting dialogues are radically opposed to each other, and we bring this out by formalizing them in terms of QAS. DIALOGUE 1: ARISTOTLE VS. V	ADIVEDA S	URI 1. Q: Do you accept p? [a1(p) = 1?] 2. A: Yes, I accept p. [a1(p) = 1] 3. Q: Therefore you reject ∼p? [a2(p) = 0 ?] 4. A: No, I do not reject ∼p. [a2(p) 6= 0] 5. Q: Does it mean that you also accept ∼p? [a2(p) = 1 ?] 6. A: Yes, I also accept ∼p. [a2(p) = 1] 7. Q: Therefore you accept p and ∼p? [a1(p ∧ ∼p) = 1 ?] 8. A: Yes, I accept both. [a1(p ∧ ∼p) = 1] 9. Q: Therefore you reject ∼(p ∧ ∼p)? [a2(p ∧ ∼p) = 0 ?] 10. A: No, I don't reject ∼(p ∧ ∼p). [a2(p ∧ ∼p) 6= 0] 11. Q: Does it mean that you also accept ∼(p ∧ ∼p)? [a2(p ∧ ∼p) = 1 ?] 29The order of attitudes and their statements can be reformulated in terms of iterated modalities: the statement α is an armation and correlated belief about α, B(α); the statement I arm that α is an armation and correlated belief about the armation and correlated belief about α, B(Bα); and so on for any n-ordered statement as a sequence of n beliefs: Bn(α). The dierence between AR4 and modal logic is that iterated attitudes are not rendered as modal operators but as logical values in the former semantics. See note 31 below. 22 12. A: Yes, I also accept ∼(p ∧ ∼p). [a2(p ∧ ∼p) = 1] 13. Q: Therefore you reject ∼((p ∧ ∼p) ∧ ∼(p ∧ ∼p))? [a2(((p ∧ ∼p) ∧ ∼(p ∧ ∼p))) = 0 ?] 14. A: No, I don't reject ∼((p ∧ ∼p) ∧ ∼(p ∧ ∼p)). [a2(((p ∧ ∼p) ∧ ∼(p ∧ ∼p))) 6= 0] 15. Q: Therefore you also accept ∼((p ∧ ∼p) ∧ ∼(p ∧ ∼p))? [a1(∼((p ∧ ∼p) ∧ ∼(p ∧ ∼p))) = 1?] 16. A: Yes, I also accept ∼((p ∧ ∼p) ∧ ∼(p ∧ ∼p)) [a1(∼((p ∧ ∼p) ∧ ∼(p ∧ ∼p))) = 1] . . . It emerges from this abortive maieutic that the doctrinalist questioner fails to make the answerer his own reason: the whole answers are perfectly rational albeit inconsistent, in the light of AR4 and its non-classical logical values that are exclusively positive or negative30. THEOREM 3. For every sentence α (including p, ∼p, p ∧ ∼p, ∼(p ∧ ∼p), and so on), the answer of the Jaina in AR4 is A(α) = 〈1, 1〉. Proof : Let us assume that a1(p ∧ ∼p) = 1; then a1(p) = a1(∼p) = a2(p) = 1. And if a1(∼(p ∧ ∼p)) = 1 then a2(p ∧ ∼p) = 1, i.e. a2(p) = 1 or a1(∼p) = 1. Hence for every α, a1(α) = a2(α) = 1. Hence A(α) = 〈a1(α), a2(α)〉 = 〈1, 1〉. Let us now apply the same process to a dual dialogue between the dogmatist questioner Aristotle and his skeptic answerer. This yields the exact reversal of the preceding dialogue, given that each question about whether a given sentence is accepted becomes a question about whether it is rejected. DIALOGUE 2: ARISTOTLE VS. N	AG	ARJUNA 1. Q: Do you reject p? [a1(p) = 0?] 2. A: Yes, I reject p. 30The semantics for AR4 can be said to be bivalent in this respect: for every answer given to question qi about the sentence α, the corresponding answer is either positive (ai(α) = 1) or negative (ai(α) = 0). Tertium non datur. Concerning any positive and negative answer to one and the same question, it has been argued earlier (see note 23) that it is equally impossible in the pluralist approach of the Jains. Hence the ensuing dierence between two grades of inconsistency in AR4: a given answer A(α) is externally inconsistent if and only if a1(α) 6= a2(α); it is internally inconsistent or incoherent if and only if, for any answer x in {0,1}, ai(α) = x and ai(α) 6= x. Accordingly, there is a crucial dierence between sentential inconsistency and non-sentential inconsistency (incoherence): two sentences α and ∼α can be mutually inconsistent while the answers A(α) and A(∼α) about them are internally consistent (coherent). 23 [a1(p) = 0] 3. Q: Therefore you accept ∼p? [a2(p) = 1 ?] 4. A: No, I do not accept ∼p. [a2(p) 6= 1] 5. Q: Does it mean that you also reject ∼p? [a2(p) = 0 ?] 6. A: Yes, I also reject ∼p. [a2(p) = 0] 7. Q: Does it mean that you reject both p and ∼p? [a1(p ∨ ∼p) = 0?] 8. Yes, I reject both p and ∼p. [a1(p ∨ ∼p) = 0] 9. Q: Therefore you accept ∼(p ∨ ∼p)? [a2(p ∨ ∼p) = 1 ?] 10. A: No, I do not accept ∼(p ∨ ∼p). [a2(p ∨ ∼p) 6= 1] 11. Does it mean that you reject both (p ∨ ∼p) and ∼(p ∨ ∼p)? [a1(((p ∨ ∼p) ∨ ∼(p ∨ ∼p))) = 0 ?] 12. A: Yes, I reject both (p ∨ ∼p) and ∼(p ∨ ∼p). [a1(((p ∨ ∼p) ∨ ∼(p ∨ ∼p))) = 0] 13. Q: Therefore you accept ∼((p ∨ ∼p) ∨ ∼(p ∨ ∼p))? [a2(((p ∨ ∼p) ∨ ∼(p ∨ ∼p))) = 1?] 14. A: No, I don't accept ∼((p ∨ ∼p) ∨ ∼(p ∨ ∼p)). [a2(((p ∨ ∼p) ∨ ∼(p ∨ ∼p))) 6= 1] 15. Q: Therefore you also reject ∼((p ∨ ∼p) ∨ ∼(p ∨ ∼p))? [a2(((p ∨ ∼p) ∨ ∼(p ∨ ∼p))) = 0?] 16. A: Yes, I also reject ∼((p ∨ ∼p) ∨ ∼(p ∨ ∼p)) [a2(((p ∨ ∼p) ∨ ∼(p ∨ ∼p))) = 0] . . . Again, the doctrinalist questioner failed to make the answerer his reason: the whole is rational albeit incomplete, so long as the answerer refuses to commit in the truth of any sentence. THEOREM 4. For every sentence α (including p, ∼p, p ∨ ∼p, ∼(p ∨ ∼p), and so on), the answer of the M	adhyamika in AR4 is A(α) = 〈0, 0〉. Proof : if a1(p ∨ ∼p) = 0 then a1(p) = a1(∼p) = a2(p) = 0. And if a1(∼(p ∨ ∼p)) = 0 then a2(p ∨ ∼p) = 0, i.e. a2(p) = 0 or a1(∼p) = 0. Hence for every α, a1(α) = a2(α) = 0. Hence A(α) = 〈a1(α), a2(α)〉 = 〈0, 0〉. 24 Just as the Jains refuse exclusive acts of positive assertion and contend themselves with inconsistent armations, the M	adhyamikas refuse exclusive acts of negative assertion and contend themselves with incomplete denials. A parallel can be made here with da Costa paraconsistent logics C1Cn: these are non-truth-functional systems where contradictions are variably armed or denied according to the structural complexity of the contradictory sentences (p and ∼p, in C0; (p ∧ ∼p) and ∼(p ∧ ∼p), in C1; and so on). By the same way, a set of dual paraconsistent logics C ′ 1-C ′ n can be devised for the dialectical process of 4CN and states that alternatives are variably armed or denied according to the structural complexity of the alternative sentences: (p or ∼p, in C′0; (p ∨ ∼p) or ∼(p ∨ ∼p), in C ′ 1; and so on). But the parallel stops here, because the preceding dialogues have shown that the structural complexity of a sentence does not change the attitude of the answerer. In this respect, the Jains and M	adhyamikas are likely to be considered as two contrary attitudes or judgments in the common logic of statements AR4: the former arm everything whereas the latter deny everything. Returning to a preceding objection, it remains to consider to what extent such radical speakers can be said to arm everything (doxastic eclecticism) or deny everything (doxastic nihilism) in their dialectical games31. While the concerned texts mention dialectical games about rst-order statements only, it hardly makes sense to contend that N	ag	arjuna would have denied his own denials with respect to rst-order statements. Let us make a semantic ascent and consider the second-order statement α′: I don't arm that α (is true). A no-answer to the question q1(α ′): is α′ justiably true? would mean that the answerer denies to have denied (the truth of) α, while a yes-answer would entail that he arms to have denied α (as he did). The same objection can be made to a universally armative stance in the Jains. Likewise, the Jain would hardly give an armative answer to α′ without refusing the truth to α and thereby violating his policy of non-one-sidedness32. Actually, the preceding dialogues have already made clear that the Jain did deny three times (steps 4, 10 and 14) 31N	ag	arjuna's following stance is the key to his allegedly radical skepticism: If I had a thesis, I would be wrong. But I have no thesis. Therefore there is nothing wrong with me. (To keep one away from the vain discussions, Number 29). What is the content of the thesis at hand? It is likely to be a rst-order thesis, i.e. a statement about any given state of aairs. Whether N	ag	arjuna would have also claimed to have no thesis about his own attitudes remains unclear, however. 32This leads to the reintroduction of the law of double negation in the form of an illocutionary law of double denial : the denial of ∼α needn't entail the armation of α, given that a1(α) = 0 needn't entail that a2(α) = 1 (compare with A(α) = 〈0, 0〉); on the other hand, the denial of the denial of α entails the armation of a, given that a1(α) 6= 0 does entail that a1(α) = 1. 25 while the M	adhyamika did arm ve times (steps 2, 6, 8, 12, and 16). If so, the radically opposed attitudes of the Jainas and M	adhyamikas should nd their own limits with the sort of sentences to be questioned: denying and arming are about the nature of reality, rather than about one's own mental states. Such a limit of dialectic might be what Aristotle had in mind, when he attempted to show the attitude of Heracliteus with respect to the PNC is self-defeating. But he failed to make his point with his elenctic strategy, locating the trouble in the propositions (arming α and arming not-α) rather than his opponent's propositional attitudes (arming α and not arming α). Admittedly, these Indian logics were much more concerned with metaphysical topics and soteriological ends than having the nal word in every yes-no answer game. 7 Conclusion We have proposed a reconstruction of the Jaina and M	adhyamika logics by means of a question-answer semantics. The result of such an enterprise is a rational reading of these Indian schools through modern logical glasses, including the logical tool of many-valuedness that presented skepticism and pluralism as radically opposed to each other and separated by a middle view of judgment that is Aristotle's bivalent way of doctrinalism. Many-valuedness accounts for the seven judgments of Jaina saptabha ng	, while a more general logic of attitudes displays Jaina and N	ag	arjuna's stances within a four-valued semantics that characterizes both M	adhyamika skepticism (the value 〈0, 0〉) and Jaina pluralism (the value 〈1, 1〉). Above all, the main import of QAS is to pay attention to the dialectical role of questions and answers in the Indian approach of logic: just as the Megarics emphasized the dialogical nature of philosophical investigation in contrast to the Aristotelian monological view of truth and falsity as transcendental values, we want to keep in mind that the Indian logicians introduced their statements in the form of answers to speculative questions. Jaina metaphysical pluralism also made sense of their inconsistent judgments, while the skeptic avor of N	ag	arjuna's philosophy explains his systematic denial to any question about the nature of reality. Last, but not least: one of the most intriguing case studies has concerned the meaning of avaktavya (non-assertibility), the third basic judgment of Jaina logic. This predicate should not be confused with common self-contradiction, where a sentence and its negation are said to be both true at once and in the same respect. The commentators frequently claimed that the Jainas subscribed to PNC in their various reasonings: so non-assertibility 26 refers to another, milder view of contradiction than coexistence of incompatible properties in the same subject. Rather, we support Tripathi's interpretation of avaktavya in the sense of non-distinction: the Jaina third judgment might mean that some objects (S) cannot be predicated by any property, that is, neither of one of them (P) or any of their complementaries (not-P). Rather than a plea for self-contradiction, avaktavya seems to argue for the impossibility to predicate anything of some such absolute subjects as atman or Brahman because these would stand beyond any set of denite properties. Such a tentative explanation would match with the Hegelian alternative process of Aufhebung (or sursumption), in contrast to the predicative process of subsumption that systematically describes a subject S as falling under a given set of properties P33. References [1] Aristotle. Metaphysics. [2] A.J. Bahm. "Does Seven-Fold Predication equal Four-Cornered Negation reversed?". Philosophy East and West, 7:12730, 1958. [3] N. Belnap. "A useful four-valued logic". In Dunn J.M. and Epstein G., editors, Modern Uses of Multiple-Valued Logic, pages 837. Dordrecht: D. Reidel Publishing Company, 1977. [4] F. Bharucha and R. Kamat. "Sy	adv	ada theory of Jainism in terms of deviant logic". Indian Philosophical Quarterly, 9:1817, 1984. [5] S.S. Chakravarti. "The M	adhyamika Catus.kot.i or Tetralemma". Journal of Indian Philosophy, 8:3036, 1980. [6] J. Ganeri. "Jaina logic and the philosophical basis of pluralism". History and Philosophy of Logic, 23:26781, 2002. [7] P. Gokhale. "The logical structure of Sy	adv	ada". Journal of Indian Council of Philosophical Research, 8:7381, 1991. 33Here is an alleged description of the Brahman by himself: This whole universe is lled by me in immaterial form; all beings are in me, but I am in them. Yet those born are not within me. Behold my kingly rule: my self sustains all beings, is not in them but creates them. Just as the mighty wind everlastingly occupies the space above us and moves throughout it, so do all created beings occupy me. (Bhagavad-G	t	a: Chapter 9, verses 4-7). This seems to match with our description of S as an ultimate class. 27 [8] M.H. Gorisse. "Non-one-sideness: context-sensitivity in Jain epistemological dialogues". In A Day of Indian Logic. ILLC Technical Report X-2009-04, Amsterdam, 2009. [9] M.H. Gorisse. "The art of non-asserting: dialogue with N	ag	arjuna". In R. Ramanujam and Sarukkai S., editors, Springer Lecture Notes in Articial Intelligence, volume 5378, pages 25768. FoLLI Series, Springer, 2009. [10] L. Horn. The Natural History of Negation. University of Chicago Press, 1989. [11] L. Kei. "Ultimately and conventionally : some remarks on N	ag	arjuna's logic". In A Day of Indian Logic. ILLC Technical Report X-2009-04, 2009. [12] B.K. Matilal. "The Jaina contribution to logic". In J. Ganeri and H. Tiwari, editors, The Character of Logic in India, pages 12739. State University of New Press, 1998. [13] D. Mohanta. "The use of Four-Cornered Negation and the denial of the Law of Excluded Middle in N	ag	arjuna's logic". In A. Schumann, editor, Logic in Religious Discourse, pages 4453. Ontos Verlag, Paris & Frankfurt, 2009. [14] T. Parsons. "Assertion, denial, and the Liar Paradox". Journal of Philosophical Logic, 13:13752, 1984. [15] G. Priest. "Jaina logic: a contemporary perspective". History and Philosophy of Logic, 29:26379, 2008. [16] P.T. Raju. "The Principle of Four-Cornered Negation in Indian philosophy". Review of Metaphysics, 7:694713, 1954. [17] N. Rescher. "Quasi-truth-functional systems of propositional logic". Journal of Symbolic Logic, 27:110, 1962. [18] F. Schang. "A plea for epistemic truth: Jaina logic from a many-valued perspective". In A. Schumann, editor, Logic in religious discourse. Ontos Verlag, Paris & Frankfurt, 2009. [19] J. Searle. Speech Acts. Cambridge University Press, 1969. 28 [20] Y. Shramko and H. Wansing. "Hyper-contradiction, generalized truthvalues and logics of truth and falsehood". Journal of Logic, Language and Information, 15:40324, 2006. [21] M. Siderits. "N	ag	arjuna as antirealist". Journal of Indian Philosophy, 16:31125, 1988. [22] R. Sylvan. "A generous Jainist interpretation of core relevant logics". Bulletin of the Section of Logic, 16:5866, 1987. [23] R.K. Tripathi. "The concept of avaktavya in Jainism". Philosophy East and West, 18:18793, 1968.