Catuṣkoṭi


Catuṣkoṭi refers to logical argument of a 'suite of four discrete functions' or 'an indivisible quaternity' that has multiple applications and has been important in the Indian logic and the Buddhist logico-epistemological traditions, particularly those of the Madhyamaka school.
In particular, the catuṣkoṭi is a "four-cornered" system of argumentation that involves the systematic examination of each of the 4 possibilities of a proposition, P:
  1. P; that is being.
  2. not P; that is not being.
  3. P and not P; that is being and that is not being.
  4. not ; that is neither not being nor is that being.
These four statements hold the following properties: each alternative is mutually exclusive and that all the alternatives are together exhaustive. This system of logic not only provides a novel method of classifying propositions into logical alternatives, but also because it does so in such a manner that the alternatives are not dependent on the number of truth-values assumed in the system of logic.
An example of a Catuṣkoṭi using the arbitrary proposition, "Animals understand love" as P would be:
  1. Animals understand love
  2. Animals do not understand love
  3. Animals both do and do not understand love
  4. Animals neither do nor do not understand love

    History

Nasadiya Sukta

Statements similar to the tetra lemma seems to appear in the Nasadiya Sukta, trying to inquire on the question of creation but was not used as a significant tool of logic before Buddhism.

Early Buddhism

, as remembered by Ānanda and codified in the Brahmajala Sutta 2.27, when expounding the sixteenth wrong view, or the fourth wrong view of the 'Eel-Wrigglers', the non-committal equivocators who adhered to Ajñana, the sceptical philosophy, though the grammatical structure is identical to the Catuṣkoṭi, the intentionality of the architecture employed by Nagarjuna is not evident, as rendered into English by Walshe :

'What is the fourth way? Here, an ascetic or Brahmin is dull and stupid. Because of his dullness and stupidity, when he is questioned he resorts to evasive statements and wriggles like an eel: "If you ask me whether there is another world. But I don't say so. And I don't say otherwise. And I don't say it is not, and I don't not say it is not." "Is there no other world?..." "Is there both another world and no other world?..."Is there neither another world nor no other world?..." "Are there spontaneously-born beings?..." "Are there not...?" "Both...? "Neither...?" "Does the Tathagata exist after death? Does he not exist after death? Does he both exist and not exist after death? Does he neither exist nor not exist after death?..." "If I thought so, I would say so...I don't say so...I don't say it is not." This is the fourth case.'

Pyrrhonism

makes a case for mutual iteration and pervasion between Pyrrhonism and Madhyamika:

An extraordinary similarity, that has long been noticed, between Pyrrhonism and Mādhyamika is the formula known in connection with Buddhism as the fourfold negation and which in Pyrrhonic form might be called the fourfold indeterminacy.

Nagarjuna

The Catuṣkoṭi was employed particularly by Nagarjuna who developed it and engaged it as a 'learning, investigative, meditative' portal to realize the 'openness', of Shakyamuni's Second Turning of the Dharmacakra, as categorized by the Sandhinirmocana Sutra.
Robinson, building on the foundations of Liebenthal to whom he gives credit, states:

What Nagarjuna wishes to prove is the irrationality of Existence, or the falsehood of reasoning which is built upon the logical principle that A equals A.... Because two answers, assertion and denial, are always possible to a given question, his arguments contain two refutations, one denying the presence, one the absence of the probandum. This double refutation is called the Middle Path.

Śūnyatā is the ninth 'view', the viewless view, a superposition of the eight possible arrays of proposition P .
The eight arrays or octaves of the iconographic Dharmacakra represent drishti or traditional views that Shakyamuni countered. These eight arrays may be plotted as coordinates on a multidimensional field which may be rendered as a sphere, a mandala, a multidimensional shunya or zero where shunyata denotes zero-ness. The eight arrays are in a concordant relationship where they each constitute a chord to the sphere. The coordinates are equidistant from the epicentre of shunya where the array of the positive configuration and the array of the negative configuration constitute two polar radii or diametrical complements, a diameter in sum. These are the 'eight limits' of 'openness', where śūnyatā is amplified by 'freedom from constructs' or 'simplicity'. Karmay conveys that 'spros bral' is a homologue of 'thig le', where 'spros bral' is literally "without amplification", understood as "that which cannot be displayed".
  1. P is true ``1 P is not true or Not P is true
  2. Not P is true ``2. Not is true i.e. P is true
  3. Both P and Not P are true i.e. the universal set `` 3 Neither P nor not P are true i.e. it is a null set
  4. Neither P nor not P are true it is a null set `` 4. Not = both P and not P are true which is the universal set.
Thus, a four-fold configuration resurfaces, with positive and negative configurations being mere rewritten alternatives. For example, if one replaces P with Not P, then the positive configuration set for Not P will be functionally equivalent to the negative configuration of P.
Sanjaya Belatthiputta, a 6th century BCE Indian ascetic whose teachings are similar to that of Nagarjuna are compared to that of a "theory of eel-wrigglers" in the famous Samannaphala Sutta. Sanjaya is recorded as saying:

Catuṣkoṭi post-Nagarjuna

The Catuṣkoṭi, following Nagarjuna, has had a profound impact upon the development of Buddhist logic and its dialectical refinement of Tibetan Buddhism.
Robinson qualifies the import of Nagarjuna's work due to the embedded noise in the scholarly lineage: "Certainly some of Nagarjuna's ancient opponents were just as confused as his modern interpreters...". This noise may also have co-arisen with Nagarjuna, following the work of Jayatilleke.

Modern interpretations

Robinson holds that Stcherbatsky, opened a productive period in Madhyamaka studies. Schayer made a departure into the rules of inference employed by early Buddhist dialecticians and examines the Catuskoti as an attribute of propositional logic and critiques Stcherbatsky. Robinson states that "Schayers criticisms of Stcherbatsky are incisive and just." Murti makes no mention of the logical contribution of Schayer. According to Robinson, Murti furthered the work of Stcherbatsky amongst others, and brought what Robinson terms "the metaphysical phase of investigation" to its apogee though qualifies this with: "Murti has a lot to say about 'dialectic,' but practically nothing to say about formal logic." Robinson opines that Nakamura, developed Schayer's methodology and defended and progressed its application.
Robinson opines that the 'metaphysical approach' evident foremost in Murti was not founded in a firm understanding of the 'logical structure of the system', i.e. catuskoti, for example:

Several fundamental limitations of the metaphysical approach are now apparent. It has tried to find comprehensive answers without knowing the answers to the more restricted questions involved - such questions as those of the epistemological and logical structure of the system.

Robinson conveys his focus and states his methodology, clearly identifying the limitations in scope of this particular publication, which he testifies is principally built upon, though divergent from, the work of Nakamura:

In considering the formal structure of Nagarjuna's argumentation, I exclude epistemology, psychology, and ontology from consideration.... Such extra-logical observations as emerge will be confined to the concluding paragraphs...

Exegesis

Puhakka charts the stylized reification process of a human sentient being, the spell of reality, a spell dispelled by the Catuṣkoṭi:

We are typically not aware of ourselves as taking something as real. Rather, its reality "takes us," or already has us in its spell as soon as we become aware of its identity. Furthermore, it's impossible to take something to be real without, at least momentarily, ignoring or denying that which it is not. Thus the act of taking something as real necessarily involves some degree of unconsciousness or lack of awareness. This is true even in the simple act of perception when we see a figure that we become aware of as "something." As the German gestalt psychologists demonstrated, for each figure perceived, there is a background of which we remain relatively unaware. We can extend this to texts or spoken communications. For every text we understand there is a context we are not fully cognizant of. Thus, with every figure noticed or reality affirmed, there is, inevitably, unawareness. Is this how a spell works? It takes us unawares.

Catuṣkoṭi paradox: a simple complex

Wayman proffers that the Catuṣkoṭi may be employed in different ways and often these are not clearly stated in discussion nor the tradition. Wayman holds that the Catuṣkoṭi may be applied in suite, that is all are applicable to a given topic forming a paradoxical matrix; or they may be applied like trains running on tracks. This difference in particular establishes a distinction with the Greek tradition of the Tetralemma. Also, predicate logic has been applied to the Dharmic Tradition, and though this in some quarters has established interesting correlates and extension of the logico-mathematical traditions of the Greeks, it has also obscured the logico-grammatical traditions of the Dharmic Traditions of Catuṣkoṭi within modern English discourse.