Second-order logic

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In logic and mathematics, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory.

First-order logic quantifies only variables that range over individuals (elements of the domain of discourse); second-order logic, in addition, quantifies over relations. For example, the second-order sentence ${\displaystyle \mathrm{\forall }P\mathrm{\forall }x(Px\vee \mathrm{\neg }Px)}$ says that for every formula P, and every individual x, either Px is true or not(Px) is true (this is the law of excluded middle). Second-order logic also includes quantification over sets, functions, and other variables (see section below). Both first-order and second-order logic use the idea of a domain of discourse (often called simply the "domain" or the "universe"). The domain is a set over which individual elements may be quantified.

Examples

Graffiti in Neukölln (Berlin) showing the simplest second-order sentence admitting nontrivial models, “∃φ φ”.

First-order logic can quantify over individuals, but not over properties. That is, we can take an atomic sentence like Cube(b) and obtain a quantified sentence by replacing the name with a variable and attaching a quantifier:

∃x Cube(x)

However, we cannot do the same with the predicate. That is, the following expression:

∃P P(b)

is not a sentence of first-order logic, but this is a legitimate sentence of second-order logic. Here, P is a predicate variable and is semantically a set of individuals.

As a result, second-order logic has greater expressive power than first-order logic. For example, there is no way in first-order logic to identify the set of all cubes and tetrahedrons. But the existence of this set can be asserted in second-order logic as:

∃P ∀x (Px ↔ (Cube(x) ∨ Tet(x))).

We can then assert properties of this set. For instance, the following says that the set of all cubes and tetrahedrons does not contain any dodecahedrons:

∀P (∀x (Px ↔ (Cube(x) ∨ Tet(x))) → ¬ ∃x (Px ∧ Dodec(x))).

Second-order quantification is especially useful because it gives the ability to express reachability properties. For example, if Parent(x, y) denotes that x is a parent of y, then first-order logic cannot express the property that x is an ancestor of y. In second-order logic we can express this by saying that every set of people containing y and closed under the Parent relation contains x:

∀P ((Py ∧ ∀a ∀b ((Pb ∧ Parent(a, b)) → Pa)) → Px).

It is notable that while we have variables for predicates in second-order-logic, we don't have variables for properties of predicates. We cannot say, for example, that there is a property Shape(P) that is true for the predicates P Cube, Tet, and Dodec. This would require third-order logic.

Syntax and fragments

The syntax of second-order logic tells which expressions are well formed formulas. In addition to the syntax of first-order logic, second-order logic includes many new sorts (sometimes called types) of variables. These are:

• A sort of variables that range over sets of individuals. If S is a variable of this sort and t is a first-order term then the expression t ∈ S (also written S(t), or St to save parentheses) is an atomic formula. Sets of individuals can also be viewed as unary relations on the domain.
• For each natural number k there is a sort of variables that ranges over all k-ary relations on the individuals. If R is such a k-ary relation variable and t₁,...,t_k are first-order terms then the expression R(t₁,...,t_k) is an atomic formula.
• For each natural number k there is a sort of variables that ranges over all functions taking k elements of the domain and returning a single element of the domain. If f is such a k-ary function variable and t₁,...,t_k are first-order terms then the expression f(t₁,...,t_k) is a first-order term.

Each of the variables just defined may be universally and/or existentially quantified over, to build up formulas. Thus there are many kinds of quantifiers, two for each sort of variables. A sentence in second-order logic, as in first-order logic, is a well-formed formula with no free variables (of any sort).

It's possible to forgo the introduction of function variables in the definition given above (and some authors do this) because an n-ary function variable can be represented by a relation variable of arity n+1 and an appropriate formula for the uniqueness of the "result" in the n+1 argument of the relation. (Shapiro 2000, p. 63)

Monadic second-order logic (MSO) is a restriction of second-order logic in which only quantification over unary relations (i.e. sets) is allowed. Quantification over functions, owing to the equivalence to relations as described above, is thus also not allowed. The second-order logic without these restrictions is sometimes called full second-order logic to distinguish it from the monadic version. Monadic second-order logic is particularly used in the context of Courcelle's theorem, an algorithmic meta-theorem in graph theory. The MSO theory of the complete infinite binary tree (S2S) is decidable. By contrast, full second order logic over any infinite set (or MSO logic over for example (${\displaystyle \mathbb{N}}$,+)) can interpret the true second-order arithmetic.

Just as in first-order logic, second-order logic may include non-logical symbols in a particular second-order language. These are restricted, however, in that all terms that they form must be either first-order terms (which can be substituted for a first-order variable) or second-order terms (which can be substituted for a second-order variable of an appropriate sort).

A formula in second-order logic is said to be of first-order (and sometimes denoted ${\displaystyle {\mathrm{\Sigma }}_0^1}$ or ${\displaystyle {\mathrm{\Pi }}_0^1}$) if its quantifiers (which may be universal or existential) range only over variables of first order, although it may have free variables of second order. A ${\displaystyle {\mathrm{\Sigma }}_1^1}$ (existential second-order) formula is one additionally having some existential quantifiers over second order variables, i.e. ${\displaystyle \mathrm{\exists }{R}_0\dots \mathrm{\exists }{R}_m\varphi }$, where ${\displaystyle \varphi }$ is a first-order formula. The fragment of second-order logic consisting only of existential second-order formulas is called existential second-order logic and abbreviated as ESO, as ${\displaystyle {\mathrm{\Sigma }}_1^1}$, or even as ∃SO. The fragment of ${\displaystyle {\mathrm{\Pi }}_1^1}$ formulas is defined dually, it is called universal second-order logic. More expressive fragments are defined for any k > 0 by mutual recursion: ${\displaystyle {\mathrm{\Sigma }}_{k+1}^1}$ has the form ${\displaystyle \mathrm{\exists }{R}_0\dots \mathrm{\exists }{R}_m\varphi }$, where ${\displaystyle \varphi }$ is a ${\displaystyle {\mathrm{\Pi }}_k^1}$ formula, and similar, ${\displaystyle {\mathrm{\Pi }}_{k+1}^1}$ has the form ${\displaystyle \mathrm{\forall }{R}_0\dots \mathrm{\forall }{R}_m\varphi }$, where ${\displaystyle \varphi }$ is a ${\displaystyle {\mathrm{\Sigma }}_k^1}$ formula. (See analytical hierarchy for the analogous construction of second-order arithmetic.)

Semantics

The semantics of second-order logic establish the meaning of each sentence. Unlike first-order logic, which has only one standard semantics, there are two different semantics that are commonly used for second-order logic: standard semantics and Henkin semantics. In each of these semantics, the interpretations of the first-order quantifiers and the logical connectives are the same as in first-order logic. Only the ranges of quantifiers over second-order variables differ in the two types of semantics (Väänänen 2001).

In standard semantics, also called full semantics, the quantifiers range over all sets or functions of the appropriate sort. A model with this condition is called a full model, and these are the same as models in which the range of the second-order quantifiers is the powerset of the model's first-order part. (2001) Thus once the domain of the first-order variables is established, the meaning of the remaining quantifiers is fixed. It is these semantics that give second-order logic its expressive power, and they will be assumed for the remainder of this article.

Leon Henkin (1950) defined an alternative kind of semantics for second-order and higher-order theories, in which the meaning of the higher-order domains is partly determined by an explicit axiomatisation, drawing on type theory, of the properties of the sets or functions ranged over. Henkin semantics is a kind of many-sorted first-order semantics, where there are a class of models of the axioms, instead of the semantics being fixed to just the standard model as in the standard semantics. A model in Henkin semantics will provide a set of sets or set of functions as the interpretation of higher-order domains, which may be a proper subset of all sets or functions of that sort. For his axiomatisation, Henkin proved that Gödel's completeness theorem and compactness theorem, which hold for first-order logic, carry over to second-order logic with Henkin semantics. Since also the Skolem–Löwenheim theorems hold for Henkin semantics, Lindström's theorem imports that Henkin models are just disguised first-order models.

For theories such as second-order arithmetic, the existence of non-standard interpretations of higher-order domains isn't just a deficiency of the particular axiomatisation derived from type theory that Henkin used, but a necessary consequence of Gödel's incompleteness theorem: Henkin's axioms can't be supplemented further to ensure the standard interpretation is the only possible model. Henkin semantics are commonly used in the study of second-order arithmetic.

Jouko Väänänen (2001) argued that the distinction between Henkin semantics and full semantics for second-order logic is analogous to the distinction between provability in ZFC and truth in V, in that the former obeys model-theoretic properties like the Lowenheim-Skolem theorem and compactness, and the latter has categoricity phenomena. For example, "we cannot meaningfully ask whether the ${\displaystyle V}$ as defined in ${\displaystyle \mathrm{Z}\mathrm{F}\mathrm{C}}$ is the real ${\displaystyle V}$ . But if we reformalize ${\displaystyle \mathrm{Z}\mathrm{F}\mathrm{C}}$ inside ${\displaystyle \mathrm{Z}\mathrm{F}\mathrm{C}}$, then we can note that the reformalized ${\displaystyle \mathrm{Z}\mathrm{F}\mathrm{C}}$ ... has countable models and hence cannot be categorical."

Expressive power

Second-order logic is more expressive than first-order logic. For example, if the domain is the set of all real numbers, one can assert in first-order logic the existence of an additive inverse of each real number by writing ∀x ∃y (x + y = 0) but one needs second-order logic to assert the least-upper-bound property for sets of real numbers, which states that every bounded, nonempty set of real numbers has a supremum. If the domain is the set of all real numbers, the following second-order sentence (split over two lines) expresses the least upper bound property:

(∀ A) ([(∃ w) (w ∈ A) ∧ (∃ z)(∀ u)(u ∈ A → u ≤ z)]

→ (∃ x)(∀ y)([(∀ w)(w ∈ A → w ≤ x)] ∧ [(∀ u)(u ∈ A → u ≤ y)] → x ≤ y))

This formula is a direct formalization of "every nonempty, bounded set A has a least upper bound." It can be shown that any ordered field that satisfies this property is isomorphic to the real number field. On the other hand, the set of first-order sentences valid in the reals has arbitrarily large models due to the compactness theorem. Thus the least-upper-bound property cannot be expressed by any set of sentences in first-order logic. (In fact, every real-closed field satisfies the same first-order sentences in the signature ${\displaystyle ⟨+,\cdot ,\le ⟩}$ as the real numbers.)

In second-order logic, it is possible to write formal sentences that say "the domain is finite" or "the domain is of countable cardinality." To say that the domain is finite, use the sentence that says that every surjective function from the domain to itself is injective. To say that the domain has countable cardinality, use the sentence that says that there is a bijection between every two infinite subsets of the domain. It follows from the compactness theorem and the upward Löwenheim–Skolem theorem that it is not possible to characterize finiteness or countability, respectively, in first-order logic.

Certain fragments of second-order logic like ESO are also more expressive than first-order logic even though they are strictly less expressive than the full second-order logic. ESO also enjoys translation equivalence with some extensions of first-order logic that allow non-linear ordering of quantifier dependencies, like first-order logic extended with Henkin quantifiers, Hintikka and Sandu's independence-friendly logic, and Väänänen's dependence logic.

Deductive systems

A deductive system for a logic is a set of inference rules and logical axioms that determine which sequences of formulas constitute valid proofs. Several deductive systems can be used for second-order logic, although none can be complete for the standard semantics (see below). Each of these systems is sound, which means any sentence they can be used to prove is logically valid in the appropriate semantics.

The weakest deductive system that can be used consists of a standard deductive system for first-order logic (such as natural deduction) augmented with substitution rules for second-order terms. This deductive system is commonly used in the study of second-order arithmetic.

The deductive systems considered by Shapiro (1991) and Henkin (1950) add to the augmented first-order deductive scheme both comprehension axioms and choice axioms. These axioms are sound for standard second-order semantics. They are sound for Henkin semantics restricted to Henkin models satisfying the comprehension and choice axioms.

Non-reducibility to first-order logic

One might attempt to reduce the second-order theory of the real numbers, with full second-order semantics, to the first-order theory in the following way. First expand the domain from the set of all real numbers to a two-sorted domain, with the second sort containing all sets of real numbers. Add a new binary predicate to the language: the membership relation. Then sentences that were second-order become first-order, with the formerly second-order quantifiers ranging over the second sort instead. This reduction can be attempted in a one-sorted theory by adding unary predicates that tell whether an element is a number or a set, and taking the domain to be the union of the set of real numbers and the power set of the real numbers.

But notice that the domain was asserted to include all sets of real numbers. That requirement cannot be reduced to a first-order sentence, as the Löwenheim–Skolem theorem shows. That theorem implies that there is some countably infinite subset of the real numbers, whose members we will call internal numbers, and some countably infinite collection of sets of internal numbers, whose members we will call "internal sets", such that the domain consisting of internal numbers and internal sets satisfies exactly the same first-order sentences as are satisfied by the domain of real numbers and sets of real numbers. In particular, it satisfies a sort of least-upper-bound axiom that says, in effect:

Every nonempty internal set that has an internal upper bound has a least internal upper bound.

Countability of the set of all internal numbers (in conjunction with the fact that those form a densely ordered set) implies that that set does not satisfy the full least-upper-bound axiom. Countability of the set of all internal sets implies that it is not the set of all subsets of the set of all internal numbers (since Cantor's theorem implies that the set of all subsets of a countably infinite set is an uncountably infinite set). This construction is closely related to Skolem's paradox.

Thus the first-order theory of real numbers and sets of real numbers has many models, some of which are countable. The second-order theory of the real numbers has only one model, however. This follows from the classical theorem that there is only one Archimedean complete ordered field, along with the fact that all the axioms of an Archimedean complete ordered field are expressible in second-order logic. This shows that the second-order theory of the real numbers cannot be reduced to a first-order theory, in the sense that the second-order theory of the real numbers has only one model but the corresponding first-order theory has many models.

There are more extreme examples showing that second-order logic with standard semantics is more expressive than first-order logic. There is a finite second-order theory whose only model is the real numbers if the continuum hypothesis holds and that has no model if the continuum hypothesis does not hold (cf. Shapiro 2000, p. 105). This theory consists of a finite theory characterizing the real numbers as a complete Archimedean ordered field plus an axiom saying that the domain is of the first uncountable cardinality. This example illustrates that the question of whether a sentence in second-order logic is consistent is extremely subtle.

Additional limitations of second-order logic are described in the next section.

Metalogical results

It is a corollary of Gödel's incompleteness theorem that there is no deductive system (that is, no notion of provability) for second-order formulas that simultaneously satisfies these three desired attributes:

• (Soundness) Every provable second-order sentence is universally valid, i.e., true in all domains under standard semantics.
• (Completeness) Every universally valid second-order formula, under standard semantics, is provable.
• (Effectiveness) There is a proof-checking algorithm that can correctly decide whether a given sequence of symbols is a proof or not.

This corollary is sometimes expressed by saying that second-order logic does not admit a complete proof theory. In this respect second-order logic with standard semantics differs from first-order logic; Quine (1970, pp. 90–91) pointed to the lack of a complete proof system as a reason for thinking of second-order logic as not logic, properly speaking.

As mentioned above, Henkin proved that the standard deductive system for first-order logic is sound, complete, and effective for second-order logic with Henkin semantics, and the deductive system with comprehension and choice principles is sound, complete, and effective for Henkin semantics using only models that satisfy these principles.

The compactness theorem and the Löwenheim–Skolem theorem do not hold for full models of second-order logic. They do hold however for Henkin models.

History and disputed value

Predicate logic was introduced to the mathematical community by C. S. Peirce, who coined the term second-order logic and whose notation is most similar to the modern form (Putnam 1982). However, today most students of logic are more familiar with the works of Frege, who published his work several years prior to Peirce but whose works remained less known until Bertrand Russell and Alfred North Whitehead made them famous. Frege used different variables to distinguish quantification over objects from quantification over properties and sets; but he did not see himself as doing two different kinds of logic. After the discovery of Russell's paradox it was realized that something was wrong with his system. Eventually logicians found that restricting Frege's logic in various ways—to what is now called first-order logic—eliminated this problem: sets and properties cannot be quantified over in first-order logic alone. The now-standard hierarchy of orders of logics dates from this time.

It was found that set theory could be formulated as an axiomatized system within the apparatus of first-order logic (at the cost of several kinds of completeness, but nothing so bad as Russell's paradox), and this was done (see Zermelo–Fraenkel set theory), as sets are vital for mathematics. Arithmetic, mereology, and a variety of other powerful logical theories could be formulated axiomatically without appeal to any more logical apparatus than first-order quantification, and this, along with Gödel and Skolem's adherence to first-order logic, led to a general decline in work in second (or any higher) order logic.

This rejection was actively advanced by some logicians, most notably W. V. Quine. Quine advanced the view that in predicate-language sentences like Fx the "x" is to be thought of as a variable or name denoting an object and hence can be quantified over, as in "For all things, it is the case that . . ." but the "F" is to be thought of as an abbreviation for an incomplete sentence, not the name of an object (not even of an abstract object like a property). For example, it might mean " . . . is a dog." But it makes no sense to think we can quantify over something like this. (Such a position is quite consistent with Frege's own arguments on the concept-object distinction). So to use a predicate as a variable is to have it occupy the place of a name, which only individual variables should occupy. This reasoning has been rejected by George Boolos.

In recent years second-order logic has made something of a recovery, buoyed by Boolos' interpretation of second-order quantification as plural quantification over the same domain of objects as first-order quantification (Boolos 1984). Boolos furthermore points to the claimed nonfirstorderizability of sentences such as "Some critics admire only each other" and "Some of Fianchetto's men went into the warehouse unaccompanied by anyone else", which he argues can only be expressed by the full force of second-order quantification. However, generalized quantification and partially ordered (or branching) quantification may suffice to express a certain class of purportedly nonfirstorderizable sentences as well and these do not appeal to second-order quantification.

Relation to computational complexity

The expressive power of various forms of second-order logic on finite structures is intimately tied to computational complexity theory. The field of descriptive complexity studies which computational complexity classes can be characterized by the power of the logic needed to express languages (sets of finite strings) in them. A string w = w₁···w_n in a finite alphabet A can be represented by a finite structure with domain D = {1,...,n}, unary predicates P_a for each a ∈ A, satisfied by those indices i such that w_i = a, and additional predicates that serve to uniquely identify which index is which (typically, one takes the graph of the successor function on D or the order relation <, possibly with other arithmetic predicates). Conversely, the Cayley tables of any finite structure (over a finite signature) can be encoded by a finite string.

This identification leads to the following characterizations of variants of second-order logic over finite structures:

• REG (the regular languages) is the set of languages definable by monadic, second-order formulas (Büchi-Elgot-Trakhtenbrot theorem, 1960)
• NP is the set of languages definable by existential, second-order formulas (Fagin's theorem, 1974).
• co-NP is the set of languages definable by universal, second-order formulas.
• PH is the set of languages definable by second-order formulas.
• PSPACE is the set of languages definable by second-order formulas with an added transitive closure operator.
• EXPTIME is the set of languages definable by second-order formulas with an added least fixed point operator.

Relationships among these classes directly impact the relative expressiveness of the logics over finite structures; for example, if PH = PSPACE, then adding a transitive closure operator to second-order logic would not make it any more expressive over finite structures.

Buddhist logico-epistemology

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Buddhist_logico-epistemology

Buddhist logico-epistemology is a term used in Western scholarship to describe Buddhist systems of pramāṇa (epistemic tool, valid cognition) and hetu-vidya (reasoning, logic). While the term may refer to various Buddhist systems and views on reasoning and epistemology, it is most often used to refer to the work of the "Epistemological school" (Sanskrit: Pramāṇa-vāda), i.e. the school of Dignaga and Dharmakirti which developed from the 5th through 7th centuries and remained the main system of Buddhist reasoning until the decline of Buddhism in India.

The early Buddhist texts show that the historical Buddha was familiar with certain rules of reasoning used for debating purposes and made use of these against his opponents. He also seems to have held certain ideas about epistemology and reasoning, though he did not put forth a logico-epistemological system.

The Theravada Kathāvatthu contains some rules on debate and reasoning. The first Buddhist thinker to discuss logical and epistemic issues systematically was Vasubandhu in his Vāda-vidhi (A Method for Argumentation). A mature system of Buddhist logic and epistemology was founded by the Buddhist scholar Dignāga (c. 480–540 CE) in his magnum opus, the Pramāṇa-samuccaya. Dharmakirti further developed this system with several innovations in his Pramanavarttika ("Commentary on Valid Cognition"). His work was influential on all later Buddhist philosophical systems as well as on numerous Hindu thinkers. It also became the main source of epistemology and reasoning in Tibetan Buddhism.

Definition

Scholars such as H.N. Randle and Fyodor Shcherbatskoy (1930s) initially employed terms such as “Indian Logic” and “Buddhist Logic” to refer to the Indian tradition of inference (anumāna), epistemology (pramana), and "science of causes" (hetu-vidyā). This tradition developed in the orthodox Hindu tradition known as Nyaya as well as in Buddhist philosophy. Logic in classical India, writes Bimal Krishna Matilal, is "the systematic study of informal inference-patterns, the rules of debate, the identification of sound inference vis-à-vis sophistical argument, and similar topics." As Matilal notes, this tradition developed out of systematic debate theory (vadavidyā):

Logic as the study of the form of correct arguments and inference patterns, developed in India from the methodology of philosophical debate. The art of conducting a philosophical debate was prevalent probably as early as the time of the Buddha and the Mahavira (Jina), but it became more systematic and methodical a few hundred years later.

"Indian Logic" is a different system than modern derivatives of classical logic (such as modern predicate calculus): anumāna-theory, a system in its own right. "Indian Logic" was also influenced by the study of grammar, whereas Classical Logic—which principally informed modern Western Logic—was influenced by the study of mathematics.

A key difference between Western Logic and Indian Logic is that certain epistemological issues are included within Indian Logic, whereas in modern Western Logic they are deliberately excluded. Indian Logic includes general questions regarding the "nature of the derivation of knowledge," epistemology, from information supplied by evidence, evidence which in turn may be another item of knowledge. For this reason, other scholars use the term "logico-epistemology" to refer to this tradition, emphasizing the centrality of the epistemic project for Indian logical reasoning. According to Georges Dreyfus, while Western logic tends to be focused on formal validity and deduction:

The concern of Indian "logicians" is quite different. They intend to provide a critical and systematic analysis of the diverse means of correct cognition that we use practically in our quest for knowledge. In this task, they discuss the nature and types of pramana. Although Indian philosophers disagree on the types of cognition that can be considered valid, most recognize perception and inference as valid. Within this context, which is mostly epistemological and practically oriented, topics such as the nature and types of correct reasoning that pertain to logic in the large sense of the word are discussed.

Pramana

Pramāṇa (Tib. tshad ma) is often translated as "valid cognition" or "instrument of knowledge" and refers to epistemic ways of knowing. Epistemological justification distinguishes Buddhist pramana from orthodox Hindu philosophy. All schools of Indian logic recognize various sets of "valid justifications for knowledge" or pramana. Buddhist logico-epistemology was influenced by the Nyāya school's methodology, but where the Nyaya recognised a set of four pramanas—perception, inference, comparison, and testimony—the Buddhists (i.e. the school of Dignaga) only recognized two: perception and inference. For Dignaga, comparison and testimony are just special forms of inference.

Most Indic pramanavada accept "perception" (Sanskrit: pratyakṣa) and "inference" (Sanskrit: anumāna), but for some schools of orthodox Hinduism the "received textual tradition" (Sanskrit: āgamāḥ) is an epistemological category equal to perception and inference. The Buddhist logical tradition of Dignaga and Dharmakirti accept scriptural tradition only if it accords with pratyakṣa and anumāna. This view is thus in line with the Buddha's injunction in the Kalama Sutta not to accept anything on mere tradition or scripture.

Early Buddhist background

Epistemology

The time of the Gautama Buddha was a lively intellectual culture with many differing philosophical theories. KN Jayatilleke, in his "Early Buddhist Theory of Knowledge", uses the Pali Nikayas to glean the possible epistemological views of the historical Buddha and those of his contemporaries. According to his analysis of the Saṅgārava-sutta (AN 3.60), during the Buddha's time, Indian views were divided into three major camps with regards to knowledge:

• The Traditionalists (anussavika) who regarded knowledge as being derived from scriptural sources (the Brahmins who upheld the Vedas).
• The Rationalists (takki vimamsi) who only used reasoning or takka (the skeptics and materialists).
• The "Experientialists" who held that besides reasoning, a kind of supra-normal yogic insight was able to bring about unique forms of knowledge (the Jains, the middle and late Upanishadic sages).

The Buddha rejected the first view in several texts such as the Kalama sutta, arguing that a claim to scriptural authority (sadda) was not a source of knowledge, as was claimed by the later Hindu Mimamsa school. The Buddha also seems to have criticized those who used reason (takka). According to Jayatilleke, in the Pali Nikayas, this term refers "primarily to denote the reasoning that was employed to construct and defend metaphysical theories and perhaps meant the reasoning of sophists and dialecticians only in a secondary sense". The Buddha rejected metaphysical speculations, and put aside certain questions which he named the unanswerables (avyakatas), including questions about the soul and if the universe is eternal or not.

The Buddha's epistemological view has been a subject of debate among modern scholars. Some such as David Kalupahana, have seen him first and foremost as an empiricist because of his teaching that knowledge required verification through the six sense fields (ayatanas). The Kalama sutta states that verification through one's own personal experience (and the experiences of the wise) is an important means of knowledge.

However, the Buddha's view of truth was also based on the soteriological and therapeutic concern of ending suffering. In the "Discourse to Prince Abhaya" (MN.I.392–4) the Buddha states that a belief should only be accepted if it leads to wholesome consequences. This has led scholars such as Mrs Rhys Davids and Vallée-Poussin to see the Buddha's view as a form of Pragmatism. This sense of truth as what is useful is also shown by the Buddha's parable of the arrow.

K. N. Jayatilleke sees Buddha's epistemological view as a kind of empiricism which also includes a particular view of causation (dependent origination): "inductive inferences in Buddhism are based on a theory of causation. These inferences are made on the data of perception. What is considered to constitute knowledge are direct inferences made on the basis of such perceptions." Jayatilleke argues the Buddhas statements in the Nikayas tacitly imply an adherence to some form of correspondence theory, this is most explicit in the Apannaka Sutta (MN 60). He also notes that Coherentism is also taken as a criterion for truth in the Nikayas, which contains many instances of the Buddha debating opponents by showing how they have contradicted themselves. He also notes that the Buddha seems to have held that utility and truth go hand in hand, and therefore something which is true is also useful (and vice versa, something false is not useful for ending suffering).

Echoing this view, Christian Coseru writes:

canonical sources make quite clear that several distinct factors play a crucial role in the acquisition of knowledge. These are variously identified with the testimony of sense experience, introspective or intuitive experience, inferences drawn from these two types of experience, and some form of coherentism, which demands that truth claims remain consistent across the entire corpus of doctrine. Thus, to the extent that Buddhists employ reason, they do so primarily in order further to advance the empirical investigation of phenomena.

Debate and analysis

The Early Buddhist Texts show that during this period many different kinds of philosophers often engaged in public debates (vivada). The early texts also mention that there was a set procedure (patipada) for these debates and that if someone does not abide by it they are unsuitable to be debated. There also seems to have been at least a basic conception of valid and invalid reasoning, including, according to Jayatilleke, fallacies (hetvabhasah) such as petitio principii. Various fallacies were further covered under what were called nigrahasthana or "reasons for censure" by which one could lose the debate. Other nigrahasthanas included arthantaram or "shifting the topic", and not giving a coherent reply.

According to Jayatilleke, 'pure reasoning' or 'a priori' reasoning is rejected by the Buddha as a source of knowledge. While reason could be useful in deliberation, it could not establish truth on its own.

In contrast to his opponents, the Buddha termed himself a defender of 'analysis' or 'vibhajjavada'. He held that after proper rational analysis, assertions could be classified in the following way:

• Those assertions which can be asserted or denied categorically (ekamsika)
• Those which cannot be asserted or denied categorically (anekamsika), which the Buddha further divided into:
  • Those which after analysis (vibhajja-) could be known to be true or false.
  • Those like the avyakata-theses, which could not be thus known.

This view of analysis differed from that of the Jains, which held that all views were anekamsika and also were relative, that is, they were true or false depending on the standpoint one viewed it from (anekantavada).

The early texts also mention that the Buddha held there to be "four kinds of explanations of questions."

• a question which ought to be explained categorically
• a question which ought to be answered with a counter question
• a question which ought to be set aside (thapaniya)
• a question which ought to be explained analytically

The Buddha also made use of various terms which reveal some of his views on meaning and language. For example, he held that many concepts or designations (paññatti) could be used in conventional everyday speech while at the same time not referring to anything that exists ultimately (such as the pronouns like "I" and "Me"). Richard Hayes likewise points to the Potthapada sutta as an example of the Early Buddhist tendency towards a nominalist perspective on language and meaning in contrast to the Brahmanical view which tended to see language as reflecting real existents.

The Buddha also divided statements (bhasitam) into two types with regards to their meaning: those which were intelligible, meaningful (sappatihirakatam) and those meaningless or incomprehensible (appatihirakatam). According to Jayatilleke, "in the Nikayas it is considered meaningless to make a statement unless the speaker could attach a verifiable content to each of its terms." This is why the Buddha held that statements about the existence of a self or soul (atman) were ultimately meaningless, because they could not be verified.

The Buddha, like his contemporaries, also made use of the "four corners" (catuṣkoṭi) logical structure as a tool in argumentation. According to Jayatilleke, these "four forms of predication" can be rendered thus:

1. S is P, e.g. atthi paro loko (there is a next world).
2. S is not P, e.g. natthi paro loko (there is no next world).
3. S is and is not P, e.g. atthi ca natthi ca paro loko (there is and is no next world).
4. S neither is nor is not P, e.g. n'ev'atthi na natthi paro loko (there neither is nor is there no next world)

The Buddha in the Nikayas seems to regard these as "'the four possible positions or logical alternatives that a proposition can take". Jayatilleke notes that the last two are clearly non-Aristotelian in nature. The Buddhists in the Nikayas use this logical structure to analyze the truth of statements and classify them. When all four were denied regarding a statement or question, it was held to be meaningless and thus set aside or rejected (but not negated).

Two levels of Truth

The early texts mention two modes of discourse used by the Buddha. Jayatilleke writes:

when he is speaking about things or persons we should not presume that he is speaking about entities or substances; to this extent his meaning is to be inferred (neyyattha-). But when he is pointing out the misleading implications of speech or using language without these implications, his meaning is plain and direct and nothing is to be inferred (nitattha-). This is a valid distinction which certainly holds good for the Nikäyas at least, in the light of the above-statement.

The later commentarial and Abhidharma literature began to use this distinction as an epistemic one. They spoke of two levels of truth, the conventional (samutti), and the absolute (paramattha). This theory of double truth became very influential in later Buddhist epistemic discourse.

Kathāvatthu

The Theravada Kathāvatthu (Points of Controversy) is a Pali Buddhist text which discusses the proper method for critical discussions on doctrine. Its date is debated by scholars but it might date to the time of Ashoka (C. 240 BC). Western scholarship by St. Schayer and following him A. K. Warder, have argued that there is an "anticipations of propositional logic" in the text. However, according to Jonardon Ganeri "the leading concern of the text is with issues of balance and fairness in the conduct of a dialogue and it recommends a strategy of argumentation which guarantees that both parties to a point of controversy have their arguments properly weighed and considered."

In the Kathāvatthu, a proper reasoned dialogue (vadayutti) is structured as follows: there is a point of contention – whether A is B; this is divided into several 'openings' (atthamukha):

1. Is A B?
2. Is A not B?
3. Is A B everywhere?
4. Is A B always?
5. Is A B in everything?
6. Is A not B everywhere?
7. Is A not B always?
8. Is A not B in everything?

These help clarify the attitude of someone towards their thesis in the proceeding argumentative process. Jonardon Ganeri outlines the process thus:

Each such ‘opening’ now proceeds as an independent dialogue, and each is divided into five stages: the way forward (anuloma), the way back (patikamma), the refutation (niggaha), the application (upanayana) and the conclusion (niggamana). In the way forward, the proponent solicits from the respondent the endorsement of a thesis and then tries to argue against it. On the way back, the respondent turns the tables, soliciting from the proponent the endorsement of the counter-thesis, and then trying argue against it. In the refutation, the respondent, continuing, seeks to refute the argument that the proponent had advanced against the thesis. The application and conclusion repeat and reaffirm that the proponent’s argument against the respondent’s thesis is unsound, while the respondent’s argument against the proponent’s counter-thesis is sound.

Milindapanha

Another Buddhist text which depicts the standards for rational debate among Buddhists is the Milindapanha ("Questions of Menander", 1st century BCE) which is a dialogue between the Buddhist monk Nagasena and an Indo-Greek King. In describing the art of debate and dialogue, Nagasena states:

When scholars talk a matter over one with another, then is there a winding up, an unravelling, one or other is convicted of error, and he then acknowledges his mistake; distinctions are drawn, and contra-distinctions; and yet thereby they are not angered.

The various elements outlined here make up the standard procedure of Buddhist debate theory. There is an 'unravelling' or explication (nibbethanam) of one's thesis and stances and then there is also a 'winding up' ending in the censure (niggaho) of one side based on premises he has accepted and the rejoinders of his opponent.

Abhidharma views

The Buddhist Abhidharma schools developed a classification of four types of reasoning which became widely used in Buddhist thought. The Mahayana philosopher Asanga in his Abhidharma-samuccaya, outlines these four reasons (yukti) that one may use to inquire about the nature of things. According to Cristian Coseru these are:

1. The principle of dependence (apeksāyukti), which takes into account the fact that conditioned things necessarily arise in dependence upon conditions: it is a principle of reason, for instance, that sprouts depend on seeds.
2. The principle of causal efficacy (kāryakāranayukti), which accounts for the difference between things in terms of the different causal conditions for their apprehension: it is a principle of reason, thus, that, in dependence upon form, a faculty of vision, and visual awareness, one has visual rather than, say, auditory or tactile experiences.
3. The realization of evidence from experience (sāksātkriyāsādhanayukti). We realize the presence of water from moisture and of fire from smoke.
4. The principle of natural reasoning, or the principle of reality (dharmatāyukti), which concerns the phenomenal character of things as perceived (for instance, the wetness and fluidity of water).

According to Coseru "what we have here are examples of natural reasoning or of reasoning from experience, rather than attempts to use deliberative modes of reasoning for the purpose of justifying a given thesis or arguing for its conditions of satisfaction."

Nyaya influences

Main article: Nyaya

The Nyaya school considers perception, inference, comparison/analogy, and testimony from reliable sources as four means to correct knowledge, holding that perception is the ultimate source of such knowledge.

The Nyāya Sūtras of Gotama (6th century BC – 2nd century CE) is the founding text of the Nyaya school. The text systematically lays out logical rules for argumentation in the form of a five-step schema and also sets forth a theory of epistemology. According to Jonardon Ganeri, the Nyaya sutra brought about a transformation in Indian thinking about logic. First, it began a shift away from interest in argumentation and debate towards the formal properties of sound inference. Secondly, the Nyaya sutra led a shift to rule-governed forms of logical thinking.

B.K. Matilal outlines the five steps or limbs of the Nyaya method of reasoning as follows:

1. There is fire on the hill. [thesis]
2. For there is smoke. [reason]
3. (Wherever there is smoke, there is fire), as in the kitchen. [example]
4. This is such a case (smoke on the hill).
5. Therefore, it is so, i.e., there is fire on the hill.

Later Buddhist thinkers like Vasubandhu would see several of these steps as redundant and would affirm that only the first two or three were necessary.

The Naiyayikas (the Nyaya scholars) also accepted four valid means (pramaṇa) of obtaining valid knowledge (pramana) - perception (pratyakṣa), inference (anumāna), comparison (upamāna) and word/testimony of reliable sources (śabda).

The systematic discussions of the Nyaya school influenced the Medieval Buddhist philosophers who developed their own theories of inferential reasoning and epistemic warrant (pramana). The Nyaya became one of the main opponents of the Buddhists.

Mahayana Buddhist philosophy

Nagarjuna's Madhyamaka

Nagarjuna (c. 150 – c. 250 CE), one of the most influential Buddhist thinkers, defended the theory of the emptiness (shunyata) of phenomena and attacked theories that posited an essence or true existence (svabhava) to phenomena in his magnum opus The Fundamental Verses on the Middle Way. He used the Buddhist catuṣkoṭi ("four corners" or "four positions") to construct reductio ad absurdum arguments against numerous theories which posited essences to certain phenomena, such as causality and movement. In Nagarjuna's works and those of his followers, the four positions on a particular thesis are negated or ruled out (pratiṣedha) as exemplified by the first verse of Nagarjuna's Middle way verses which focuses on a critique of causation:

Entities of any kind are not ever found anywhere produced from themselves, from another, from both [themselves and another], and also from no cause.

Nagarjuna also famously relied upon refutation based argumentation (vitanda) drawing out the consequences (prasaṅga) and presuppositions of his opponents' own theories and showing them to be self refuting. Because the vaitandika only seeks to disprove his opponents arguments without putting forward a thesis of his own, the Hindu Nyaya school philosophers such as Vatsyayana saw it as unfair and also irrational (because if you argue against P, you must have a thesis, mainly not-P). According to Matilal, Nagarjuna's position of not putting forth any implied thesis through his refutations would be rational if seen as a form of illocutionary act.

Nagarjuna's reductions and the structure of the catuṣkoṭi became very influential in the Buddhist Madhyamaka school of philosophy which sees itself as a continuation of Nagarjuna's thought. Nagarjuna also discusses the four modes of knowing of the Nyaya school, but he is unwilling to accept that such epistemic means bring us ultimate knowledge.

Nagarjuna's epistemic stance continues to be debated among modern scholars, his skepticism of the ability of reason and language to capture the nature of reality and his view of reality as being empty of true existence have led some to see him as a skeptic, mystic, nihilist or agnostic, while others interpret him as a Wittgensteinian analyst, an anti-realist, or deconstructionist.

Nagarjuna is also said to be the author of the Upāyaśṛdaya one of the first Buddhist texts on proper reasoning and argumentation. He also developed the Buddhist theory of two truths, defending ultimate truth as the truth of emptiness.

Vasubandhu

Vasubandhu was one of the first Buddhist thinkers to write various works on sound reasoning and debate, including the Vādavidhi (Methods of Debate), and the Vādavidhāna (Rules of Debate).

Vasubandhu was influenced by the system of the Nyaya school. Vasubandhu introduced the concept of 'logical pervasion' (vyapti). He also introduced the trairūpya (triple inferential sign).

The trairūpya is a logical argument that contains three constituents which a logical ‘sign’ or ‘mark’ (linga) must fulfill to be 'valid source of knowledge' (pramana):

1. It should be present in the case or object under consideration, the ‘subject-locus' (pakṣa)
2. It should be present in a ‘similar case’ or a homologue (sapakṣa)
3. It should not be present in any ‘dissimilar case’ or heterologue (vipakṣa)

Pramāṇavāda

See also: Dignaga and Dharmakirti

Dignāga

Dignaga. A statue in Elista, Russia.

Buddhist epistemology holds that perception and inference are the means to correct knowledge.

Dignāga (c. 480 – 540 CE) is the founder of an eponymous tradition of Buddhist logic and epistemology which was widely influential in Indian philosophy due to the introduction of unique epistemological questions.  According to B.K. Matilal, Dignāga "was perhaps the most creative logician in medieval (400-1100) India."

Dignāga's tradition of Buddhist logic is sometimes called the "School of Dignāga" or "The School of Dinnāga and Dharmakīrti". In Tibetan, it is often called “those who follow reasoning” (Tibetan: rigs pa rjes su ‘brang ba); in modern literature, it is sometimes known by the Sanskrit "pramāṇavāda", often translated as "the epistemological school" or "the logico-epistemological school."

Dignāga defended the validity of only two pramāṇas (instruments of knowledge), perception and inference, in his magnum opus, the Pramāṇa-samuccaya.

His theory does not "make a radical distinction between epistemology and the psychological processes of cognition." As noted by Cristian Coseru, Dignāga's theory of knowledge is strongly grounded in perception "as an epistemic modality for establishing a cognitive event as knowledge".

Since perception is information that is acquired through the senses, it is not susceptible to error. However, there is susceptibility to error in processes of interpretation, including mental construction and inferential thinking.

Dignāga also wrote on language and meaning. His "apoha" (exclusion) theory of meaning was widely influential. For Dignāga, a word can express its own meaning only by repudiating other meanings. The word 'cow' gives its own meaning only by the exclusion of all those things which are other than cow.

Dharmakīrti

Following Dignāga, Dharmakīrti (c. 7th century), contributed significantly to the development and application of Buddhist pramana theory. Dharmakīrti's Pramāṇavārttika, remains in Tibet as a central text on pramana and was widely commented on by various Indian and Tibetan scholars.

Dharmakīrti's theory of epistemology differed from Dignāga's by introducing the idea that for something to be a valid cognition it must "confirm causal efficacy" (arthakriyāsthiti) which "consists in [this cognition’s] compliance with [the object’s capacity to] perform a function" (Pramāṇavārttika 2.1ac).

He was also one of the primary theorists of Buddhist atomism, according to which the only items considered to exist or be ultimately real are momentary particulars (svalakṣaṇa) including material atoms and momentary states of consciousness (dharmas). Everything else is considered to be only conventional (saṃvṛtisat) and thus he has been seen as a nominalist, like Dignāga.

Vincent Eltschinger has argued that Buddhist epistemology, especially Dharmakīrti's, was an apologetic response to attacks by hostile Hindu opponents and thus was seen by Buddhists as "that which, by defeating the outsiders, removes the obstacles to the path towards liberation." Coseru meanwhile simply notes the inseparability of epistemic concerns from spiritual praxis for Buddhist epistemologists such as Dharmakīrti:

It is this praxis that leads a representative thinker such as Dharmakīrti to claim that the Buddha, whose view he and his successors claim to propound, is a true embodiment of the sources of knowledge. Thus, far from seeing a tension between empirical scrutiny and the exercise of reason, the Buddhist epistemological enterprise positions itself not merely as a dialogical disputational method for avoiding unwarranted beliefs, but as a practice aimed at achieving concrete, pragmatic ends. As Dharmakīrti reminds his fellow Buddhists, the successful accomplishment of any human goal is wholly dependent on having correct knowledge.

Later figures of the tradition

The Buddhist philosophers who are part of this pramāṇavāda tradition include numerous other figures who followed Dignāga and Dharmakīrti. They developed their theories further, commented on their works and defended their theories against Hindu and Buddhist opponents.

Fyodor Stcherbatsky divided the followers and commentators on Dharmakirti into three main groups:

• The philological school of commentators, these figures (such as Devendrabuddhi and Śākyabuddhi) focused on "exactly rendering the direct meaning of the commented text without losing oneself in its deeper implications". They all commented on the Pramāṇavārttika.
• The Kashmiri school of philosophy, which sought to "disclose the deep philosophic contents of the system of Dignāga and Dharmakīrti, regarding it as a critical system of logic and epistemology." Its founding figure was Dharmottara (8th century), a philosopher from Kashmir who wrote some independent works and also a commentary on Dharmakīrti's Nyāyabindu and on his Pramanaviniscaya.
• The religious school of commentators, who sought to "disclose the profound meaning of Dharmakirti's works and to reveal their concealed ultimate tendency." Unlike the Kashmiri school, which saw Dharmakīrti's work as primarily focused on epistemology and reasoning, the "religious" school used Dharmakīrti in order to develop and comment on the entirety of the metaphysics of Mahayana Buddhism. The founder of this school was the layman Prajñakaragupta (740–800 C.E.), an important and original thinker who introduced various new perspectives into the Pramāṇavāda tradition, such as backwards causation. He is the author of the large commentary, the Pramāṇavārttikālaṅkāra ("Ornament of the Pramāṇavārttikā").

Some of the other figures of the epistemological school include:

• Īśvarasena, a disciple of Dignāga, and teacher of Dharmakīrti
• Śaṅkarasvāmin, wrote an introduction to Dignāga's logic
• Jinendrabuddhi (7th or 8th century), a commentator on Dignāga's Pramanasamuccaya
• Bāhuleya, a commentator on Dignāga's Nyāyamukha
• Śubhakara (650–750), was particularly noteworthy because he composed a work which aimed at proving the objective reality of external things and thus attempted to disprove Vijñānavāda (the doctrine of consciousness, idealism)
• Śākyabuddhi (ca. 700 C.E.), wrote a commentary on Dharmakīrti's Pramāṇavārttika
• Chandragomin, purported author of the *Nyāyasiddhyāloka
• Anandavardhana, wrote a sub commentary to Dharmottara's Pramana-viniscaya commentary.
• Vinītadeva (8th century), wrote a commentary on Dharmakīrti's Nyāyabindu
• Śāntabhadra, wrote a commentary on Dharmakīrti's Nyāyabindu
• Jinamitra, wrote a commentary on Dharmakīrti's Nyāyabindu
• Devendrabuddhi (7th century), wrote various commentaries, including one on Dharmakīrti's Pramāṇavārttika
• Karṇakagomin, wrote a commentary on Dharmakīrti's Pramāṇavārttika
• Manorathanandin, wrote a commentary on Dharmakīrti's Pramāṇavārttika
• Śakyamati, wrote a commentary on Dharmakīrti's Pramāṇavārttika
• Arcaṭa, wrote a commentary on Dharmakīrti's Hetubindu
• Jina, a follower of Prajñakaragupta
• Ravigupta, a follower of Prajñakaragupta
• Yamari, a follower of Prajñakaragupta
• Śubhagupta (720–780), was a Vaibhāṣika writer on pramana who wrote a proof of the external world
• Śaṅkaranandana (10th century), a prolific author of at least 17 texts, known as "the second Dharmakīrti."
• Jñanasrimitra (975–1025), a "gate-scholar" at Vikramashila who wrote several original works
• Paṇḍita Aśoka (980–1040)
• Jñanasribhadra (1000–1100), wrote a commentary on the Pramāṇaviniścaya (Dharmakīrti)
• Jayanta (1020–1080), author of the Pramāṇavārttikālaṅkāraṭīkā, a commentary on Prajñakaragupta's text.
• Jitāri or Jetāri (940–1000), teacher of Atisha and author of numerous pramana texts.
• Durvekamiśra (970–1030), a disciple of Jitāri
• Ratnakīrti (11th century), a student of Jñanasrimitra
• Mokṣākaragupta (11th–12th centuries), author of the Tarkabhāṣā
• Vidyākaraśānti (1100–1200), author of the Tarkasopāna
• Śākyaśrībhadra, a Kashmiri pandita who was the teacher of the Tibetan Sakya Pandita

Influence and reception

Dignāga also influenced non-Buddhist Sanskrit thinkers. According to Lawrence J. McCrea, and Parimal G. Patil, Dignāga set in motion an "epistemic turn" in Indian philosophy:

In the centuries following Dignāga’s work, virtually all philosophical questions were reconfigured as epistemological ones. That is, when making any claim at all, it came to be seen as incumbent on a philosopher to situate that claim within a fully developed theory of knowledge. The systematic articulation and interrogation of the underlying presuppositions of all knowledge claims thus became the central preoccupation of most Sanskrit philosophers.

The Hindu philosophers, especially those of the Nyāya, Vaiseshika and Vedanta schools, were in constant debate with the Buddhist epistemologists, developing arguments to defend their realist position against the nominalism of the Buddhists. Nyāya-Vaiseshika thinkers such as Uddyotakara and Prashastapada critiqued the views of Dignaga as they developed their own philosophy.

Vācaspati Miśra's Nyāya-vārtika-tātparya-tikā is almost entirely focused on outlining and defeating the arguments of the Buddhist epistemologists. Prabhākara (active c. 6th century) meanwhile, may have been influenced by Buddhist reasoning to move away from some of the realistic views of older Mīmāṃsā thought. The Vedanta scholar Śrīharṣa who attacked the realism of Nyāya may have been influenced by the Buddhists as well. Even the "New Reason" (Navya Nyāya) scholar Gaṅgeśa Upādhyāya shows an influence from the Buddhist epistemological school, in his arrangement of his Tattvacintāmaṇi.

Svātantrika Mādhyamika

Bhāvaviveka

Ācārya Bhāviveka Converts a Nonbeliever to Buddhism, Gelug 18th-century Qing painting in the Philadelphia Museum of Art^[39]

Bhāvaviveka (c. 500 – c. 578) appears to be the first Buddhist logician to employ the "formal syllogism" (Wylie: sbyor ba'i tshig; Sanskrit: prayoga-vākya) in expounding the Mādhyamaka view, which he employed to considerable effect in his commentary to Nagarjuna's Mūlamadhyamakakārikā entitled the Prajñāpradīpa. To develop his arguments for emptiness, Bhāvaviveka drew on the work of Dignāga which put forth a new way of presenting logical arguments.

Bhāvaviveka was later criticized by Chandrakirti (540-600) for his use of these positive logical arguments. For Chandrakirti, a true Mādhyamika only uses reductio ad absurdum arguments and does not put forth positive arguments. Chandrakirti saw in the logico-epistemic tradition a commitment to a foundationalist epistemology and essentialist ontology. For Chandrakirti, a Mādhyamika's job should be to just deconstruct concepts which presuppose an essence.

Yogācāra-svātantrika Mādhyamika

In spite of Chandrakirti's critique, later Buddhist philosophers continued to explain Madhyamaka philosophy through the use of formal syllogisms as well as adopting the conceptual schemas of the Dignaga-Dharmakirti school along with those of the closely related Yogacara school). These figures include Jñanagarbha (700–760), Śāntarakṣita (725–788), Kamalaśīla, Haribhadra and Ratnākaraśānti (c.1000). Another thinker who worked on both pramana and Madhyamaka was the Kashmiri pandita Parahitabhadra.

This tendency within Madhyamaka is termed Svātantrika, while Chandrakirti's stance is termed Prasangika. The Svatantrika-Prasaṅgika distinction is a central topic of debate in Tibetan Buddhist philosophy. In Tibetan Buddhism, those who follow this method and also make use of Yogācāra doctrines are called Yogācāra-Svātantrika Mādhyamika (Tibetan: Rnal ’byor spyod pa’i dbu ma rang rgyud pa).

Probably the most influential figure in this tradition is Śāntarakṣita. According to James Blumenthal

Śāntarakṣita attempted to integrate the anti-essentialism of Nāgārjuna with the logico-epistemological thought of Dignāga (ca. 6th c.) and Dharmakīrti (ca. 7th c.) along with facets of Yogācāra/Cittamātra thought into one internally consistent, yet fundamentally Madhyamaka system.

This synthesis is one of the last major developments in Indian Buddhist thought, and has been influential on Tibetan Buddhist philosophy.

In the Tibetan tradition

Ngok Loden Sherab

Tom Tillemans, in discussing the Tibetan translation and assimilation of the logico-epistemological tradition, identifies two currents and transmission streams:

The first is the tradition of the Kadampa scholar Ngok Lodzawa Loden Shayrap (1059–1109) and Chapa Chögyi Sengge (1109–69) and their disciples, mainly located at Sangpu Neutok. Chapa's Tshad ma’i bsdus pa (English: "Summaries of Epistemology and Logic") became the groundwork for the "Collected Topics" (Tibetan: Düra; Wylie: bsdus grwa) literature, which in large part furnished the Gelugpa-based logical architecture and epistemology. These two scholars (whose works are now lost) strengthened the influence of Dharmakirti in Tibetan Buddhist scholarship.

There is also another tradition of interpretation founded by Sakya Pandita (1182–1251), who wrote the Tshad-ma rigs-gter (English: "Treasury of Logic on Valid Cognition"). Sakya pandita secured the place of Dharmakirti's Pramanavarttika as the foundational text on epistemology in Tibet. Later thinkers of the Gelug school such as Gyeltsap and Kaydrup attempted a synthesis of the two traditions, with varying results. This is because the views of Chapa were mostly that of Philosophical realism, while Sakya pandita was an anti-realist.

Lexicon

This article should specify the language of its non-English content, using {{lang}}, {{transliteration}} for transliterated languages, and {{IPA}} for phonetic transcriptions, with an appropriate ISO 639 code. Wikipedia's multilingual support templates may also be used. See why. (April 2023)

• Argument: Vada, rtsod pa
• Basis of cognition: Alambana
• Characteristic: laksana, mtshan nid
• Condition: pratyaya, rkyen
• Causal function, purpose: arthakriyā
• Debate: Vivada
• Demonstrandum: sadhya, bsgrub par bya ba
• Demonstrator: sadhaka, grub byed
• Dialectician: tartika, rtog ge ba
• Dialectics: tarka, rtog ge
• Direct perception: pratyaksa, mngon sum
• Event: dharma, chos
• Event-associate: dharmin, chos can
• Exclusion: Apoha, sel ba (Anya-apoha: gzhan sel ba)
• Exemplification: drstanta, dpe
• Inference: anumana, rjes su dpag pa
  • Inference for oneself, reasoning: svārthānumāna
  • Inference for others, demonstration: parārthānumāna
• Interference: vyavakirana, 'dres pa
• Invariable concomitance: avinabhava, med na mi 'byun ba
• Judgment: prajnanana, shes-rab
• Justification: hetu, gtan-tshigs
• Means of valid cognition: pramana, tshad ma
• Means of evidence: linga, rtags
• Particular: svalakṣaṇa
• Pervading/pervasion/logical pervasion: vyapti, khyab pa
• Perception, Sensation: pratyaksa
• Universal, General attribute: Samanyalaksana