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The Fifth Corner of Four: An Essay on Buddhist Metaphysics and the Catuṣkoṭi
Graham Priest presents an exploration of Buddhist metaphysics, drawing on texts which include those of Nãgãrjuna and Dõgen. The development of Buddhist metaphysics is viewed through the lens of the catuṣkotị. At its simplest, and as it appears in the earliest texts, this is a logical metaphysical principle which says that every claim is true, false, both, or neither; but the principle itself evolves, assuming new forms, as the metaphysics develops. An important step in the evolution incorporates ineffability. Such things make no sense from the perspective of a logic which endorses the principles of excluded middle and non-contradiction, which are standard fare in Western logic. However, the book shows how one can make sense of them by applying the techniques of contemporary non-classical logic, such as those of First Degree Entailment, and Plurivalent Logic. An important issue that emerges as the book develops is the notion of non-duality and its transcendence. This allows many of the threads of the book to be drawn together at its end. All matters are explained, in as far as possible, in a way that is accessible to those with no knowledge of Buddhist philosophy or contemporary non-classical logic.
172 pages, Hardcover
Published November 8, 2018
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December 25, 2018
Unspeakably Interesting
This book is a first for me, for there was a lot of new material that I was not familiar with. However, Priest made an effort in explaining and making said material accessible for someone new to this area of philosophy.
The book features an overview of the development of Buddhist metaphysics, and especially the ‘Catuskoti’ from early India (6th or 5th BCE) up to the works of the Japanese philosopher Dōgen Kigen (13th AD). The Catuskoti itself consists of four corners (hence the title), and by four corners it means four possibilities. For the Catuskoti, something can be true, false, both, or neither. This obviously goes against the orthodoxy of Western metaphysics where something can either be true or false, but not both and certainty not neither. In other words, the Catuskoti then seems to violate the principle of non-contradiction and the principle of the excluded middle.
This is not an issue though, and Priest shows how the Catuskoti can hold. In addition, he tries to see what role it plays in Buddhist philosophy as a whole, as Buddhist metaphysics in essence view the reality as a duality, namely that the world we live in is a world that can be spoken of, but ultimate reality is ineffable. The fifth corner then is ineffability, for all other corners can be spoken of.
What I found most interesting of all is how similar discussions have been happening in Western thought too; we may be on to something if different traditions arrive at similar conclusions regarding the effability and ineffability of the world. For example, we have Kant who distinguished the world between noumena and phenomena, Heidegger who was obsessed with being, (early) Wittgenstein who held that there are things that you cannot speak about but only show, as well as A. W. Moore who believes that there are things you can not speak about.
In conclusion, this is a book on metaphysics, and is one that can grasp the interest of a plethora of people. The novelty of Priest is bringing Buddhist thought and making it more accessible for us Westerns, while highlighting that we all have something to learn from one another in philosophy, and that such categorisations, though necessary, should not act as barriers of communication. All in all, a very good and enlightening read.
This book is a first for me, for there was a lot of new material that I was not familiar with. However, Priest made an effort in explaining and making said material accessible for someone new to this area of philosophy.
The book features an overview of the development of Buddhist metaphysics, and especially the ‘Catuskoti’ from early India (6th or 5th BCE) up to the works of the Japanese philosopher Dōgen Kigen (13th AD). The Catuskoti itself consists of four corners (hence the title), and by four corners it means four possibilities. For the Catuskoti, something can be true, false, both, or neither. This obviously goes against the orthodoxy of Western metaphysics where something can either be true or false, but not both and certainty not neither. In other words, the Catuskoti then seems to violate the principle of non-contradiction and the principle of the excluded middle.
This is not an issue though, and Priest shows how the Catuskoti can hold. In addition, he tries to see what role it plays in Buddhist philosophy as a whole, as Buddhist metaphysics in essence view the reality as a duality, namely that the world we live in is a world that can be spoken of, but ultimate reality is ineffable. The fifth corner then is ineffability, for all other corners can be spoken of.
What I found most interesting of all is how similar discussions have been happening in Western thought too; we may be on to something if different traditions arrive at similar conclusions regarding the effability and ineffability of the world. For example, we have Kant who distinguished the world between noumena and phenomena, Heidegger who was obsessed with being, (early) Wittgenstein who held that there are things that you cannot speak about but only show, as well as A. W. Moore who believes that there are things you can not speak about.
In conclusion, this is a book on metaphysics, and is one that can grasp the interest of a plethora of people. The novelty of Priest is bringing Buddhist thought and making it more accessible for us Westerns, while highlighting that we all have something to learn from one another in philosophy, and that such categorisations, though necessary, should not act as barriers of communication. All in all, a very good and enlightening read.
October 26, 2023
Ok, yes, I liked the logic, but first, the Buddhism. I think Priest misses the point that Buddhist thinkers are pretty aware of contradictions, and especially that they are aware of the paradox of ineffability. Like, the issue raised with the catuskoti is more so that the statements are not predicated on anything, even if the argument for this is somewhat of a dialectical movement. Priest even mentions such several times, in favor of putting forth his reading as an engagement with the texts regardless of scholarship as such. It was also kind of funny to see passing mentions of Kant or Hegel.
I will admit, Priest lost me somewhat as he travels between different Indian Buddhist texts to more Chinese Buddhist texts. One example is the mereological reductionism and metaphysical foundationalism of the Abhidharma school of early Buddhism. I understand his use of first degree entailment for the early history of the catuskoti, but then his introduction of the ineffible truth value to first degree entailment with motivation from emptiness in the Perfection of Wisdom sutras and Nāgārjuna’s writings lost me. When he moves to the Jizang hierarchy where the successive universal truth is the negation of the conventional truth, in turn becoming conventional truth, that also seemed only tangential to the catuskoti. The, he uses graph theory to explain Indra’s Net and interpenetration, which seems to get even further from the catuskoti. In the end, he claims Dogen, who talks about meditating and daily living, ties together all of the topics so far, but this was not the effect for me.
Some logical information (even in the so-called technical appendices) seemed a bit glossed over. I read his notorious Logic of Paradox (1979) article which clarified some or at least familiarized me with Priest's method a bit more. For example, I accepted that consequence relations are someone of a black box or a tool to be used rather than picked apart. Priest also does well in distinguishing between names of propositions with angle brackets and the T-schema ( so that T) as well as states of affairs (of propositions) with underlining. These are typographically difficult, but roughly, the correspondence theory of truth boils down to “the sentence named A is true, if the state of affairs that A describes is a fact.
Priest also carefully describes (1) many-valued logics, (2) first-degree entailment (a propositional language that is paraconsistent but uses validity of a sequent to relate formulas in K3 Kleene’s strong logic, and LP Priest’s logic of paradox to determine what is true or untrue with respect to contradiction and excluded middle respectively), and (3) plurivalent logic (which is how Priest works the ineffible truth value into FDE). Priest relates these three aspects of non-canonical logic using the elementary methods of sequential deductions and truth matrices, which he also explains for anyone unfamiliar.
Here is his exposition on many valued logics (pp 30):
"Given some propositional language with a set of connectives, C, a logic is defined by a structure (V, D, {fc : c ∈ C}). V is the set of truth values: it may have any number of members (≥1). D is a subset of V, and is the set of designated values. For every connective, c, fc is the corresponding truth function. Thus, if c is an n-place connective, fc is an n-place function with inputs and outputs in V. An interpretation for the language is a map, ν, from the set of propositional parameters, P, into V. This is extended to a map from all formulas of the language to V by applying the appropriate truth functions recursively. Thus, if c is an n-place connective, ν(c(A1, ... , An)) = fc(ν(A1), ... , ν(An)). Finally, if Σ is a set of formulas, Σ ⊨ A iff there is no interpretation, ν, such that for all B ∈ Σ , ν(B) ∈ D, but ν(A) ∉ D. A is a logical truth iff ∅ ⊨ A, i.e., iff for every interpretation, ν, ν(A) ∈ D."
Here is his exposition on the propositional language of first degree entailment (pp 30):
"The connectives of the language are ¬, ∧, and ∨. A ⊃ B may be defined as ¬A ∨ B. The language may be augmented with quantifiers and additional non-extensional operators, such as modal operators, and a conditional operator. But these play no role in this book, so we may ignore them here. In FDE, V is {t, f, b, n}, and D = {t, b}. f¬ is a function which maps: t to f, f to t, b to b, and n to n. f∧(x, y) is the greatest lower bound of x and y, and f∨(x, y) is the least upper bound of x and y. An equivalent way to set up FDE is, not as a many-valued logic, but as a relational logic. Specifically, an evaluation is now thought of as a relation, ρ, between the set of propositional parameters, P, and {1, 0}. We can define what it is for a formula to be true ⊢+ and false ⊢-, with respect to an evaluation, ρ, as follows. (A formula may be both or neither.)
• ⊢+ p iff pρ1
• ⊢- p iff pρ0
• ⊢+ ¬A iff - A
• ⊢- ¬A iff + A
• ⊢+ A ∧ B iff + A and + B
• ⊢- A ∧ B iff - A or - B
• ⊢+ A ∨ B iff + A or + B
• ⊢- A ∨ B iff - A and - B
• Σ ⊢A iff for all ρ, if ⊢+ B for all B ∈ Σ , ⊢+ A."
Finally, here is his exposition on Plurivalent logic:
"As in many-valued logic, a plurivalent logic is defined by a structure (V, D, {fc : c ∈ C}). But in a plurivalent logic, an interpretation is a one-many relation, ⊳, between propositional parameters and V. That is, every propositional parameter relates to at least one value in V. The relation ⊳ is extended to a relation between all formulas and values in V pointwise. That is:
• c(A1, ... ,An) ⊳ v iff ∃v1, ... vn(A1 ⊳ v1, ... , An ⊳ vn and v = fc(v1, ... , vn))
Since every parameter relates to at least one value, so does every formula. We will write the plurivalent consequence relation as ⊢p. Let us say that ⊳ designates A iff for some v such that A ⊳ v, v ∈ D. Then:
• Σ ⊢p A iff for all ⊳, if ⊳ designates every member of Σ, ⊳ designates A"
I will admit, Priest lost me somewhat as he travels between different Indian Buddhist texts to more Chinese Buddhist texts. One example is the mereological reductionism and metaphysical foundationalism of the Abhidharma school of early Buddhism. I understand his use of first degree entailment for the early history of the catuskoti, but then his introduction of the ineffible truth value to first degree entailment with motivation from emptiness in the Perfection of Wisdom sutras and Nāgārjuna’s writings lost me. When he moves to the Jizang hierarchy where the successive universal truth is the negation of the conventional truth, in turn becoming conventional truth, that also seemed only tangential to the catuskoti. The, he uses graph theory to explain Indra’s Net and interpenetration, which seems to get even further from the catuskoti. In the end, he claims Dogen, who talks about meditating and daily living, ties together all of the topics so far, but this was not the effect for me.
Some logical information (even in the so-called technical appendices) seemed a bit glossed over. I read his notorious Logic of Paradox (1979) article which clarified some or at least familiarized me with Priest's method a bit more. For example, I accepted that consequence relations are someone of a black box or a tool to be used rather than picked apart. Priest also does well in distinguishing between names of propositions with angle brackets and the T-schema ( so that T) as well as states of affairs (of propositions) with underlining. These are typographically difficult, but roughly, the correspondence theory of truth boils down to “the sentence named A is true, if the state of affairs that A describes is a fact.
Priest also carefully describes (1) many-valued logics, (2) first-degree entailment (a propositional language that is paraconsistent but uses validity of a sequent to relate formulas in K3 Kleene’s strong logic, and LP Priest’s logic of paradox to determine what is true or untrue with respect to contradiction and excluded middle respectively), and (3) plurivalent logic (which is how Priest works the ineffible truth value into FDE). Priest relates these three aspects of non-canonical logic using the elementary methods of sequential deductions and truth matrices, which he also explains for anyone unfamiliar.
Here is his exposition on many valued logics (pp 30):
"Given some propositional language with a set of connectives, C, a logic is defined by a structure (V, D, {fc : c ∈ C}). V is the set of truth values: it may have any number of members (≥1). D is a subset of V, and is the set of designated values. For every connective, c, fc is the corresponding truth function. Thus, if c is an n-place connective, fc is an n-place function with inputs and outputs in V. An interpretation for the language is a map, ν, from the set of propositional parameters, P, into V. This is extended to a map from all formulas of the language to V by applying the appropriate truth functions recursively. Thus, if c is an n-place connective, ν(c(A1, ... , An)) = fc(ν(A1), ... , ν(An)). Finally, if Σ is a set of formulas, Σ ⊨ A iff there is no interpretation, ν, such that for all B ∈ Σ , ν(B) ∈ D, but ν(A) ∉ D. A is a logical truth iff ∅ ⊨ A, i.e., iff for every interpretation, ν, ν(A) ∈ D."
Here is his exposition on the propositional language of first degree entailment (pp 30):
"The connectives of the language are ¬, ∧, and ∨. A ⊃ B may be defined as ¬A ∨ B. The language may be augmented with quantifiers and additional non-extensional operators, such as modal operators, and a conditional operator. But these play no role in this book, so we may ignore them here. In FDE, V is {t, f, b, n}, and D = {t, b}. f¬ is a function which maps: t to f, f to t, b to b, and n to n. f∧(x, y) is the greatest lower bound of x and y, and f∨(x, y) is the least upper bound of x and y. An equivalent way to set up FDE is, not as a many-valued logic, but as a relational logic. Specifically, an evaluation is now thought of as a relation, ρ, between the set of propositional parameters, P, and {1, 0}. We can define what it is for a formula to be true ⊢+ and false ⊢-, with respect to an evaluation, ρ, as follows. (A formula may be both or neither.)
• ⊢+ p iff pρ1
• ⊢- p iff pρ0
• ⊢+ ¬A iff - A
• ⊢- ¬A iff + A
• ⊢+ A ∧ B iff + A and + B
• ⊢- A ∧ B iff - A or - B
• ⊢+ A ∨ B iff + A or + B
• ⊢- A ∨ B iff - A and - B
• Σ ⊢A iff for all ρ, if ⊢+ B for all B ∈ Σ , ⊢+ A."
Finally, here is his exposition on Plurivalent logic:
"As in many-valued logic, a plurivalent logic is defined by a structure (V, D, {fc : c ∈ C}). But in a plurivalent logic, an interpretation is a one-many relation, ⊳, between propositional parameters and V. That is, every propositional parameter relates to at least one value in V. The relation ⊳ is extended to a relation between all formulas and values in V pointwise. That is:
• c(A1, ... ,An) ⊳ v iff ∃v1, ... vn(A1 ⊳ v1, ... , An ⊳ vn and v = fc(v1, ... , vn))
Since every parameter relates to at least one value, so does every formula. We will write the plurivalent consequence relation as ⊢p. Let us say that ⊳ designates A iff for some v such that A ⊳ v, v ∈ D. Then:
• Σ ⊢p A iff for all ⊳, if ⊳ designates every member of Σ, ⊳ designates A"
June 21, 2023
An amazing extended essay on Buddhist philosophy through the lens of formal logic. The author, Graham Priest is a professor of philosophy, but a mathematician by training. He wrote THE textbook on non-classical logic, and is a well-known proponent of paraconsistent logic - a logic system that allows contradictions.
The idea that there can be true contradictions and false contradictions is a compelling one. The book opens with Dickens' opening lines of "It was the best of times, it was the worst of times..." Clearly (at least to me personally), some contradictions are truer than others. Turns out that there are multiple ways to define a 'proper' logic system that allows for contradictory statements such as, 'this statement is false' without reducing the entire system to a triviality. What I found surprising is that one such logic system called chatushkoti (the four corners) was very popular amongst ancient Indian and Buddhist philisophers! Chatushkoti allows a statement to be either true or false or both or none - four possible truth values in total.
Anyone who is an admirer of the mystical will have encountered the paradoxical nature of truth/reality in one form or the other. Questions on enlightenment (nirvana) or the ultimate nature of reality (brahman or sunyata) always seem to involve paradoxical statements that somehow escape rigorous logical analysis. A certain notion of 'ineffability' is also almost always attributed to 'God' or 'the ultimate truth'. For instance, take the Vedic mahavakya, 'You are that. (tat tvam asi)' What is 'that'? And why can't it be put in words? This book shows how ancient Indian and Chinese buddhist philosophers tackled such problems. The discussion is fascinating, highly original, and even inspiring at times.
The idea that there can be true contradictions and false contradictions is a compelling one. The book opens with Dickens' opening lines of "It was the best of times, it was the worst of times..." Clearly (at least to me personally), some contradictions are truer than others. Turns out that there are multiple ways to define a 'proper' logic system that allows for contradictory statements such as, 'this statement is false' without reducing the entire system to a triviality. What I found surprising is that one such logic system called chatushkoti (the four corners) was very popular amongst ancient Indian and Buddhist philisophers! Chatushkoti allows a statement to be either true or false or both or none - four possible truth values in total.
Anyone who is an admirer of the mystical will have encountered the paradoxical nature of truth/reality in one form or the other. Questions on enlightenment (nirvana) or the ultimate nature of reality (brahman or sunyata) always seem to involve paradoxical statements that somehow escape rigorous logical analysis. A certain notion of 'ineffability' is also almost always attributed to 'God' or 'the ultimate truth'. For instance, take the Vedic mahavakya, 'You are that. (tat tvam asi)' What is 'that'? And why can't it be put in words? This book shows how ancient Indian and Chinese buddhist philosophers tackled such problems. The discussion is fascinating, highly original, and even inspiring at times.
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