Graham Priest's "The Fifth Corner of Four"

A truth that's told with bad intent \ Beats all the lies you can invent.

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.

Define a logic system within a propositional language. It consists of a set of truth values (V), designated values (D), and truth functions for different connectives. An interpretation (ν) assigns truth values to propositional parameters and extends this assignment to all language formulas using truth functions. A statement A is considered a logical truth if, for all interpretations, A always results in a designated value. 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."

Priest describes a logical system with connectives ¬, ∧, and ∨. He explains how A ⊃ B can be defined as ¬A ∨ B. In this system, there are specific truth values (V) like t, f, b, and n, with designated values (D) being t and b. Functions f¬, f∧, and f∨ are used to determine truth values for different logical operations. Priest also mentions that this system can be viewed as a relational logic, where evaluations are relations (ρ) between propositional parameters and {1, 0}. Formulas can be considered true (⊢+) or false (⊢-) with respect to these evaluations. He also provides rules for determining the truth value of compound statements and defines when a set of formulas (Σ) entails another formula (A). 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."

In a plurivalent logic, the system is defined by a structure (V, D, {fc : c ∈ C}), similar to many-valued logic. However, in plurivalent logic, an interpretation is a one-to-many relation (⊳) between propositional parameters and the set of truth values (V), meaning that each propositional parameter can relate to multiple values in V. This relation (⊳) is extended to relate all formulas to values in V pointwise. The plurivalent consequence relation is denoted as ⊢p, and a formula A is considered designated by ⊳ if there exists a value v such that A ⊳ v. Therefore, the statement Σ ⊢p A holds if, for all possible interpretations ⊳, if every member of Σ is designated by ⊳, then A is also designated by ⊳. 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 am sorry, if you want a satisfying conclusion. Priest should have talked about Buddhist demons like Mara who is used metaphorically, psychologically, and literally in different traditions to embody mental states called Kleshas that cloud the mind and prevent wholesome actions, as well as the entirety of conditioned existence, and also death. That surely could be related to non-classical logic and even the catuskoti just as much as his historical accounts and discussion of adjacent metaphysics like interpenetration with respect to Indra’s Net. I stand by the influence of esoteric or mystical elements on logical and metaphysical formulations on the symbolic and imaginary levels.