Summary: although Boolean algebra works perfectly well for singular subjects, things in the world are not singular: they have multiple parts. We argue that the application of a Boolean predicate to a compound subject (i.e. a physical object) should result in a compound truth value. The four compound truth values are {true}. {false}, {true,false}, and {}. Interestingly, these truth values correspond to a set of logical outcomes used in ancient India called the catuskoti.

Boolean logic is a fantastic symbolic calculus, and we will not challenge its basic tenets. However, it is not sufficient to adequately describe all of our cognitive operations: the human mind is composed of more than just the rational mind.

In particular, Boolean logic applies to singular (unitary) subjects, and from a Madhyamaka point of view, there are no non-composite subjects in reality. Although the conceptual mind imputes singularity, all existing objects are composed of a plurality of parts. Boolean logic, for all of its merit, was not designed to apply to non-singular subjects.

So from the Buddhist point of view, modern logic is not incorrect, but it is a logic which applies only to concepts. A more general logical system with four outcomes, called the catuskoti (meaning “four corners”) has been used for debate in India before Buddhism began.

The catuskoti offers a plurality of logical outcomes: {true}, {false}, {true,false} and {} (i.e. neither true nor false). Some things are also said to be beyond this analysis (ineffable), but we will of course not speak of those. A modern logician would of course balk at the notion of “true and false” and “neither true nor false” as logical outcomes, since they are not logical outcomes with a Boolean algebra: they are contradictory within a logical system in which the premise must resolve to a single truth value.

Boolean algebra is appropriate to synthetic (a priori) statements or deductions. To use a classical example, if all men are mortal, and Socrates is a man, then we may conclude with certainty that Socrates is mortal. Statements about the world, however, are less certain. For example, the way that the name of an object corresponds to its object poses a problem known in philosophical circles as the problem of identity. To put it simply, if a name is taken to refer to a particular thing, and that thing changes while the name does not, then we have some uncertainty with respect to the correspondence of the name and the object.

Let us take an egg as an our subject of analysis. Conceptuality, the egg is a single thing, but of course it corresponds to a region of space filled with egg-matter which is multiple (or at least divisible). We then apply the logical predicate “yellow” to the subject, which corresponds to the question: is the egg yellow? On a classical Boolean analysis, we have the following possible answers:

- True: the inside of the egg (the yolk) is yellow.
- False: the outside of the egg (the egg white or shell) is not yellow.

If we are determined to reduce this answer to a single truth value, we might say that these two answers should be combined with one of the Boolean operators, either AND or OR:

- True: the egg is yellow because some part of it is yellow (true OR false).
- False: the egg is not yellow because some part of it is not yellow (true AND false).

There is a problem with this logic, however. Suppose we have another predicate, not-yellow, and ask the same question:

- True: the egg is not-yellow because some part of it is not-yellow (true OR false).
- False: the egg is not not-yellow because some part of it is not not-yellow (true AND false).

The problem should be clear: we obtain the same answer asking if the egg is yellow that we do when we ask if it is not-yellow, which entails a contradiction (yellow, because it is a non-compound predicate, is subject to the law of the excluded middle). There is simply no way to simplify the situation (consistently and nonparadoxically) when we start with a subject that is non-uniform with respect to the answer.

A mereological logic (which is better suited to traditional Buddhist analysis) a would answer the question in terms of parts of the object, in which case we have:

- {True, False}: the egg is yellow because a part of it is yellow, and it is not yellow because a part of it is not yellow.

Again, the answer {true,false} is the result of applying a binary (logical) predicate to a *compound* subject: an object which has the parts {yellow, not-yellow}. In traditional Buddhist texts, this answer is expressed as the combination of a predicate and its negation, or “yellow and not yellow”.

There is one more outcome to the catuskoti argument, which is “neither yellow nor not-yellow”. One example of a subject which results in this answer is a word that does not correspond to any actual object. Take, for example, the horns of a rabbit:

- The horns of a rabbit are not yellow
- The horns of a rabbit are yellow

Both of these statements are equally meaningless: a rabbit does not have horns. Therefore, there is no basis to say anything about them: since they do not have a color whatsoever, we cannot say that this non-existant color is yellow or not yellow.

When something and it’s nonexistence

Both are absent from before the mind,

No other option does the latter have:

It comes to perfect rest, from concepts free.

~Shantideva

**Bibliography**:

- Graham Priest, Beyond True and False
- Shantideva, Way of the Bodhisattva

I believe this corresponds to Sir Roger Penrose’s non-algorithmic theory of consciousness explicated in his book Shadows of the Mind. He proposes a real scientific theory for how the mind does not work like a computer by using mathematical theorems such as Godel’s incompleteness theorem. The mechanism for which he thinks this works is controversial, but I think the abstract point still stands.

Also, as a Buddhist, job well done!

Forgive me for posting on such an old article, but it is very interesting so I couldn’t resist. Within Western logic, some people have had problems with vacuous truth (“the rabbit’s horns are yellow” being considered True), and so a system called Intuitionistic Logic was developed, which avoids vacuous truth as well as double negation and proof by contradiction.

I learned about this mostly through inquiry in basic-level classes, so I’m unfamiliar with the relation between intuitionistic logic, modal logic, and catuskoti.

However, from my experience, which perhaps someone can clarify or add on, it seems that modal logic seeks to generally help logic fit better into natural language by adding tense (will be, was) as well as possibility (might be).

It seems “might be” is not the same as either {true, false} or {}, though the Catuskoti mentioned here, like Modal Logic, seeks to help logic live up better to the world and human language by adding more options for truth values.

In terms of the specific situations addressed, such as ambiguous (vacuous) situations, it seems Catuskoti and Intuitionistic Logic both seek to avoid claiming vacuous truth, though Intu.. simply avoids stating it (and situations that would imply it, such as double negations), while Catuskoti positively states neither True nor False when nothing can be claimed, correct?

I’m happy you posted. This work is a bit old, and I have a better treatment on the way, but in the mean time:

* I agree with what you say about intuitionistic logic, and I’m trying to do a similar thing with mathematics in general (for example, point-sets are not a good mental model for space; see the Wikipedia entry on point-free topology).

* Modal logic adds possibility and necessity; I was not aware that it added tense. It is thus an approximation of fuzzy logic, which uses non-binary truth values.

* You state that “It seems ‘might be’ is not the same as either {true, false} or {}, though the Catuskoti mentioned here, like Modal Logic, seeks to help logic live up better to the world and human language by adding more options for truth values.” Modal logic is probably better seen as adding modifiers to truth values rather than adding entirely new truth values. The catuskoti definitely adds two new truth values; both and neither.

* With respect to the catuskoti, it is a very old logic, and I am not an expert in it. However, I have studied the philosophical context from which it arises. That context was strongly influenced by subjective observations (how do we know things) and belief that things in the world are not singular.

So I think {} could be treated as the truth value Unknown; this is done in some ternary logics such as Priest or Kleene logics. It may be a good fit for intuitionistic logic, although I do not know the formal definition of vacuity in that logic. This accommodates subjectivity within Boolean logic.

To accommodate non-singular predicates, as the madhyamika logicians who invented the catuskoti would want to do, is what requires the truth value {true,false}. As far as I can tell, though, I am alone in backing that hypothesis (the professional logicians I’ve written to about it have not responded).

Personally, I think logic as taught in our schools should have an Unknown value. In that context, an affirming negation operates as usual, turning True and False values into each other, but there is also an opportunity for non-affirming negation, which turns True and False values into the value Unknown. Whether we need the additional complexity of fuzziness/modality and compound predicates probably depends on the abilities of those to whom it is taught.

Sorry for being pedantic, but I want to clarify should anyone else happen upon this. Intuitionistic logic does not affirm double negation elimination (which is equivalent to the Law of Excluded Middle). I assume this is what you meant, but there is nothing preventing discussing “double negation” in intuitionistic logic.

Also, in classical logic, direct proof of negation (i.e. suppose P, get a contradiction, conclude not P) and proof by contradiction (i.e. suppose not P, get a contradiction, therefore not not P, use double negation elimination to conclude P) are both typically referred to as proof by contradiction. The latter is clearly not intuitionistically valid because the final step relies upon applying excluded middle.