All is one: a logical/mystical explanation

Image by the beautiful, spiritually awake artist Alex Grey.

This post concerns the idea that all is one, that all distinctions are illusory, yet also real. I humbly consider this to be the deepest spiritual and philosophical truth I’m aware of. I believe that mystical experience can reveal this truth to humans, in the sense of providing sense, evidence, and justification for it. I believe this idea, which appears logically absurd, reconciles beautifully and completely with logic, and can in fact be justified using logic alone.

I am presenting the idea from two angles: a mystical angle and a logical angle. The key phenomena being studied are mystical experiences and logical paradoxes.

This post can be motivated from a logical perspective and a mystical perspective.

  1. A logical motivation is to provide an intuitively satisfying explanation of logical paradoxes, to complement and extend the technically and practically adequate solution to logical paradoxes of my previous post, Paradoxes and the rules of logic.
  2. A mystical motivation is to reconcile and marry with logic the proposition that all is one. This proposition is a formulation of nondualism, which is a common species of philosophical idea coming from mysticism.

I’ll explain how I view mystical experience as providing justification for the idea that all is one. I’ll also provide a logical, philosophical argument for the same conclusion, in the section titled “An argument for the Law of One.” That argument is not at all dependent on mysticism; it relies only on logical considerations.

The assumptions and conclusions of this post won’t appeal to everybody. If something about this doesn’t sit right with you, then I advise just leaving it where you found it.

 

The Law of One

In this post I’m concerned with one particular idea which I think can be repeatably supported by mystical experience.

It is the idea of the unity of all things.

It is called the Law of One.

It is called nondualism.

It is called the Tao.

It can be stated as, “all is one.”

It can be rationalized as follows.

Real world systems are highly complex and entangled. Things affect each other in so many ways that to predict the behavior of any part of the universe, you must ultimately be able to predict the behavior of the whole universe. A complete and wholly accurate point of view would have to consider the whole universe as one single and inseparable thing which is not a simple sum of its parts.

The universe is fractally self-similar. If you look throughout nature, you can find similar structures at different levels. A tree’s branches resemble a neuron’s branching dendrites. Some computer-generated pictures of the universe bear a certain resemblance to artistic images of networks of neurons (such as below), and to computer-generated maps of the Internet.

dendrites

Each cell in a human body contains instructions (DNA) for producing another human body. Each human in a society contains an incomplete copy of that society’s body of learning and experience, in the form of language ability, education, socialization, shared memories (history), acquired skills, and so forth.

There are a few examples of how the universe is fractally self-similar. Behind these few and paltry examples, is there a deeper orderliness to the universe that human science has yet to understand? In any case, the idea of the fractal self-similarity of the universe provides a way of illustrating the idea of the unity of all things.

The logical extreme of the idea of the unity of all things states that any two distinct things are in reality identically the same thing. Each part of the universe is one and the same as every other part of the universe. If so, any appearance of separation and distinctness is some form of illusion, and in reality there is only unity.

I believe the truth of the Law of One, the idea of the unity of all things, has been revealed to me in mystical experience. I believe the idea in all forms just described, including the logical extreme of the idea of the Law of One, that any two distinct things are identically the same thing. I believe mystical experience has revealed the truth of these forms to me.

If any two distinct things are identically the same thing, then in my opinion it follows that every statement is true. Consider the following argument. Let A and B be statements, and let A be a true statement. For example, let A be the statement “the sun has energy” and B be the statement “the sun has no energy.” By the assumption that any two distinct things are the same thing, A and B are the same thing. In other words, B is A. Since A is true, and B is A, B is true. That is, the sun has no energy. By the same reasoning pattern one can arrive at the conclusion that every statement is true. I believe this follows from the Law of One and that in the final analysis of reality, every statement is true.

How can I, somebody who has studied the cutting edge in logic and philosophy of logic, believe that every statement is true? How can this square with logic, common sense, or anything? If you believe the Law of One, then it’s a paradox that the Law of One entails every statement. I apply my general method of solving paradoxes to this problem. For more information on how my perspective on logic integrates with my perspective on the Law of One, you can skip to the section titled “An argument for the Law of One.”

 

Epistemic status of this post for me

As stated already, I believe the perspective I’m articulating here. It is what I have arrived at after seeking truth on the relevant topics to the best of my ability for around seven years. I don’t think everybody will share the intuitions that make me believe the perspective. Some people just won’t agree with me, maybe for reasons neither of us can explain.

If you don’t agree with what I’m saying and you can explain why, I’m interested in hearing it.

On the other hand, as far as I can tell after years of thinking about it, the perspective articulated in this post is irrefutable. That doesn’t mean I can prove it; it just means that as far as I can tell, nobody can refute it. I don’t expect people to be able to use words to sway me from the view, and I don’t expect my words will sway everybody towards the view. Yet I remain open to being surprised.

Suppose I’m right that the perspective of this post is irrefutable. In such a situation, the disagreeing parties are likely to feel they have little choice but to agree to disagree. Maybe one or the other is right, or maybe neither is right, but the parties don’t necessarily have any way to resolve the disagreement. I don’t assume this is the case between me and everybody who disagrees with me on this. But I think there are many people with whom I would be in such a philosophical stalemate if we were to discuss this with each other. I rationalize my co-existence with humans of such thoroughly conflicting perspectives by observing that I am fallible, others are fallible, and life is still a great mystery to all of us. We all have our opinions, but none of us know everything, and in my opinion absolute certainty about anything is beyond the ability of humans to attain.

 

Mysticism, mystical experiences, and mystical revelation

Mysticism has various interpretations and aspects. For the purposes of this post I am concerned with mystical experiences, practices intended to create them, and philosophical ideas that grow up around mystical experiences. Those are the aspects of mysticism that will play a role in this post’s discussion.

Mystical experiences are a variety of subjectively powerful experiences. I won’t try to give a definition of “mystical experience.” For examples of the types of experiences people call mystical, see The Mystical Experience Registry.

My first mystical experience occurred the first time I used LSD. I experienced a form of consciousness which felt so deeply, intensely real that by comparison all my prior experiences seemed unreal. I later learned to reproduce similar states of consciousness at will, through meditation and other mystical disciplines. This immediate and subjectively irrefutable sense of touching on a deeper reality is a hallmark of mystical experiences for me.

Mystical experiences are frequently interpreted in religious, spiritual, or philosophical terms. People who have mystical experiences often take the experiences to reveal to them something about themselves, their lives, and/or the world. For example, here is a quote from the Protestant mystic Jacob Boehme, via The Mystical Experience Registry:

The gate was opened to me that in one quarter of an hour I saw and knew more than if I had been many years together at a university…For I saw and knew the being of all beings…I saw in myself all the three worlds, namely the divine…the dark…and the external and visible world..And I saw and knew the whole working essence, in the evil and the good and the original and the existence of each of them…

Can mystical experiences be taken to reveal reality? I think the answer is that certainly they can, at least to the extent any experiences can be taken to reveal reality. Mystical experiences, to me, provide a raw view of reality, at a higher level of concentration than the level of “ordinary,” sober experiences. For me there is a continuity between mystical experiences and “ordinary,” sober experiences. Mystical experiences, for me, are distinguished by the greatest vividness, the greatest density, the greatest intensity and certainty of awareness, and therefore I assume my mystical experiences to constitute particularly rich subjective views into reality.

Everybody who recalls having mystical experiences has the responsibility, if they choose to accept it, to figure out what if anything the experiences tell them about the world.

I think mystical experiences can appropriately be used as evidence to support philosophical conclusions. However, anybody who lacks the type of mystical experience used to support a conclusion is likely to find this type of argument for the conclusion unpersuasive.

The evidence (if any) which mystical experiences provide for conclusions seldom has much influence on people who did not themselves have the relevant experience(s). If I have a mystical experience A which appears to me to support philosophical conclusion X, and you yourself haven’t had an experience like A, does my report of having such an experience as A provide any evidence, for you, for conclusion X? Maybe so, maybe not. I certainly think you are free to conclude that it doesn’t provide evidence for you.

Can mystical experiences form the evidentiary basis for conclusions which transfer from person to person in a repeatable fashion? I think so. I think it requires that people repeatably be able to obtain mystical experiences which support the conclusions in question. That is, it requires chains of people inspiring each other to reach the same conclusions on the basis of reproducible types of experiences. I believe that religious and spiritual memes are often spread by means of processes much like this. Perhaps the same is true of some philosophical memes.

There is no doubt in my mind that sometimes people reach false conclusions on the basis of mystical experience. I assume it’s possible to find examples where different people have reached different, contradictory conclusions on the basis of mystical experience. This is true, for example, if somebody has had a mystical experience which they took to reveal that the Catholic Church teaches the only true religion of God, and somebody else has had a mystical experience which they took to reveal that Sunni Islam teaches the only true religion of God.

The proposition, that mystical experiences can lead people to false conclusions, does not in my view undermine the idea that mystical experiences can be regarded (along with other experiences) as views into the truth. People are able to misinterpret their experiences and overreach their evidence in all kinds of ways to arrive at false conclusions. I don’t think mystical experiences are any different in this regard, and I think this can explain why people arrive at false conclusions on the basis of mystical experiences.

What is harder to explain is how to tell when a mystical experience can reasonably be assumed to provide evidence for a conclusion. Mystical experiences are subjective. They can’t be adequately described in words. Their meaning can’t be analyzed with the machinery of logic. As such, if mystical experiences convey truth, one might assume that that truth can’t be adequately expressed in words or adequately analyzed with the machinery of logic. I think the Law of One is a truth of this nature; as I’ve interpreted the principle, it defies logic and renders words useless by entailing that every statement is true.

When all is said, whether or not a mystical experience supports a conclusion is going to be a matter of subjective judgment and personal opinion.

 

Verifying the Law of One

I believe the reader might arrive at the conclusion that the Law of One is true by producing and observing appropriate mystical experiences. I have met several people who have reached the same or similar conclusions as mine, inspired by mystical experiences of their own.

The quickest and easiest way you might try to get experiential evidence of the Law of One would be to induce a mystical state of consciousness and reflect on questions like, “who am I?” and “what is all this I am aware of?” and “what is that which is witnessing all this?” Contemplate the possible truth of the equations God = I, You = I, Subject = Object. Picture the universe as a single, indivisible object.

If you don’t know how to induce a mystical state of consciousness, the quickest and easiest way may be to take a hallucinogenic drug, such as (for example) LSD, magic mushrooms, nitrous oxide, DXM, or DMT, with an appropriate set and setting, I would do this exercise when you are alone and in a peaceful frame of mind.

This procedure is not perfect. Our sober selves are not necessarily ready to believe the conclusions of our drug-affected selves, and perhaps that skepticism is warranted. Therefore I offer no warranty of suitability for purpose for the quick and dirty method just described for verifying the Law of One.

I would suggest the following steps to a general individual wishing to embark on a laborious and life-encompassing effort of spiritual growth which I hypothesize will probably lead them to experiences confirming the truth of the Law of One, if it is true and the individual wants the truth of the matter. Following these steps entails a commitment to a life of spiritual growth which trends to color all moments. If you faithfully follow these steps and you do not receive confirmation of the Law of One, nonetheless I would suggest the thought that by faithfully following these steps you can hardly avoid receiving great spiritual, emotional, and intellectual rewards in this life (to say nothing of the afterlife). Therefore I suggest it would not be wasted effort even if you do not find my hypothesis to be true in your case.

  1. Take what steps may be needed to be in good physical fitness, as much as practicable.
  2. Shun dishonesty and immorality. As much as practicable, leave any situations you may be in which compel you to be morally corrupt, dishonest, or immoral.
  3. Lead a well-examined life marked by continual and deep self-scrutiny and moral reflection.
  4. Accept and love yourself, and accept and love those in your life, those in your thoughts, and all of existence.
  5. Continually work towards peace, progress, and higher levels of awareness in all aspects of your life.
  6. With faith and determination, practice meditation, concentration, and the deliberate raising of consciousness. Let this practice integrate ever more deeply and pervasively into your life.
  7. Do whatever things inspire the spirit in you.
  8. Let yourself go through life in a natural and unstudied manner.
  9. Reflect seriously and with single-pointed concentration on questions like:
    1. Who am I?
    2. What is all this that I am aware of?
    3. What is that which is witnessing all this?

Each of the steps 1-8 is a type of preparation which I believe will contribute to success in the spiritual exercise listed as step 9. Step 9 is an activity I hope will bring you to realization of the Law of One, if you are open to the possibility, and if the practice of the exercise is in the context of a path of seeking spiritual truth which encompasses and transforms one’s life and being.

The procedure I’m suggesting eventually requires total commitment of the self. Spiritual growth transforms one’s whole being. If it fails to do so, then I expect spiritual growth to be stopped. Spiritual growth transforms the mind and body, and in time it colors all moments.

The length of time the procedure takes to yield success at realizing the Law of One may vary. Such a realization might happen shortly after starting the procedure; or in a matter of months; or in a matter of years; or perhaps not at all.

Unfortunately I can’t guarantee this procedure will cause you to realize the Law of One. I merely predict that it should typically do so, if you’re genuinely interested in the truth of the matter, and you’re prepared and seeking to experience that truth.

Commitment to spiritual growth and practice led me to mystical consciousness which now colors all my experience. Today, by concentration I am able to reproduce at will the intuitions upon which I believe in the Law of One. For me these intuitions are a readily accessible mental proof of the Law of One. The intuitions are a synthesis of years of mystical and intellectual experience around the Law of One, in which I have questioned the idea intensely and from many angles.

Everybody who replicates these subjective insights will come to them by a different path, and I don’t doubt that different people’s subjective insights will always be subjectively different. My path to these insights has been by a spiritual growth process featuring an approximation of total commitment of the self. As awkward as it may be, I cannot prescribe a procedure for reproducing my conclusion which does not resemble the process I myself followed.

I do believe there are other ways of arriving at the conclusion which require less than whole-being commitment to spiritual growth. People may have mystical experiences spontaneously, and through the use of drugs, and as a consequence of difficult emotions, and from many other causes. A single mystical experience might cause somebody to believe in the Law of One. A more convincing kind of evidence is repeated mystical experiences evidencing the Law of One, and the ability to produce at will the intuitions that motivate the belief. Some people might have that type of evidence naturally.

For some readers, it might not be too hard to follow my steps because they are already following most of them. Such readers may already believe the Law of One. Or, they may be in a good position to try out my proposed exercises in step 9, and see if they obtain insight into the Law of One. They might do all this and conclude that the Law of One is false in one or more of the formulations I’ve mentioned.

I believe that the Law of One is a repeatable mystical insight, because the insight appears to have been repeated again and again in variants throughout history, and I have met several people of diverse histories who have had variants of the insight. I have never come across anybody who put the Law of One into words in just the way I have, interpreting it as logically entailing that every statement is true, but I don’t think that means the fundamental insight is new to me. The source which has given me the greatest philosophical inspiration around the Law of One is The Law of One, also known as the Ra material. This text, even if it is a work of fiction, is to me the greatest and most educational philosophical and psychological work I have experienced.

In all of this I see no way of reproducing the conclusion that the Law of One is true, which I can guarantee will work for anybody. I also see no way of falsifying or refuting the Law of One. I can understand the frustration this may cause Law of One skeptics to feel. I am unsure what comfort I can offer such a person. All I can say is that I’m honestly trying to convey reality as I see it, and I recognize in fairness that I may be in error, but if so I don’t and probably won’t recognize my error.

At the end of the day, those who want to believe the Law of One will believe it, and those who want to reject the Law of One will reject it. Whether the Law of One is true is not a question that has been settled by the scientific method. From my current perspective, it is a spiritual and philosophical question, to be decided by individuals, who will base their conclusions on subjective feelings at the end of the day.

I am curious to hear about anybody’s thoughts and experiences around the Law of One and/or mystical revelation.

 

An argument for the Law of One

The rest of this post provides another way of reaching the conclusion that the Law of One is true, by logical analysis and philosophical argumentation instead of by mysticism. The argument I’ll give does not force the reader to accept its conclusion. There are other alternative conclusions the reader can draw. I will present such alternative conclusions. I am not sure how many people will find this philosophical argument for the Law of One compelling in the absence of mystical experience of the Law of One.

I am interested to hear about your reaction to the following argument. I don’t know what to expect people to think about it.

The following argument for the Law of One can alternately be viewed as an explanation of why logical paradoxes happen.

In a nutshell, the perspective of the argument/explanation states: paradoxes allow us to prove the Law of One, and the Law of One explains why we observe paradoxes.

Explaining this requires some background on logic. Here I will present this in simplistic broad strokes. More detail can be found starting from the companion post, Paradoxes and the rules of logic.

 

Background on logic

A “statement” is a piece of language which can be true or false.

An “argument” consists of a series of statements, called premises, intended to support or provide evidence for another statement, the conclusion of the argument.

A “valid” argument is an argument such that it is logically necessary that if its premises are true, then its conclusion is true.

A central question in logic is: what arguments are valid?

The orthodox answer to this question is called classical logic. Classical logic is a species of rules for logic, which allows a wide class of ordinary language and formal language arguments to be analyzed as valid or not.

There are other species of rules for logic, which differently answer the question of what arguments are valid. These are so-called “non-classical logics.” For most non-classical logics, the set of arguments they deem valid is a subset of the set of arguments classical logic deems valid.

Classical logic has a consequence known as the principle of explosion. This is the fact that according to classical logic, from a contradictory set of premises one may validly infer any conclusion.

A contradiction is a statement of the form “P and not P.” For example, “I am alive and I am not alive.” According to classical logic, if a set of premises entails a contradiction, then those premises entail any statement. From the premise “I am alive and not alive,” classical logic says I can infer “the sky is purple.”

The orthodox understanding of this phenomenon is that contradictions cannot ever be true, and so if a set of premises entails a contradiction, then some of those premises are false. Given false premises, logic allows you to infer false conclusions. Garbage in, garbage out.

 

Logical paradoxes

Committing oneself to classical logic and the orthodox understanding of the principle of explosion leads to challenges in dealing with logical paradoxes.

Here is a logical paradox. Consider the statement “this statement is false.” Is it true or false? If it’s true, then it’s false. If it’s false, then it’s false that it’s false, so it’s true. Classical logic says the statement is either true or false, and therefore it’s both true and false, and therefore every statement is true. This argument can be called the liar paradox.

This is a stunningly short argument for the conclusion that every statement is true, phrased in simple English, predicated on the background assumption that classical logic, as straightforwardly applied to English, is correct.

This argument might be taken as demonstrating that something about the background assumptions is wrong. This is the usual way of looking at things in the academic literature I’ve seen. One can say that classical logic is not correct, or that classical logic can’t be straightforwardly applied to English (but perhaps it can be cleverly applied to English in a way that deals with paradoxes).

In Paradoxes and the rules of logic I gave a refinement of the background assumptions which tries to use as little cleverness as possible. In short, I said that we should view the rules of classical logic not as being totally exceptionless, but as having occasional exceptions, in particular in the vicinity of paradoxes. I said these paradox-avoiding exceptions could safely be produced in an ad hoc manner, whenever one encounters contradiction-producing paradoxes such as the liar paradox.

 

Logical paradoxes as proofs of the Law of One

Now I wish to put forward a different way of looking at the meaning of the liar paradox. This is to view the liar paradox as proving that everything is true.

If the Law of One is true, in the extreme formulation which says that any two distinct things are identical, then everything is true. This conditional statement can be reached by a variety of arguments. Here is one. Suppose the Law of One is true. Let A be any statement. Let B be some true statement, e.g., “water contains hydrogen.” By the Law of One, A and B are the same thing. Since B is true, A is true. In other words, every statement is true.

The Law of One (in its logically extreme formulation), and the proposition that every statement is true, are logically interchangeable propositions, in the sense that each entails the other.

Belief in the Law of One, and belief in the liar paradox as a valid argument to the conclusion that every statement is true, cohere with each other. They provide two different routes (one mystical-logical and one purely logical) towards the conclusion that everything is true.

However, if one accepts every statement as true in every context, then it defeats the purpose of language. Statements are part of language. Language exists to serve life. Life has experiences of separateness and partiality. If separateness is an illusion, still it is an inescapable illusion for us.

Any notion of achieving a purpose supposes that there is something which now is not and later could be. Therefore all purposeful action supposes a lack of belief in the necessary future actual existence of some state of affairs.

For those reasons, one can’t always accept the perspective that everything is true. In some contexts, such as contexts where one is trying to achieve some purpose in the material world, it is appropriate not to assume that everything is true, but to assume that for the purposes at hand, only some things are true.

It is in such contexts — including most contexts — that I apply the perspective of Paradoxes and the rules of logic, according to which I reject the arbitrary statements that can be inferred from paradoxes, to retain a context where not everything is considered true.

It is in meditation that I apply the perspective that all is one and everything is true.

The energy of the thought of the unity of all things colors and enlivens my life from moment to moment.

 

Theoretical options

That concludes my explanation of the perspective that logical paradoxes allow us to prove the Law of One, and the Law of One explains why we observe logical paradoxes.

There are many ways to look at the situation other than the one presented here. You can reject the Law of One. You can accept some way of looking at paradoxes that doesn’t involve assuming everything is true. You can ignore the issue of paradoxes.

I don’t claim to be proving this perspective. I’m offering this perspective for consideration. It appears to me to be the most elegant solution I’ve seen for solving and explaining paradoxes. It appears to me to be a successful theoretical integration of mystical intuitions which are very powerful in me, with a logical system of thought which appeared hard to reconcile with those intuitions. This perspective appears to me to be true to the best of my ability to discern truth, but that judgment is subjective.

The perspective of Paradoxes and the rules of logic, with its ad hoc method of rejecting contradictions, is free-standing from the perspective of this post. You can accept that post’s theory without accepting this post’s theory. I think that by taking that route you lose an intuitively grounded story about why we observe paradoxes.

Personally, I favor the perspective of this post because of my mystical intuitions. For me this whole investigation into paradoxes and logic has been motivated by the desire to better understand the Law of One and how it can be reconciled with logic. I have figured out to my own satisfaction how to reconcile the Law of One with logic. Are you satisfied?

I consider this project to have value from a mystical perspective, for bolstering the analytic philosophical defensibility of nondualism. I doubt if my reasoning will convert any skeptics who firmly don’t want to believe in mysticism, but I hope it will provide inspiration and clarity to people who perceive truth in nondualism.

 

Mystical arrogance and chauvinism

To my mind, there is one objection to the perspective I’ve laid out which is the biggest and stickiest. This perspective asserts the primacy of mystical consciousness — the distinctive consciousness which mystics are supposed to experience — as the best source of fundamental metaphysical truth available to humans. This can be viewed as chauvinist and arrogant.

Ayn Rand lays out the objection in the final chapter of Introduction to Objectivist Epistemology:

In the history of philosophy — with some very rare exceptions — epistemological theories have consisted of attempts to escape one or the other of the two fundamental questions which cannot be escaped. Men have been taught either that knowledge is impossible (skepticism) or that it is available without effort (mysticism). These two positions appear to be antagonists, but are, in fact, two variants on the same theme, two sides of the same fraudulent coin: the attempt to escape the responsibility of rational cognition and the absolutism of reality — the attempt to assert the primacy of consciousness over existence.

How do I respond to this objection?

It’s everybody’s choice how they regard mysticism, and how they regard the mystical philosophy I’ve shared in particular. People can choose to perceive the Law of One as true, or to perceive it as untrue. Those who don’t believe in mysticism are free to choose to view mystical philosophy as chauvinist and arrogant. Of course I may feel discouraged if people make this choice, but I don’t begrudge them their freedom.

Do I view mystical philosophy as at all chauvinist or arrogant? I think it easily can be. Perhaps it is always at least a little bit of each.

I have tried to minimize the arrogance and chauvinism of my mystical philosophy. I have spent the past seven years thinking about what I believe on the topic. I have tried to avoid arrogantly rushing to a conclusion. I am an ideologue without a doubt, but I have tried my best to be a stable, reasonable ideologue. I have sincerely and seriously tried to minimize the arrogance and chauvinism of my philosophy. I could have done more, by abandoning mysticism, but that’s not the choice I made. My success is yours to judge.

 

Closing song

My belief in nondualism is fundamentally based on experience, on a type of feeling which feels like evidence. In another attempt to communicate this species of feeling, I want to point the reader to a song which to me conveys the sense of nondualism, in hopes the reader finds a similar perspective on the song.

The song is Joni Mitchell’s “Both Sides Now.” I myself prefer Carly Rae Jepsen’s cover version. Here are the lyrics.

Please enjoy the mystery of existence, and share your thoughts in the comments!

Paradoxes and the rules of logic

Consider the statement “this statement is false.” Is it true or false? Some reflection will show that if it’s true, then it’s false, and if it’s false, then it’s true. What should we make of that? Is it both true and false? Or else what’s going on?

This is a classic logical problem known as the “liar paradox.” The liar paradox, and paradoxes like it, have been subject to immense philosophical attention stretching back thousands of years.

In this post I’m going to present a solution to this classic problem. In a nutshell, I say, ignore it, like almost everybody does. The length of the post is spent in making this solution rigorous and explaining the basis of my opinion that this is the best way of solving the liar paradox and similar paradoxes, for most purposes.

The workings of the solution lead to a picture of logic which is potentially surprising and new to people. In this picture, the rules of logic have exceptions, the rules of logic are subject to legitimate subjective differences of opinion, and the rules of logic are not completely rigorous and formal, but are made up as we go along to at least some extent. In the course of the work I hope to convey a deeper explanation of this picture.

I’m not trying to provide a full picture of how the practice of logic should work under the assumption that the rules of logic are not completely rigorous and formal. I’m providing a solution to paradoxes which exploits that assumption. I hope for other work which builds out a bigger picture of how logic can work when it’s not completely rigorous and formal. I have sketched only a little fragment of what I imagine to be a big space of potentially interesting problems and solutions.

 

The problem

In this post, by “paradox” I will mean, “an argument which proceeds from apparently true premises to a contradictory conclusion.” A contradictory conclusion is a statement of the form (P and (not P)), asserting that a statement P is both true and false.

Here is an example of a paradox: the liar paradox. Consider the statement “this statement is false.” I’ll refer to this statement as L. If L is true, then L is false. If L is false, then L is true. L is either true or false. Therefore L is true and L is false, or in other words, (L and (not L)) is true. The preceding argument, which I call the liar paradox, has a contradictory conclusion, so it is a paradox.

Other examples of paradoxes include Russell’s paradox and the Sorites paradox. In this post I will focus mainly on the liar paradox as a chosen example of a paradox.

This post explains a general method of solving paradoxes.

What does it mean to solve a paradox? Every paradox presents a problem: it appears to imply that something false is true, which is impossible, at least on a traditional understanding of logic. Nothing impossible can occur, but paradoxes appear to be cases where something impossible occurs. In that sense, paradoxes are intellectual, theoretical problems. To solve a paradox is to solve this problem, for example by pointing out which of the paradox’s premises are not true, or by explaining why its conclusion does not follow from its premises.

 

This theory, this post, and their history

I don’t claim the theory I’m giving is fundamentally new. I think it is basically a technical extrapolation of the common way of responding to paradoxes like the liar paradox, according to which these paradoxes can safely be ignored.

I didn’t base this theory on anybody else’s work.

I first described this post’s method of solving paradoxes in my book Winning Arguments. This post aims to describe the method in a quicker and more surveyable way, putting together thoughts that are scattered throughout a wide span of text in Winning Arguments.

This post is aimed at a general philosophical audience. It is not very technical and it doesn’t engage deeply with the scholarly literature on paradoxes. There are over 2,000 scholarly works about the liar paradox cited on PhilPapers. This post talks about one solution to the liar paradox, which is in competition with a huge number of solutions discussed in the weighty corpus of paradox research. I have kept the comparison with competing solutions brief and superficial, relative to the extent of the literature.

I studied the topic of paradoxes in the University of Connecticut Philosophy PhD program, where I learned about paradoxes among lovely and intelligent philosophers including, for example, world paradox experts Jc Beall and David Ripley. I published research related to paradoxes in two of the top journals related to the topic: the Journal of Philosophical Logic and the Review of Symbolic Logic.

The solution to paradoxes which I’m describing in this post is something I began conceiving around 2011, during my undergraduate studies at Arizona State University. I have been developing the idea since then. By the time I left UConn in 2015, I had all the pieces of the theory conceived. It took me until 2017 to put the pieces together, to believe the resulting theory, and to put it down in writing.

This post does not provide my full theory of paradoxes. It only provides the first piece of the theory. This piece of theory is called the ad hoc method of rejecting incorrigible paradoxes, or the ad hoc method, for short.

The ad hoc method deals only with the technical aspect of solving paradoxes. The other part of the theory, to come in a later post, deals with the more philosophical question of why paradoxes happen and how we can explain them.

 

Summary of the ad hoc method of rejecting incorrigible paradoxes

Some arguments have apparently true premises and contradictory conclusions, but some of their premises, though apparently true, are false. An appropriate method of solving such paradoxes is to identify which of the premises are false.

In contrast, this post is mainly concerned with incorrigible paradoxes. An incorrigible paradox, by definition, is an argument which leads from true premises to a contradictory conclusion via correct rules of deductive logical inference.

The notion of an incorrigible paradox is implicitly relative to: (a) some language in which arguments can be formulated, (b) some opinion about what rules of deductive logical inference are correct, and (c) some opinion about what statements are true. You will get different, sometimes irreconciliable notions of what constitutes an incorrigible paradox by starting from different choices of (a), (b), and (c).

If a paradox is not incorrigible, then either it has at least one false premise, or its conclusion does not follow logically from its premises, and solving the paradox is a matter of finding a false premise or using logical methods to demonstrate that the conclusion doesn’t follow from the premises.

If a paradox is incorrigible, then the ad hoc method of rejecting incorrigible paradoxes recommends rejecting the conclusion of the paradox and all further inferences from it, understanding this rejection to be an ad hoc modification/exception to the rules of logic one employs.

 

Assumptions

The notion of an incorrigible paradox is implicitly relative to some language, some opinion about what rules of logical inference are correct, and some opinion about what statements are true.

In this post, I’m going to choose English as the language in which arguments can be formulated.

I’m going to assume that the rules of classical logic, as described for example by the system LK, are correct rules of deductive logical inference.

I’m going to assume a lot of common sense about what’s true. I will assume that the following statements, which I construe as premises of the liar paradox, are true:

  1. The statement L = “This statement is false” is an English statement.
  2. If L is true, then L is false.
  3. If L is false, then L is true.

From these premises it follows by the rules of classical logic that L is true and L is false, and that (L and (not L)) is true.

The rules of classical logic are always described relative to some formal language, such as the language of first-order logic used in the system LK. In this post I’m assuming the rules of classical logic can be construed as rules governing use of the English language.

Since English isn’t based on a set of precise rules that cover all cases, it is practically impossible to give a precise description of the rules of classical logic as they apply to English.

When logicians apply the rules of classical logic to English, they generally rely on a precise understanding of the rules of classical logic as understood through some formal description such as the system LK, and on a more artistic understanding of how to translate between English and a formal language such as first-order logic. This mapping between English and the formal language yields an imprecise method for applying the rules of classical logic to English. This imprecise method is precise enough to be unproblematic for typical purposes, and for our purposes.

According to the assumptions of this post, an incorrigible paradox is an English-language argument whose premises are true and whose conclusion follows from the premises by the rules of classical logic. By assumptions, the liar paradox is an example of an incorrigible paradox.

The solution to paradoxes of this post will work for basically any choice of language, rules of logic, and concept of truth. The assumptions of this post, as described in this section, are primarily for the purpose of making the discussion concrete. If you dislike the assumptions chosen in this section, you can still use the solution to paradoxes I’m describing, replacing English with your language of choice, replacing classical logic with your logic of choice, and using your choice of assumptions about what statements are true.

 

The ad hoc method of rejecting incorrigible paradoxes

To solve any paradox by the method this theory prescribes, you start by determining whether or not the paradox is incorrigible. This requires answering two questions: whether the premises are true, and whether the conclusion follows logically from the premises. Answering whether the conclusion of an argument follows from the premises by the rules of classical logic is basically a routine process if you are familiar with appropriate techniques. Answering whether the premises are true is in general more complex, and I don’t provide any general method for doing that.

If you show that a paradox is not incorrigible, then you’ve solved the paradox. Either you’ve shown that its premises or not all true, or you’ve shown that the conclusion doesn’t follow logically from the premises. Either way, you’ve shown that there is no problem.

If your analysis concludes that a paradox is incorrigible, then the next step of the method is to reject the conclusion of the argument. One makes the stipulation that the paradoxical argument is to be rejected, even though its premises are true and its logical inferences are valid. This is an ad hoc modification/exception to the rules of logic which one employs.

One ends up having not only positive rules of logic, according to which certain inferences are correct, but also negative rules of logic, which supersede the positive rules by stating that inferences which are valid in general are invalid in some particular cases (the cases of incorrigible paradoxes).

This method leaves the rules of logic one employs perpetually incomplete. If you employ this method, then the rules of logic you follow are unlike (for example) the rules of the system LK in that you lack a precise and complete description of the rules of logic you follow. If you employ this method, then you have the expectation that if you encounter new incorrigible paradoxes in the future, then you will need to extend the rules of logic you employ, by adopting new negative rules to block the new incorrigible paradoxes.

In classical logic, it is possible to infer any statement from a contradiction. If you can infer a statement of the form (P and (not P)) from a given set of premises, then you can infer any statement whatsoever from those premises. This is called the principle of explosion.

Because of the principle of explosion, the ad hoc method of rejecting incorrigible paradoxes also involves rejecting any variants on a paradoxical argument which can (in other variations) be used to prove any statement. For every argument whose conclusion is a contradiction, there are many variations with conclusions including all statements. All such variants are rejected, in the method.

People may arrive at differing results by this method. Differences on what constitutes an incorrigible paradox may arise from different opinions on what rules of logic are correct, different opinions on what statements are true, and different judgments about what constitutes a variation on an incorrigible paradox, among other possible sources of differences. Different judgments about what constitutes a variation on an incorrigible paradox may arise because I don’t offer any general method for making such judgments; rather, I assume there to be some ineliminable element of artistry involved in making such judgments. This assumption is justified later.

These sources of potential variability in people’s results from this method may be viewed as a consideration against the use of this method. Of course, people arrive at different results in reasoning due to all kinds of causes. I view the mentioned consideration against this method as a valid one, to be weighed against other considerations (and of course I believe this method has considerations in its favor that outweigh the considerations against, for general purposes).

According to the ad hoc method of rejecting incorrigible paradoxes, one augments the rules of logic one follows on an ad hoc basis by adding negative rules which exclude the inferences to the conclusions of incorrigible paradoxes. One does this as needed. One also rejects all variations on incorrigible paradoxes which can be used (in further variations) to prove arbitrary statements. I don’t provide strict rules on how to draw the line between such variations on incorrigible paradoxes, and other arguments; I assume there is an ineliminable element of artistry in drawing the line. In other words, I assume that the technique of drawing the line can’t be made completely formal.

Some appropriate questions:

  1. How does the ad hoc method of rejecting incorrigible paradoxes compare to other ways of solving paradoxes? Why use this method over competing methods?
  2. What’s the reason for the assumption that there is an ineliminable element of artistry in identifying variations on incorrigible paradoxes which can be used (in further variations) to prove arbitrary statements?
  3. What are the risks involved in using the ad hoc method of rejecting incorrigible paradoxes?
  4. Is there any intuitive picture that explains why this method is correct?

Subsequent sections will speak to these questions. Here is a summary of the subsequent sections.

 

Non-classical logic and its successes and limitations with paradoxes reviews some of the successes and limitations of some competing approaches to solving paradoxes, which are based on restricting the rules of classical logic in some formally defined way, in the form of a so-called non-classical logic. This section looks (in a very high level way) at the pros and cons of approaches using non-classical logic, as compared to the ad hoc method of rejecting incorrigible paradoxes. This section speaks to questions 1, 2, and 3.

Implications for reasoning talks at a high level about how this approach to paradoxes works in the context of everyday practical reasoning. This section speaks to question 3.

Implications for math talks at a high level about how this approach to paradoxes works in the context of math. This section speaks to question 3.

I will speak to question 4 in a subsequent post.

 

Non-classical logic and its successes and limitations with paradoxes

Classical logic is the prevailing perspective on logic. Classical logic is a system of rules of inference, which exists in many variants, and which is used, taught, and studied far more often than any other paradigm of rules of logic.

A non-classical logic is any system of rules for logical reasoning which is not a form of classical logic. Non-classical logics are a huge and diverse category. PhilPapers cites over 6,000 works on non-classical logics.

An early and important type of non-classical logic is intuitionistic logic. Roughly, this is the logic you get by removing from classical logic the principle that every statement is either true or false.

Another important type of non-classical logic is relevance logic. Systems of logic known as relevance logics aim to disallow inferences known as “paradoxes of relevance,” such as the inference (valid in classical logic) from the premise “the sky is blue” to the conclusion “if the sky is red then the sky is blue.”

Neither intuitionistic logic, nor most relevance logics, solve the liar paradox and similar paradoxes.

I’ll focus on two categories of non-classical logics which get traction in solving paradoxes such as the liar paradox. These are paraconsistent/paracomplete logics, and substructural logics.

As far as I’m aware, these are the best studied categories of paradox-solving logics, with no competing categories of comparable prominence in the academic literature. Each of these categories includes many different formal systems of logic, and the literature around each of them is extensive and technical. I’ll only provide a very high-level survey of what is going on in these subfields of logic, without going into technical details about any of the systems.

For a more detailed survey of the topic of non-classical logics, I would recommend Graham Priest’s An Introduction to Non-Classical Logic. That book does not discuss substructural logics. For an entry point into the field of substructural logics, I would recommend Greg Restall’s article in the Stanford Encyclopedia of Philosophy. Now I’ll proceed to my nutshell survey of these fields.

A paraconsistent logic is a system of logic in which some statements can be both true and false, without every statement being true, and without every statement being false.

An example of a paraconsistent logic is LP, the Logic of Paradox. LP is a system of logic in which there are three possible “truth values:” true, false, and paradoxical (meaning both true and false). Every statement is either true, false, or paradoxical.

LP provides simple rules for determining the truth value of any statement as a function of the truth values of the sub-statements that make it up, provided one knows the truth values of the smallest sub-statements. For example, a sentence of the form “A or B” is has the following truth value:

  • “True,” if A is true and B is true.
  • “False,” if at least one of A or B is false and neither is paradoxical.
  • “Paradoxical,” if at least one of A or B is paradoxical.

Notice that these three possibilities cover all the possibilities, so that the truth value of “A or B” is always determined by the truth value of A, the truth value of B, and the given rules.

LP is similar in design to classical logic, but in LP, you can let some statements be both true and false, without all statements being so. LP allows this by having rules that are less restrictive than those of classical logic; they let you infer less from the same premises, so that in particular the principle of explosion is not valid in LP.

A major problem with LP is that its rules are too weak to allow the development of math in the usual manner of proving theorems from a limited set of axioms. I have proven facts about LP which illustrate this general phenomenon. That LP is too weak to allow the development of math in the usual manner is such a broad and imprecise statement that I doubt it can be stated completely and precisely, much less proven.

LP is an example of a paraconsistent logic, meaning it allows for true contradictions without every statement being true.

Moving on to another type of non-classical logic: a paracomplete logic is a logic in which some statements can be neither true nor false.

K3, or Kleene 3-valued logic, is an example of a paracomplete logic. In K3, there are three possible truth values: true, false, and neither. K3 has simple rules, exactly parallel to LP’s, for determining the truth value of any statement from the truth value of the statements making it up, provided one is given the truth values of the smallest statements.

Kripke’s “Outline of a theory of truth” is an example of applying K3 to solve paradoxes like the liar paradox. Kripke’s solution works as far as it goes, but it doesn’t go as far as giving an operational explanation of how to reason in the presence of paradoxes, and it’s not clear how to extend it to work for paradoxes of set theory such as Russell’s paradox.

In any case, K3 is an example of a paracomplete logic, one which allows for statements which are neither true nor false.

Some logics are both paraconsistent and paracomplete. FDE, or First-Degree Entailment, is a simple example. It has four truth values: true, false, both, and neither. It works like a fusion of LP and K3.

Substructural logics are another class of non-classical logics, usually defined in the form of a sequent calculus. A sequent calculus is a manner of defining a system of rules of logical inference. It is based on patterns called “sequents.”

A sequent is a pattern of the form “A1,…,An entails B1,…,Bm,” where A1,…,An is a sequence of zero or more statements, and B1,…,Bm is another sequence of zero or more statements.  Usually the word “entails” is written as a turnstile. A basic way of reading the sequent “A1,…,An entails B1,…,Bm” is as saying “if A1,…,An are all true, then at least one of B1,…,Bm is true.”

A sequent calculus tells you how to produce all sequents that are “valid,” by repeatedly applying a small set of rules. The system LK is an example of a sequent calculus, one which defines the rules of classical logic.

The system LK tells you what arguments are valid according to classical logic, if you can translate the arguments into the language of first-order logic. Take an argument to consist of one or premises A1,…,An, and a single conclusion B, with the claim being that the premises are true and B follows from them. Suppose A1,…,An and B are written in the language of first-order logic. Then the argument in question is valid according to classical logic, meaning its conclusion follows logically from its premises, if and only if “A1,…,An entails B” is a valid sequent in the sense that it can be produced according to the rules of the system LK.

Sequent calculi have rules known as “structural rules.” In the system LK, there are the following structural rules:

  • Weakening: If you add statements to the premise or conclusion side of a valid sequent, then you get another valid sequent.
  • Contraction: If a statement occurs repeatedly on the premise side or conclusion side of a valid sequent, then you can drop extra occurrences and still have a valid sequent.
  • Permutation: You can change the order of the premises and conclusions in a valid sequent, and get a valid sequent.

There is also the rule of cut, which is not always classified as a structural rule. It is harder to summarize, but you can look at it in Wikipedia’s presentation of LK.

Substructural logics are logics which relax/weaken some of the structural rules of a sequent calculus, with the system LK being a prototypical starting point.

One reason some people are interested in substructural logics is that some people see them as a promising avenue for solving paradoxes.

One example of an approach to paradoxes based on substructural logic is an approach advanced by Dave Ripley and others, articulated for example in Ripley’s paper Paradoxes and failures of cut. This approach starts with an LK-like sequent calculus, and removes the rule of cut.

This approach avoids the problem seen in the case of LP, that the weakening of logic makes too many things unprovable, so you can’t get things like math off the ground. It avoids the problem because the rule of cut is not necessary for the completeness of the system LK. One can remove the rule of cut from the system LK and exactly the same sequents are valid (derivable) in the resulting system. This is Gentzen’s cut elimination theorem. The catch is that sequent derivations not using the cut rule may need to be much longer than derivations of the same sequent using the cut rule.

The interesting observation is that by starting with this cut-free sequent calculus, one can add a concept of truth which gives rise to the liar paradox, without causing the system to explode the way that classical logic does. Ripley’s theory shows how to reason in the presence of the liar paradox, in a way that works and is operationally understandable.

Ripley has also applied his approach to other types of paradox, including paradoxes of vagueness and paradoxes of set theory. Perhaps it can be deemed a generally successful approach to paradoxes. In my opinion, the only part of Ripley’s approach that is not evidently successful is the approach to the paradoxes of set theory. It remains to be seen how capable Ripley’s set theory is of proving the usual theorems of math. Many failures and difficulties have been encountered in this area. Similarities between Ripley’s set theory and LP set theory may be a warning sign of limitations similar to those found in LP set theory. However, as far as I’m aware, it remains to be seen either way whether or not this set theory can be used effectively as a foundation for math. Resolving this uncertainty would shed light on how generally Ripley’s cut-free approach can be applied to solve paradoxes. Does it work on all the hardest paradoxes, or does it fail to apply to some of the hardest paradoxes?

I have discussed only a small and biased selection of approaches to solving paradoxes using non-classical logic. I have discussed each approach only to a superficial extent. We have only skimmed the surface of this subject. I, personally, have not read most of the academic literature on paradoxes. I am limited by the desire to keep this post’s length reasonable and this section not overly laborious or time-consuming for the reader.

Now I will make some general comments on how the ad hoc method of rejecting incorrigible paradoxes compares to approaches to solving paradoxes based on non-classical logic.

I am not aware of any approach to paradoxes based on non-classical logic which solves all of the hardest paradoxes, including the liar paradox, the Russell paradox, etc., which approach is uniform across such paradoxes, and which approach results in a method that is practically usable. I do not rule out the possibility that such an approach can be constructed. However, my own suspicion is that it won’t happen. My own suspicion is that the paradoxes are an intellectual trick of God which no finite system can completely contain or address. I have no proof of that opinion. My opinion is that the opinion can’t be proven, it can’t be proven that it can’t be proven, and so forth.

All currently existing non-classical logic solutions to paradoxes that I’m aware of, insofar as they are successful, are successful in a limited area. On the other hand, the method I’ve proposed can be applied uniformly to all paradoxes. I’m not aware of a non-classical logic approach to paradoxes which can claim the same.

All non-classical logic approaches to paradoxes involve the complexity inherent in non-classical logic. Learning how to apply the full formal rules of classical logic, as expressed e.g. through the system LK, is a lot to ask of people. Non-classical logics provide more stuff to learn, and they are usually more complex and/or less intuitive and/or harder to apply than classical logic. In most non-classical logic solutions to paradoxes I have seen, there is no clear idea of how to operationalize and apply the ideas. In the cases I’ve seen and can recall, when there is an idea of how to operationalize the solution in principle, there is still not a fleshed out idea of how to do it practically.

In contrast, the method I’ve proposed is easy to apply. Indeed, I think the ad hoc method of rejecting incorrigible paradoxes is the method which most people instinctively apply when presented with the liar paradox. The method is so easy to execute that it doesn’t even need to be taught.

Considering these things, I think it’s fair to say that from a practical perspective, the method I’ve proposed has distinctive advantages over non-classical logic approaches to paradoxes. I think most people will probably be sympathetic to my assessment that at least at this time, the method I’ve proposed is more suitable for most practical purposes, compared to non-classical logic approaches.

Where the non-classical logic approaches are better than the ad hoc method of rejecting incorrigible paradoxes is mainly in the theoretical qualities of exactness, formality, and certainty. The non-classical logic approaches provide systems of rules that are not ad hoc, but in principle handle all cases, in their domains of applicability. Meta-logical theorems can provide assurances that certain undesired outcomes will not result from applying the rules.

On the other hand, the method I’ve proposed is inherently inexact, and there is the possibility that applying it will fail to protect one from falsehoods that can be inferred from paradoxes, if one somehow unintentionally invokes a paradox in one’s reasoning. This is an area where non-classical logic approaches have the advantages.

In summary, the ad hoc method of rejecting incorrigible paradoxes has the advantages that it is easy to learn and apply and it works uniformly across all cases, whereas the non-classical logic approaches have the advantages of exactness, formality, and certainty.

The question becomes, which considerations are more important? This depends what is important to you.

If your interest in paradoxes is theoretical, and an exact solution to the problems is simply what you want, then you’ll deem that more important than ease of learning and having a general, easily usable solution on the table.

If you’re interested in this topic for some extrinsic reason, e.g. because you want to understand how to reason in general for all purposes, then you will likely favor the ad hoc approach I’ve presented, because you can apply it today to all your paradoxes. Naturally, it’s up to you how you approach these problems, and how much time you spend thinking about them. I would not discourage you from taking whatever approach to paradoxes seems to you to suit your goals.

In my view, the approaches to paradoxes based on non-classical logic have had such tough going because they are going against a general principle, one I call the law of no perfect system.

In general, the law of no perfect system is that there is no perfect system for doing anything. Whatever your problem is, if it’s above a certain level of complexity, then there is no finite and exact set of rules which solves the problem optimally in all cases.

As applied to this case, there is no perfect set of rules for logic which has everything it can correctly have and nothing else, defined formally and finitely.

Gödel’s first incompleteness theorem can be construed as the law of no perfect system as applied to axioms for math. It implies that there is no set of axioms for math, which can be listed by a computer program, which prove everything true and nothing false about math.

The law of no perfect system, as a general rule of thumb, tells you that it is proper for rules to have exceptions. Because no system of rules is able to handle everything, each particular situation should be considered as a particular situation where exceptions to rules might properly apply.

I think the general law of no perfect system is fairly common sense, but as applied to logic, it goes against the grain of the academic literature. I’m not aware of other defenses, besides the present one, of the law of no perfect system as applied to logic. Please let me know of any you’re aware of.

 

Implications for reasoning

What does the ad hoc method of rejecting incorrigible paradoxes have to tell us about general reasoning for practical purposes?

As far as I can tell, the theory has virtually no implications in this area. For practical purposes, people already reason as if they were following the ad hoc method of rejecting incorrigible paradoxes.

By using the ad hoc method, one accepts the possibility that one will be led from truth to falsehood by following the rules of logic, by stumbling across some unseen paradox. I’m not aware of cases like this, but it is a theoretical possibility that this will happen to somebody and that it will matter for their purposes.

One accepts a certain level of risk and uncertainty around one’s use of logic by using the ad hoc method. To put it in perspective, most people carry a much higher level of risk and uncertainty around their use of logic, because they probably haven’t learned any set of rules for logic, and they probably don’t subscribe to any theory about how to solve paradoxes.

 

Implications for math

The ad hoc method of rejecting incorrigible paradoxes, or for short the ad hoc method, has some practical implications for some theoretical math.

Using the ad hoc method has consequences for set theory. The method lets you do math using naive set theory, in which every set you can describe exists, avoiding the need to use a set theory like ZFC whose paradox-avoiding mechanisms introduce complications and limitations.

Using the ad hoc method has consequences for category theory. Category theory has conventionally used paradox-avoiding mechanisms of varying levels of complexity in rigorous presentations. The ad hoc method tells us that there is no need to do this. Paradoxes arise naturally from category theory, but they can be handled by the usual ad hoc method, without a need for paradox-avoiding restrictions. Category theory’s paradox avoiding restrictions usually limit what categories you can talk about, preventing you from talking about sensible categories such as the category of all sets, the category of all groups, or the category of all categories. It’s nice to do away with the paradox avoiding restrictions.

Using the ad hoc method has consequences for type theory. For example, the original impredicative formulation of intuitionistic type theory gave rise to a paradox, Girard’s paradox. Subsequent systems introduced paradox avoiding restrictions which reduced the expressive flexibility of the systems. As in other cases, these paradox avoiding restrictions can be done away with if one is comfortable with the ad hoc method of rejecting incorrigible paradoxes.

This method can even be applied to dependent type theory as applied to software development. People already use logical methods in this context which bear some resemblance-like relation to the ad hoc method of rejecting incorrigible paradoxes.

For example, Idris is a computer programming language based on dependent type theory. In Idris, you can prove theorems in your code, and in particular you can apply this capability to prove that your software has desired properties. In Idris, you can prove any statement. Yet, this is unlikely to happen by accident. Arguably, Idris’ ability to prove any statement is an outcome of overall-desirable design tradeoffs. Normally you wouldn’t use that ability, and the fact that it’s there does not prevent you from drawing confidence from the proofs which you write and Idris verifies. From such a proof, formulated in a normal fashion free of prove-anything back-doors, you can draw practical certainty that the statement proven is true.

Although it can prove any statement, Idris contains paradox avoiding restrictions (a hierarchy of nested type universes). As far as I can see, those paradox avoiding restrictions are not necessary. They’re not necessary to keep the system from proving arbitrary statements, because Idris developers have built the capability to prove any statement into the system anyway. Probably there is approximately nothing to lose from dropping the paradox avoiding restrictions and including the axiom Type : Type which allows paradoxes to be proven. This modification would (arguably) make the system simpler and nicer to use.

Please comment if you’re aware of paradox theories similar to the one laid out here, and to share any paradox research that you think I might find interesting.