Anti-Paradoxes are Bullshit

Navigating Logical Trickery

TL;DR: This article explores the concepts of paradoxes, anti-paradoxes, and deepities. We'll explore the Liar's Paradox and Belnap's four-valued logic and discuss how understanding these concepts can sharpen critical thinking in everyday life and business.

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Apologies to

David Pinsof

for borrowing his signature title structure. Consider it a form of flattery – after all, imitation is the sincerest form of bullshit.

Pinsof's recent article "Deep Bullshit" struck a chord with many readers, myself included. It got me thinking about the nature of truth, belief, and the words we use to convey them. So, I'd like to take his exploration of deepities a step further and dive into the murky waters of paradoxes, anti-paradoxes, and the bullshit that lies between.

You can read David’s article here:

"Deep" Bullshit

The Liar's Paradox: When Logic Confronts Its Limits

Paradoxes have been a thorn in the side of logicians and philosophers for millennia. These mind-bending statements or situations that seem to contradict themselves have played a crucial role in developing logic, mathematics, and philosophy. From ancient Greece to modern set theory, paradoxes have pushed thinkers to refine their understanding of reasoning and language.

One of the most famous and enduring paradoxes is the Liar's Paradox. You know the one:

"This statement is false."

If it's true, it's false. If it's false, it's true. It's the philosophical equivalent of dividing by zero – it breaks the system.

The Liar's Paradox is like a linguistic black hole – it sucks in logic and reason, leaving only confusion in its wake.

This paradox has been giving logicians headaches since ancient times. The Stoics, those paragons of rational thought, were particularly vexed by it. Chrysippus, one of the most influential Stoic philosophers, devoted considerable attention to this puzzle. After all, for the Stoics, logic wasn't just an abstract intellectual exercise – it was one of the three fundamental branches of philosophy, alongside physics and ethics.

The Liar's Paradox challenges a fundamental principle of classical logic: the law of the excluded middle. This principle states that for any proposition, either that proposition is true or its negation is true. There's no middle ground. But the Liar's Paradox seems to occupy precisely that forbidden territory.

For the Stoics, this wasn't just an abstract problem. Their philosophy emphasized living in accordance with nature and reason. How could one do so when language itself seems to rebel against logic? The paradox threatened the very foundation of their rational worldview.

Chrysippus and his followers tackled this and other logical puzzles with a sophistication that, in some cases, wouldn't be matched until the 20th century. They developed a system of propositional logic based on five basic indemonstrable argument forms, which they believed could be used to reduce all valid inference forms in simple assertoric logic.

However, the Liar's Paradox remained a thorn in their side. It exemplified a class of problems that seemed to defy their logical system, challenging their (and our) intuitions about meaningful statements.

The Stoics' struggle with the Liar's Paradox highlights a crucial point: even the most robust logical systems can encounter statements that seem to defy categorization. Such linguistic constructs challenge our usual ways of processing information and making logical inferences.

Belnap to the Rescue: A New Way of Thinking

Fast forward a couple of millennia, and along comes Nuel Belnap with a four-valued logic system that might save us from paradoxical purgatory. Belnap's logic adds two new truth values to the classical true and false:

1. Both true and false (B) - accounting for Contradictions and Paradoxes;

2. Neither true nor false (N) - accounting for lacking information to determine either.

This four-valued logic system, known as A4, was originally proposed by Belnap in 1975 to contain contradictions in computer question-answering systems. However, its applicability extends beyond computer science, offering a fresh perspective on ancient philosophical problems.

Suddenly, our Liar's Paradox has a home. It's both true and false (B). Problem solved, right?

Well, not quite. But Belnap's logic does give us a framework for dealing with these troublesome statements. It's a bit like Pyrrhonism, the ancient Greek philosophy that advocated suspending judgment in the face of conflicting evidence. Pyrrhonists would have loved Belnap – his system allows us to acknowledge contradictions without our brains imploding.

Four-valued logic provides a way to formalize the Pyrrhonist approach to knowledge and belief.

Interestingly, this four-valued logic system resembles the Buddhist catuṣkoṭi, or tetralemma, a "four-cornered" argumentation system. This connection is not coincidental, as Pyrrhonism has been shown to have significant overlaps with Buddhist philosophy.

In Pyrrhonism, the concept of epoché (suspension of judgment) plays a crucial role. The Pyrrhonist approach to appearances (phantasiai) and reality mirrors the distinction between the 'Both true and false' (B) and the classical true/false values in Belnap's logic. Pyrrhonists argue that we should withhold judgment about the underlying reality, while we can acknowledge our impressions or appearances.

This four-valued logic system provides a formal structure for the Pyrrhonist practice of using modes or tropes to induce suspension of judgment. For instance, the Ten Modes of Aenesidemus, a set of skeptical arguments used by Pyrrhonists, can be seen as ways of pushing propositions from the realm of true/false into the realms of 'B' or 'N' in Belnap's system.

By embracing this more nuanced approach to truth values, we can navigate the complexities of paradoxes and contradictions without falling into the traps of dogmatism or nihilism. This aligns perfectly with the Pyrrhonist goal of achieving ataraxia (tranquility) through carefully examining our beliefs and judgments.

Anti-Paradoxes and Deepities: Logical Bullshit

Now, let's flip the script and embark on a little thought experiment together. Consider this statement:

"This statement is true."

At first glance, it seems harmlessly true, maybe even obvious. But let's dig deeper. What information does it convey?

If the speaker is telling the truth, the statement is valid. But if the speaker is lying, the statement is also consistent - a liar would claim their lie is true. In both cases, the statement works, but it tells us absolutely nothing about its truth value or anything else in the world.

We've just stumbled upon what we might call the "Truthteller's Anti-Paradox." Unlike the Liar's Paradox, which gives us too much conflicting information, the Truthteller provides us with no information at all.

The Truthteller's Anti-Paradox is like a verbal magic trick - it seems to say something, but upon reflection, it vanishes into thin air.

Let's define an anti-paradox: An anti-paradox is a statement that appears informative but, upon analysis, proves to be devoid of actual content or truth value. It's neither true nor false but empty of information.

The easiest way to consider them is whether the statement would be considered valid under both True and False conditions. If so, it's probably an anti-paradox.

In Belnap's system, we might categorize this as Neither true nor false (N). It's not that it's both true and false like the Liar's Paradox; instead, it fails to be either because it carries no actual content.

Anti-paradoxes are particularly insidious because they can easily pass for informative statements. They exploit our tendency to assume that grammatically correct sentences must have some inherent meaning. They are pieces of noise that sound like words.

Now, let's bridge the gap between anti-paradoxes and deepities. Recall that a deepity, as defined by philosopher Daniel Dennett, is a statement that seems profound but is either trivial or nonsensical. It has two possible interpretations: one that's true but trivial and another that's false but would be profound if it were true.

We can place anti-paradoxes and deepities on a spectrum of logical bullshit:

1. Anti-Paradoxes: These statements are utterly devoid of content. They work whether the speaker is truthful or lying; upon reflection, we realize they convey no information at all.

2. Strong Deepities: These are statements that are closer to anti-paradoxes. They seem to say something profound, but upon careful analysis, they become empty of actual content. For example, "Everything happens for a reason" – it's true in the trivial sense that events have causes but empty in terms of implying any more significant meaning or purpose.

3. Weak Deepities: These are statements that do have some content, but far less than they appear to at first glance. They're not completely empty, but they're not as profound as they seem. "Love is just a word" for example.

The critical difference is that anti-paradoxes, upon reflection, are recognized as completely empty, while deepities retain at least a shred of meaning, even if it's trivial. However, both serve similar functions in discourse:

1. They can be used to sound informative or profound without saying anything substantial.

2. They can shut down critical thinking by overwhelming the listener with apparent meaning and profundity.

3. They can be used to avoid committing to a clear, falsifiable statement.

Understanding this spectrum can help us become more discerning consumers and producers of language. When we encounter a statement that sounds deep, we can ask ourselves: Is this genuinely profound, or is it somewhere on the spectrum from deepity to anti-paradox?

Deepities and anti-paradoxes are like the fast food of philosophy – they look filling but leave you intellectually malnourished.

Belnap's Logic and Anti-Paradoxes: A Deeper Look

Now, let's revisit Belnap's four-valued logic and see how it applies to these tricky statements.

Recall that Belnap's system introduces two additional truth values beyond the classical true and false:

1. Both true and false (B)

2. Neither true nor false (N)

Anti-paradoxes, like our Truthteller's Anti-Paradox, fall squarely into the "Neither true nor false" (N) category. They lack the contradictory nature of paradoxes (which would be "Both true and false") and fail to have a definite true or false value.

This "N" category is particularly useful for dealing with meaningless statements or those lacking sufficient information. It allows us to formally represent the idea that some statements simply don't have a truth value—not because they're paradoxical but because they're empty of actual content.

Consider another anti-paradox: "What I'm saying is exactly what you're hearing." If we believe we are hearing what they are saying, it means the statement is valid. But if we think we are having some crisis and what he is saying is not that sentence, it would still be valid to hear that. Belnap's system lets us cleanly categorize this apparent insanity as "N" – neither true nor false, but inapplicable or meaningless.

This approach aligns well with the Pyrrhonist practice of epoché or suspension of judgment. When faced with a potential anti-paradox, instead of trying to force it into a true/false dichotomy, we can recognize it as a statement that doesn't warrant a truth value at all.

Belnap's logic thus provides a formal framework for dealing with the full spectrum of logical bullshit, from paradoxes to deepities to anti-paradoxes. It gives us a more nuanced way to categorize statements, acknowledging that logic and language are often more complex than a simple true/false dichotomy.

Believe Me! It's True! You Can Trust Me!

And now we come to this crown jewel of political and corporate bullshit: "Believe me."

At first glance, it seems like a simple request for trust. But let's break it down:

1. If the speaker is truthful, they would say, "Believe me," because they're telling the truth and want you to believe it.

2. If the speaker is lying, they would also say "Believe me" because they want you to believe their lie.

Like our Truthteller's Anti-Paradox, "Believe me" works in both cases but provides no information about whether the speaker should be believed.

It's an anti-paradox par excellence. It provides no information, no reason for belief. It's the argumentative equivalent of saying, "because I said so." In Belnap's terms, it's Neither true nor false (N) – it's not even wrong.

In business and in life, 'Believe me' is often the last refuge for those who have run out of actual arguments.

What makes anti-paradoxes so intriguing is that, unlike paradoxes that immediately boggle the mind, they appear perfectly normal and informative initially. It's only upon reflection that we realize they're entirely devoid of content. They're the stealth bombers of bullshit, flying under our logical radar.

Navigating the BS: Practical Applications

Understanding deepities and anti-paradoxes can sharpen your critical thinking skills in both personal and professional contexts:

1. In negotiations: When someone uses a deepity or an anti-paradox like "Believe me," push for more precise, more substantive communication. Ask for specific evidence or reasoning.

2. In decision-making: Be wary of seemingly profound statements that don't provide actionable information. Ask yourself: Would this statement still be valid if I was lying? If so, you might be dealing with an anti-paradox.

3. In personal growth: Recognize when you're using deepities or anti-paradoxes in your own thinking. Challenge yourself to articulate your thoughts more precisely.

4. In media literacy: Identify when politicians, pundits, or influencers use these linguistic tricks to avoid saying anything of substance. Look for statements that sound good, regardless of whether they're true or false.

Conclusion: Navigating the Logical Minefield

So, where does this leave us? How do we find solid ground in a world full of deepities, paradoxes, and anti-paradoxes?

Perhaps the answer lies in embracing a bit of Pyrrhonian skepticism tempered with Stoic rationality. We need to:

1. Recognize these linguistic traps for what they are.

2. Suspend judgment when faced with paradoxes and anti-paradoxes.

3. Seek clarity and precision in our own communication.

4. Focus on actions and observable truths rather than getting lost in wordplay.

Anti-paradoxes, in particular, remind us of the importance of critical thinking. They show us that not every grammatically correct statement carries information and that we must be vigilant in examining the actual content of what we hear and say.

A Pyrrhonist might use paradoxes and anti-paradoxes to reach ataraxia, recognizing that many statements we encounter are neither true nor false but simply empty. But for those of us navigating the practical world of business and relationships, perhaps the most Stoic response to "Believe me" is merely this: "Show me."

After all, as Marcus Aurelius wrote, "Everything we hear is an opinion, not a fact. Everything we see is a perspective, not the truth." Or did he? I'm pretty sure he did. Believe me.

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