Logical Nihilism: Rehabilitating the Concept of Neutrality in Logic

And we're back to discussing logical nihilism. Let's explore whether logical nihilism can alleviate the tension at the foundations of logical thinking and the shortcomings of neutrality in ensuring equality within a dialectical space of enquiry. In the context of the philosophy of logic, 'nihilism' is a technical term. It's distinct from the general definition of 'nihilism,' which, according to my dictionary, means "the rejection of all religious and moral principles, in the belief that life is meaningless." Logical nihilism doesn't concern religious or moral beliefs.

My thesaurus first lists synonyms for nihilism as "negativity, cynicism, pessimism; rejection, repudiation, renunciation, denial, abnegation; disbelief, non-belief, unbelief, scepticism, lack of conviction, absence of moral values, agnosticism, atheism, non-theism." The second batch of synonyms includes "anarchy, disorder, chaos, absence of government, lawlessness, mobocracy." Both sets of terms paint nihilism in a rather negative light, something humorously portrayed in The Big Lebowski. None of these seems directly relevant to addressing issues of equity in dialectical spaces. However, it's this second set that brings us closer to understanding logical nihilism, not through the concept of anarchism, which tends to inflate egotism, but rather through the idea of lawlessness. Logical nihilism posits that there are no inherent laws of logic. Each time I mention this, I instinctively wonder: “But what remains of logic if there are no laws?” considering logic is fundamentally about “the laws of thought.” This concept is deeply ingrained, asserting that logic is universally applicable, its laws holding true regardless of context, implying its neutrality.

Embracing logical nihilism means challenging the very idea that logic is neutral. But in my work, "Logic in the Wild," don't I argue that logic facilitates a neutral space for dialectical enquiry? Indeed, because logic excels at this! However, I don't assert that this space remains identical across all dialectical endeavors. Logic, acting as the guardian of coherence, applies different standards in varied contexts. In mathematics, 'consistency' measures coherence, while 'beyond reasonable doubt' sets the standard in legal proceedings. Meanwhile, probability and statistics play a crucial role in guiding policy-making, particularly in fields such as public health and economics. Thus, each context demands a unique measure of coherence, leading to distinct interpretations of neutrality.

Contrary to the belief that there's a singular, all-encompassing logic, logical nihilism advocates for a more flexible approach to logical practice, allowing for nuanced expectations regarding neutrality. By accepting the negotiability of neutrality at the core of our thinking, we reopen the possibility of addressing issues of equity. Far from being a radical rejection of logic as meaningless, logical nihilism invites us into richer, more constructive dialectical spaces of enquiry. It encourages participants to engage more deeply and constructively. That, at least, is my hope.

Dialetheism Alongside Validity as a Social Construct and Logical Nihilism

This week I argued that validity is a social construct and I defended logical nihilism. While I’m at it, I might as well tell you why I’m a dialetheist. To be a dialetheist is to believe in the existence of true contradictions. This might sound perplexing at first. Take, for instance, statements like "I am human and I am not human" or "It’s 23 degrees in Auckland and it isn’t 23 degrees in Auckland." These are clear contradictions, and it would indeed seem absurd to claim they are true. In fact, they are false. This initial sense of absurdity is precisely the challenge that dialetheism addresses.

Historically, in the European tradition, Aristotle is often recognized for formulating the law against contradictions. Rather than definitively defending this law, Aristotle articulated it in various ways, some of which are blatantly false. Despite these inaccuracies, his authority has consistently been used to deny the possibility of contradictions. This complex narrative, including Aristotle's contributions and their implications, is more deeply explored Logic in the Wild.

As I’ve discussed this week, Twentieth-century logic has been obsessively seeking the foundations of mathematics, which are supposed to provide logical underpinnings for all reasoning considered 'valid' (a notion I argue is a social construct). Therefore, asserting the existence of true contradictions in the 21st century directly challenges this long-established tradition, particularly within mathematical logic. To claim that the quest for mathematical foundations was merely a dream, disrupted by the reality of true contradictions, is naturally (and here I empathize with the reaction) perceived as an offense.

However, recognizing true contradictions does not mean we forsake all rationality. Accepting every contradiction would lead to triviality, where everything is considered true - a situation no logician accepts. Even we dialetheists, who acknowledge certain contradictions, staunchly reject this notion of triviality. Our focus lies on what I describe in "Logic in the Wild" as 'insolubles,' statements that, based on first principles, imply each other’s opposites. The Liar paradox is a classic example, asserting “I am a false sentence.” This paradox demonstrates a true contradiction through its self-referential nature: if it’s true, then what it says is true, namely that it’s false, so it’s false. But if it’s false, it’s false that it is false, so it is true (double negation, first principle). So it’s a true contradiction. Weird? Sure, but not trivial.

In the chapter “Logic in the Weird” of Logic in the Wild, you’ll discover how logic guards coherence in the face of the weird, the absurd, the paradoxical, and the insoluble. Writing this book has altered my perspective, especially regarding how logic functions within a community. It has led me to a broader viewpoint that accommodates the practices of twentieth-century mathematical logicians, who sought coherence in mathematics, while also allowing logic to uphold coherence in more perplexing contexts. Recognizing that logic isn't governed by universal laws (logical nihilism) frees it within the community, offering an alternative to full-blown emotional empathy. This, I believe, marks the beginning of logic’s rehabilitation.

Validity as a Social Construct: Insights from Logical Nihilism

In my previous blog post, I posited that validity might be considered a social construct, which naturally leads to an intriguing question: does this extend to fundamental logical principles like conjunction elimination? Conjunction elimination is a foundational logical rule stating that if a conjunction 'A & B' is true, then both A and B must be independently true. This raises an intriguing question: how can such an apparently unassailable rule be viewed as a social construct?

Gillian Russell, a prominent logician, proposes the concept of logical nihilism. This perspective is underscored by her argument that counter-examples can be identified for any logical principle. Consider the peculiar sentence: "I'm only true when part of a conjunction." When combined with another true statement, such as "My cat's name is Wilhelmina," we arrive at a true conjunction because both conjuncts are independently true. Yet, upon eliminating the conjunction, we are left with a standalone statement that is false. This contradiction is indeed puzzling.

In "Logic in the Wild," I explore various perplexing scenarios and the solutions proposed for them. Using the metaphor of the 'mailbox,' a makeshift air filter created during the Apollo 13 mission, I illustrate how insolubles continue to exist regardless of efforts to resolve them. Theoretical constructs like Tarski's "Tower of Truth" are developed to circumvent self-referential truth ascriptions. However, Tarski's Tower, akin to the mailbox, serves more as a temporary fix rather than a comprehensive resolution to the problem of insolubles.

Revisiting the concept of logical nihilism and Russell's unique counter-examples, we can imagine a range of 'mailboxes' designed to exclude such statements. Common objections might include "this isn't a legitimate statement" (but why shouldn't it be?) or "a statement shouldn't alter its truth value when combined with others" (but why not? The example I provided does exactly this). However, these objections falter upon closer examination, as demonstrated by the example discussed previously. It's crucial to note that such solutions are akin to mailboxes unless they possess independent motivation and aren't merely ad hoc or arbitrary constructs devised to sidestep a problematic scenario we find disagreeable.

Russell's advocacy for logical nihilism posits that there are no universally applicable logical principles. This does not render logic futile. On the contrary, it's a sobering acknowledgment that logic transcends mere mathematical formalism. The failure of 20th-century attempts to completely formalize logic lies not in an incomplete understanding of the mathematics but in the inherent limitations of such an endeavor. The misconception that validity is solely what 20th-century mathematicians constructed (akin to a mailbox) is where I see validity as a social construct.

My proposal in "Logic in the Wild" is to view logic as the 'guardian of coherence.' This approach reconciles logical nihilism by not asserting the existence of universal logical principles, while still valuing 20th-century logic as a significant means of maintaining coherence through mathematical consistency. So, there is no cause for alarm; we are better positioned to appreciate and utilize logic now more than ever.