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.
